Equations of motion In mathematical physics , equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of time . More specifically , the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables : normally spatial coordinates and time are used , but others are also possible , such as momentum components and time . The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system . The functions are defined in a Euclidean space in classical mechanics , but are replaced by curved spaces in relativity . If the dynamics of a system is known , the equations are the solutions to the differential equations describing the motion of the dynamics . There are two main descriptions of motion : dynamics and kinematics . Dynamics is general , since momenta , forces and energy of the particles are taken into account . In this instance , sometimes the term refers to the differential equations that the system satisfies ( e.g. , Newton 's second law or Euler -- Lagrange equations ) , and sometimes to the solutions to those equations . However , kinematics is simpler as it concerns only variables derived from the positions of objects , and time . In circumstances of constant acceleration , these simpler equations of motion are usually referred to as the SUVAT equations , arising from the definitions of kinematic quantities : displacement ( s ) , initial velocity ( u ) , final velocity ( v ) , acceleration ( a ) , and time ( t ) . Equations of motion can therefore be grouped under these main classifiers of motion . In all cases , the main types of motion are translations , rotations , oscillations , or any combinations of these . A differential equation of motion , usually identified as some physical law and applying definitions of physical quantities , is used to set up an equation for the problem . Solving the differential equation will lead to a general solution with arbitrary constants , the arbitrariness corresponding to a family of solutions . A particular solution can be obtained by setting the initial values , which fixes the values of the constants . To state this formally , in general an equation of motion M is a function of the position r of the object , its velocity ( the first time derivative of r , v = dr / dt ) , and its acceleration ( the second derivative of r , a = d r / dt ) , and time t . Euclidean vectors in 3D are denoted throughout in bold . This is equivalent to saying an equation of motion in r is a second order ordinary differential equation ( ODE ) in r , M ( r ( t ) , r ̇ ( t ) , r ̈ ( t ) , t ) = 0 , ( \ displaystyle M \ left ( \ mathbf ( r ) ( t ) , \ mathbf ( \ dot ( r ) ) ( t ) , \ mathbf ( \ ddot ( r ) ) ( t ) , t \ right ) = 0 \ , , ) where t is time , and each overdot denotes one time derivative . The initial conditions are given by the constant values at t = 0 , r ( 0 ) , r ̇ ( 0 ) . ( \ displaystyle \ mathbf ( r ) ( 0 ) \ , , \ quad \ mathbf ( \ dot ( r ) ) ( 0 ) \ , . ) The solution r ( t ) to the equation of motion , with specified initial values , describes the system for all times t after t = 0 . Other dynamical variables like the momentum p of the object , or quantities derived from r and p like angular momentum , can be used in place of r as the quantity to solve for from some equation of motion , although the position of the object at time t is by far the most sought - after quantity . Sometimes , the equation will be linear and is more likely to be exactly solvable . In general , the equation will be non-linear , and can not be solved exactly so a variety of approximations must be used . The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions . Contents ( hide ) 1 History 2 Kinematic equations for one particle 2.1 Kinematic quantities 2.2 Uniform acceleration 2.2. 1 Constant translational acceleration in a straight line 2.2. 2 Constant linear acceleration in any direction 2.2. 3 Applications 2.2. 4 Constant circular acceleration 2.3 General planar motion 2.4 General 3D motion 3 Dynamic equations of motion 3.1 Newtonian mechanics 3.2 Applications 4 Analytical mechanics 5 Electrodynamics 6 General relativity 6.1 Geodesic equation of motion 6.2 Spinning objects 7 Analogues for waves and fields 7.1 Field equations 7.2 Wave equations 7.3 Quantum theory 8 See also 9 References History ( edit ) Historically , equations of motion first appeared in classical mechanics to describe the motion of massive objects , a notable application was to celestial mechanics to predict the motion of the planets as if they orbit like clockwork ( this was how Neptune was predicted before its discovery ) , and also investigate the stability of the solar system . It is important to observe that the huge body of work involving kinematics , dynamics and the mathematical models of the universe developed in baby steps -- faltering , getting up and correcting itself -- over three millennia and included contributions of both known names and others who have since faded from the annals of history . In antiquity , notwithstanding the success of priests , astrologers and astronomers in predicting solar and lunar eclipses , the solstices and the equinoxes of the Sun and the period of the Moon , there was nothing other than a set of algorithms to help them . Despite the great strides made in the development of geometry made by Ancient Greeks and surveys in Rome , we were to wait for another thousand years before the first equations of motion arrive . The exposure of Europe to the collected works by the Muslims of the Greeks , the Indians and the Islamic scholars , such as Euclid 's Elements , the works of Archimedes , and Al - Khwārizmī 's treatises began in Spain , and scholars from all over Europe went to Spain , read , copied and translated the learning into Latin . The exposure of Europe to Arabic numerals and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe . By the 13th century the universities of Oxford and Paris had come up , and the scholars were now studying mathematics and philosophy with lesser worries about mundane chores of life -- the fields were not as clearly demarcated as they are in the modern times . Of these , compendia and redactions , such as those of Johannes Campanus , of Euclid and Aristotle , confronted scholars with ideas about infinity and the ratio theory of elements as a means of expressing relations between various quantities involved with moving bodies . These studies led to a new body of knowledge that is now known as physics . Of these institutes Merton College sheltered a group of scholars devoted to natural science , mainly physics , astronomy and mathematics , of similar in stature to the intellectuals at the University of Paris . Thomas Bradwardine , one of those scholars , extended Aristotelian quantities such as distance and velocity , and assigned intensity and extension to them . Bradwardine suggested an exponential law involving force , resistance , distance , velocity and time . Nicholas Oresme further extended Bradwardine 's arguments . The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion . For writers on kinematics before Galileo , since small time intervals could not be measured , the affinity between time and motion was obscure . They used time as a function of distance , and in free fall , greater velocity as a result of greater elevation . Only Domingo de Soto , a Spanish theologian , in his commentary on Aristotle 's Physics published in 1545 , after defining `` uniform difform '' motion ( which is uniformly accelerated motion ) -- the word velocity was n't used -- as proportional to time , declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles , without his proving these propositions or suggesting a formula relating time , velocity and distance . De Soto 's comments are shockingly correct regarding the definitions of acceleration ( acceleration was a rate of change of motion ( velocity ) in time ) and the observation that during the violent motion of ascent acceleration would be negative . Discourses such as these spread throughout Europe and definitely influenced Galileo and others , and helped in laying the foundation of kinematics . Galileo deduced the equation s = 1 / 2gt in his work geometrically , using the Merton rule , now known as a special case of one of the equations of kinematics . He could n't use the now - familiar mathematical reasoning . The relationships between speed , distance , time and acceleration was not known at the time . Galileo was the first to show that the path of a projectile is a parabola . Galileo had an understanding of centrifugal force and gave a correct definition of momentum . This emphasis of momentum as a fundamental quantity in dynamics is of prime importance . He measured momentum by the product of velocity and weight ; mass is a later concept , developed by Huygens and Newton . In the swinging of a simple pendulum , Galileo says in Discourses that `` every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc . '' His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion . He did not generalize and make them applicable to bodies not subject to the earth 's gravitation . That step was Newton 's contribution . The term `` inertia '' was used by Kepler who applied it to bodies at rest . ( The first law of motion is now often called the law of inertia . ) Galileo did not fully grasp the third law of motion , the law of the equality of action and reaction , though he corrected some errors of Aristotle . With Stevin and others Galileo also wrote on statics . He formulated the principle of the parallelogram of forces , but he did not fully recognize its scope . Galileo also was interested by the laws of the pendulum , his first observations was when he was a young man . In 1583 , while he was praying in the cathedral at Pisa , his attention was arrested by the motion of the great lamp lighted and left swinging , referencing his own pulse for time keeping . To him the period appeared the same , even after the motion had greatly diminished , discovering the isochronism of the pendulum . More careful experiments carried out by him later , and described in his Discourses , revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum . Thus we arrive at René Descartes , Isaac Newton , Gottfried Leibniz , et al. ; and the evolved forms of the equations of motion that begin to be recognized as the modern ones . Later the equations of motion also appeared in electrodynamics , when describing the motion of charged particles in electric and magnetic fields , the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field . With the advent of special relativity and general relativity , the theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light , and curvature of spacetime . In all these cases the differential equations were in terms of a function describing the particle 's trajectory in terms of space and time coordinates , as influenced by forces or energy transformations . However , the equations of quantum mechanics can also be considered `` equations of motion '' , since they are differential equations of the wavefunction , which describes how a quantum state behaves analogously using the space and time coordinates of the particles . There are analogs of equations of motion in other areas of physics , for collections of physical phenomena that can be considered waves , fluids , or fields . Kinematic equations for one particle ( edit ) Kinematic quantities ( edit ) Kinematic quantities of a classical particle of mass m : position r , velocity v , acceleration a . From the instantaneous position r = r ( t ) , instantaneous meaning at an instant value of time t , the instantaneous velocity v = v ( t ) and acceleration a = a ( t ) have the general , coordinate - independent definitions ; v = d r d t , a = d v d t = d 2 r d t 2 ( \ displaystyle \ mathbf ( v ) = ( \ frac ( d \ mathbf ( r ) ) ( dt ) ) \ , , \ quad \ mathbf ( a ) = ( \ frac ( d \ mathbf ( v ) ) ( dt ) ) = ( \ frac ( d ^ ( 2 ) \ mathbf ( r ) ) ( dt ^ ( 2 ) ) ) \ , \ ! ) Notice that velocity always points in the direction of motion , in other words for a curved path it is the tangent vector . Loosely speaking , first order derivatives are related to tangents of curves . Still for curved paths , the acceleration is directed towards the center of curvature of the path . Again , loosely speaking , second order derivatives are related to curvature . The rotational analogues are the `` angular vector '' ( angle the particle rotates about some axis ) θ = θ ( t ) , angular velocity ω = ω ( t ) , and angular acceleration α = α ( t ) : θ = θ n ^ , ω = d θ d t , α = d ω d t , ( \ displaystyle ( \ boldsymbol ( \ theta ) ) = \ theta ( \ hat ( \ mathbf ( n ) ) ) \ , , \ quad ( \ boldsymbol ( \ omega ) ) = ( \ frac ( d ( \ boldsymbol ( \ theta ) ) ) ( dt ) ) \ , , \ quad ( \ boldsymbol ( \ alpha ) ) = ( \ frac ( d ( \ boldsymbol ( \ omega ) ) ) ( dt ) ) \ , , ) where n̂ is a unit vector in the direction of the axis of rotation , and θ is the angle the object turns through about the axis . The following relation holds for a point - like particle , orbiting about some axis with angular velocity ω : v = ω × r ( \ displaystyle \ mathbf ( v ) = ( \ boldsymbol ( \ omega ) ) \ times \ mathbf ( r ) \ , \ ! ) where r is the position vector of the particle ( radial from the rotation axis ) and v the tangential velocity of the particle . For a rotating continuum rigid body , these relations hold for each point in the rigid body . Uniform acceleration ( edit ) The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple : the acceleration is constant , so the second derivative of the position of the object is constant . The results of this case are summarized below . Constant translational acceleration in a straight line ( edit ) These equations apply to a particle moving linearly , in three dimensions in a straight line with constant acceleration . Since the position , velocity , and acceleration are collinear ( parallel , and lie on the same line ) -- only the magnitudes of these vectors are necessary , and because the motion is along a straight line , the problem effectively reduces from three dimensions to one . v = a t + v 0 ( 1 ) ( \ displaystyle ( \ begin ( aligned ) v& = at + v_ ( 0 ) \ quad ( 1 ) \ \ \ end ( aligned ) ) ) r = r 0 + v 0 t + 1 2 a t 2 ( 2 ) ( \ displaystyle ( \ begin ( aligned ) r& = r_ ( 0 ) + v_ ( 0 ) t+ ( \ tfrac ( 1 ) ( 2 ) ) ( a ) t ^ ( 2 ) \ quad ( 2 ) \ \ \ end ( aligned ) ) ) r = r 0 + 1 2 ( v + v 0 ) t ( 3 ) v 2 = v 0 2 + 2 a ( r − r 0 ) ( 4 ) r = r 0 + v t − 1 2 a t 2 ( 5 ) ( \ displaystyle ( \ begin ( aligned ) r& = r_ ( 0 ) + ( \ tfrac ( 1 ) ( 2 ) ) \ left ( v + v_ ( 0 ) \ right ) t \ quad ( 3 ) \ \ v ^ ( 2 ) & = v_ ( 0 ) ^ ( 2 ) + 2a \ left ( r - r_ ( 0 ) \ right ) \ quad ( 4 ) \ \ r& = r_ ( 0 ) + vt - ( \ tfrac ( 1 ) ( 2 ) ) ( a ) t ^ ( 2 ) \ quad ( 5 ) \ \ \ end ( aligned ) ) ) where : r is the particle 's initial position r is the particle 's final position v is the particle 's initial velocity v is the particle 's final velocity a is the particle 's acceleration t is the time interval ( show ) Derivation Equations ( 1 ) and ( 2 ) are from integrating the definitions of velocity and acceleration , subject to the initial conditions r ( t ) = r and v ( t ) = v ; v = ∫ a d t = a t + v 0 , ( 1 ) r = ∫ ( a t + v 0 ) d t = a t 2 2 + v 0 t + r 0 , ( 2 ) ( \ displaystyle ( \ begin ( aligned ) \ mathbf ( v ) & = \ int \ mathbf ( a ) dt = \ mathbf ( a ) t+ \ mathbf ( v ) _ ( 0 ) \ , , \ quad ( 1 ) \ \ \ mathbf ( r ) & = \ int ( \ mathbf ( a ) t+ \ mathbf ( v ) _ ( 0 ) ) dt = ( \ frac ( \ mathbf ( a ) t ^ ( 2 ) ) ( 2 ) ) + \ mathbf ( v ) _ ( 0 ) t+ \ mathbf ( r ) _ ( 0 ) \ , , \ quad ( 2 ) \ \ \ end ( aligned ) ) ) in magnitudes , v = a t + v 0 , ( 1 ) r = a t 2 2 + v 0 t + r 0 . ( 2 ) ( \ displaystyle ( \ begin ( aligned ) v& = at + v_ ( 0 ) \ , , \ quad ( 1 ) \ \ r& = ( \ frac ( ( a ) t ^ ( 2 ) ) ( 2 ) ) + v_ ( 0 ) t + r_ ( 0 ) \ ,. \ quad ( 2 ) \ \ \ end ( aligned ) ) ) Equation ( 3 ) involves the average velocity v + v / 2 . Intuitively , the velocity increases linearly , so the average velocity multiplied by time is the distance traveled while increasing the velocity from v to v , as can be illustrated graphically by plotting velocity against time as a straight line graph . Algebraically , it follows from solving ( 1 ) for a = ( v − v 0 ) t ( \ displaystyle \ mathbf ( a ) = ( \ frac ( ( \ mathbf ( v ) - \ mathbf ( v ) _ ( 0 ) ) ) ( t ) ) ) and substituting into ( 2 ) r = r 0 + v 0 t + t 2 ( v − v 0 ) , ( \ displaystyle \ mathbf ( r ) = \ mathbf ( r ) _ ( 0 ) + \ mathbf ( v ) _ ( 0 ) t+ ( \ frac ( t ) ( 2 ) ) ( \ mathbf ( v ) - \ mathbf ( v ) _ ( 0 ) ) \ , , ) then simplifying to get r = r 0 + t 2 ( v + v 0 ) ( \ displaystyle \ mathbf ( r ) = \ mathbf ( r ) _ ( 0 ) + ( \ frac ( t ) ( 2 ) ) ( \ mathbf ( v ) + \ mathbf ( v ) _ ( 0 ) ) ) or in magnitudes r = r 0 + ( v + v 0 2 ) t ( 3 ) ( \ displaystyle r = r_ ( 0 ) + \ left ( ( \ frac ( v + v_ ( 0 ) ) ( 2 ) ) \ right ) t \ quad ( 3 ) ) From ( 3 ) , t = ( r − r 0 ) ( 2 v + v 0 ) ( \ displaystyle t = \ left ( r - r_ ( 0 ) \ right ) \ left ( ( \ frac ( 2 ) ( v + v_ ( 0 ) ) ) \ right ) ) substituting for t in ( 1 ) : v = a ( r − r 0 ) ( 2 v + v 0 ) + v 0 v ( v + v 0 ) = 2 a ( r − r 0 ) + v 0 ( v + v 0 ) v 2 + v v 0 = 2 a ( r − r 0 ) + v 0 v + v 0 2 v 2 = v 0 2 + 2 a ( r − r 0 ) ( 4 ) ( \ displaystyle ( \ begin ( aligned ) v& = a \ left ( r - r_ ( 0 ) \ right ) \ left ( ( \ frac ( 2 ) ( v + v_ ( 0 ) ) ) \ right ) + v_ ( 0 ) \ \ v \ left ( v + v_ ( 0 ) \ right ) & = 2a \ left ( r - r_ ( 0 ) \ right ) + v_ ( 0 ) \ left ( v + v_ ( 0 ) \ right ) \ \ v ^ ( 2 ) + vv_ ( 0 ) & = 2a \ left ( r - r_ ( 0 ) \ right ) + v_ ( 0 ) v + v_ ( 0 ) ^ ( 2 ) \ \ v ^ ( 2 ) & = v_ ( 0 ) ^ ( 2 ) + 2a \ left ( r - r_ ( 0 ) \ right ) \ quad ( 4 ) \ \ \ end ( aligned ) ) ) From ( 3 ) , 2 ( r − r 0 ) − v t = v 0 t ( \ displaystyle 2 \ left ( r - r_ ( 0 ) \ right ) - vt = v_ ( 0 ) t ) substituting into ( 2 ) : r = a t 2 2 + 2 r − 2 r 0 − v t + r 0 0 = a t 2 2 + r − r 0 − v t r = r 0 + v t − a t 2 2 ( 5 ) ( \ displaystyle ( \ begin ( aligned ) r& = ( \ frac ( ( a ) t ^ ( 2 ) ) ( 2 ) ) + 2r - 2r_ ( 0 ) - vt + r_ ( 0 ) \ \ 0& = ( \ frac ( ( a ) t ^ ( 2 ) ) ( 2 ) ) + r - r_ ( 0 ) - vt \ \ r& = r_ ( 0 ) + vt - ( \ frac ( ( a ) t ^ ( 2 ) ) ( 2 ) ) \ quad ( 5 ) \ end ( aligned ) ) ) Usually only the first 4 are needed , the fifth is optional . Here a is constant acceleration , or in the case of bodies moving under the influence of gravity , the standard gravity g is used . Note that each of the equations contains four of the five variables , so in this situation it is sufficient to know three out of the five variables to calculate the remaining two . In elementary physics the same formulae are frequently written in different notation as : v = u + a t ( 1 ) s = u t + 1 2 a t 2 ( 2 ) s = 1 2 ( u + v ) t ( 3 ) v 2 = u 2 + 2 a s ( 4 ) s = v t − 1 2 a t 2 ( 5 ) ( \ displaystyle ( \ begin ( aligned ) v& = u + at \ quad ( 1 ) \ \ s& = ut+ ( \ tfrac ( 1 ) ( 2 ) ) at ^ ( 2 ) \ quad ( 2 ) \ \ s& = ( \ tfrac ( 1 ) ( 2 ) ) ( u + v ) t \ quad ( 3 ) \ \ v ^ ( 2 ) & = u ^ ( 2 ) + 2as \ quad ( 4 ) \ \ s& = vt - ( \ tfrac ( 1 ) ( 2 ) ) at ^ ( 2 ) \ quad ( 5 ) \ \ \ end ( aligned ) ) ) where u has replaced v , s replaces r , and s = 0 . They are often referred to as the SUVAT equations , where `` SUVAT '' is an acronym from the variables : s = displacement ( s = initial displacement ) , u = initial velocity , v = final velocity , a = acceleration , t = time . Constant linear acceleration in any direction ( edit ) Trajectory of a particle with initial position vector r and velocity v , subject to constant acceleration a , all three quantities in any direction , and the position r ( t ) and velocity v ( t ) after time t . The initial position , initial velocity , and acceleration vectors need not be collinear , and take an almost identical form . The only difference is that the square magnitudes of the velocities require the dot product . The derivations are essentially the same as in the collinear case , v = a t + v 0 ( 1 ) r = r 0 + v 0 t + 1 2 a t 2 ( 2 ) r = r 0 + 1 2 ( v + v 0 ) t ( 3 ) v 2 = v 0 2 + 2 a ⋅ ( r − r 0 ) ( 4 ) r = r 0 + v t − 1 2 a t 2 ( 5 ) ( \ displaystyle ( \ begin ( aligned ) \ mathbf ( v ) & = \ mathbf ( a ) t+ \ mathbf ( v ) _ ( 0 ) \ quad ( 1 ) \ \ \ mathbf ( r ) & = \ mathbf ( r ) _ ( 0 ) + \ mathbf ( v ) _ ( 0 ) t+ ( \ tfrac ( 1 ) ( 2 ) ) \ mathbf ( a ) t ^ ( 2 ) \ quad ( 2 ) \ \ \ mathbf ( r ) & = \ mathbf ( r ) _ ( 0 ) + ( \ tfrac ( 1 ) ( 2 ) ) \ left ( \ mathbf ( v ) + \ mathbf ( v ) _ ( 0 ) \ right ) t \ quad ( 3 ) \ \ v ^ ( 2 ) & = v_ ( 0 ) ^ ( 2 ) + 2 \ mathbf ( a ) \ cdot \ left ( \ mathbf ( r ) - \ mathbf ( r ) _ ( 0 ) \ right ) \ quad ( 4 ) \ \ \ mathbf ( r ) & = \ mathbf ( r ) _ ( 0 ) + \ mathbf ( v ) t - ( \ tfrac ( 1 ) ( 2 ) ) \ mathbf ( a ) t ^ ( 2 ) \ quad ( 5 ) \ \ \ end ( aligned ) ) ) although the Torricelli equation ( 4 ) can be derived using the distributive property of the dot product as follows : v 2 = v ⋅ v = ( v 0 + a t ) ⋅ ( v 0 + a t ) = v 0 2 + 2 t ( a ⋅ v 0 ) + a 2 t 2 ( \ displaystyle v ^ ( 2 ) = \ mathbf ( v ) \ cdot \ mathbf ( v ) = ( \ mathbf ( v ) _ ( 0 ) + \ mathbf ( a ) t ) \ cdot ( \ mathbf ( v ) _ ( 0 ) + \ mathbf ( a ) t ) = v_ ( 0 ) ^ ( 2 ) + 2t ( \ mathbf ( a ) \ cdot \ mathbf ( v ) _ ( 0 ) ) + a ^ ( 2 ) t ^ ( 2 ) ) ( 2 a ) ⋅ ( r − r 0 ) = ( 2 a ) ⋅ ( v 0 t + 1 2 a t 2 ) = 2 t ( a ⋅ v 0 ) + a 2 t 2 = v 2 − v 0 2 ( \ displaystyle ( 2 \ mathbf ( a ) ) \ cdot ( \ mathbf ( r ) - \ mathbf ( r ) _ ( 0 ) ) = ( 2 \ mathbf ( a ) ) \ cdot \ left ( \ mathbf ( v ) _ ( 0 ) t+ ( \ tfrac ( 1 ) ( 2 ) ) \ mathbf ( a ) t ^ ( 2 ) \ right ) = 2t ( \ mathbf ( a ) \ cdot \ mathbf ( v ) _ ( 0 ) ) + a ^ ( 2 ) t ^ ( 2 ) = v ^ ( 2 ) - v_ ( 0 ) ^ ( 2 ) ) ∴ v 2 = v 0 2 + 2 ( a ⋅ ( r − r 0 ) ) ( \ displaystyle \ therefore v ^ ( 2 ) = v_ ( 0 ) ^ ( 2 ) + 2 ( \ mathbf ( a ) \ cdot ( \ mathbf ( r ) - \ mathbf ( r ) _ ( 0 ) ) ) ) Applications ( edit ) Elementary and frequent examples in kinematics involve projectiles , for example a ball thrown upwards into the air . Given initial speed u , one can calculate how high the ball will travel before it begins to fall . The acceleration is local acceleration of gravity g . At this point one must remember that while these quantities appear to be scalars , the direction of displacement , speed and acceleration is important . They could in fact be considered as unidirectional vectors . Choosing s to measure up from the ground , the acceleration a must be in fact − g , since the force of gravity acts downwards and therefore also the acceleration on the ball due to it . At the highest point , the ball will be at rest : therefore v = 0 . Using equation ( 4 ) in the set above , we have : s = v 2 − u 2 − 2 g . ( \ displaystyle s = ( \ frac ( v ^ ( 2 ) - u ^ ( 2 ) ) ( - 2g ) ) . ) Substituting and cancelling minus signs gives : s = u 2 2 g . ( \ displaystyle s = ( \ frac ( u ^ ( 2 ) ) ( 2g ) ) . ) Constant circular acceleration ( edit ) The analogues of the above equations can be written for rotation . Again these axial vectors must all be parallel to the axis of rotation , so only the magnitudes of the vectors are necessary , ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 θ = θ 0 + 1 2 ( ω 0 + ω ) t ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) θ = θ 0 + ω t − 1 2 α t 2 ( \ displaystyle ( \ begin ( aligned ) \ omega & = \ omega _ ( 0 ) + \ alpha t \ \ \ theta & = \ theta _ ( 0 ) + \ omega _ ( 0 ) t+ ( \ tfrac ( 1 ) ( 2 ) ) \ alpha t ^ ( 2 ) \ \ \ theta & = \ theta _ ( 0 ) + ( \ tfrac ( 1 ) ( 2 ) ) ( \ omega _ ( 0 ) + \ omega ) t \ \ \ omega ^ ( 2 ) & = \ omega _ ( 0 ) ^ ( 2 ) + 2 \ alpha ( \ theta - \ theta _ ( 0 ) ) \ \ \ theta & = \ theta _ ( 0 ) + \ omega t - ( \ tfrac ( 1 ) ( 2 ) ) \ alpha t ^ ( 2 ) \ \ \ end ( aligned ) ) ) where α is the constant angular acceleration , ω is the angular velocity , ω is the initial angular velocity , θ is the angle turned through ( angular displacement ) , θ is the initial angle , and t is the time taken to rotate from the initial state to the final state . General planar motion ( edit ) Main article : General planar motion Position vector r , always points radially from the origin . Velocity vector v , always tangent to the path of motion . Acceleration vector a , not parallel to the radial motion but offset by the angular and Coriolis accelerations , nor tangent to the path but offset by the centripetal and radial accelerations . Kinematic vectors in plane polar coordinates . Notice the setup is not restricted to 2D space , but a plane in any higher dimension . These are the kinematic equations for a particle traversing a path in a plane , described by position r = r ( t ) . They are simply the time derivatives of the position vector in plane polar coordinates using the definitions of physical quantities above for angular velocity ω and angular acceleration α . These are instantaneous quantities which change with time . The position of the particle is r = r ( r ( t ) , θ ( t ) ) = r e ^ r ( \ displaystyle \ mathbf ( r ) = \ mathbf ( r ) \ left ( r ( t ) , \ theta ( t ) \ right ) = r \ mathbf ( \ hat ( e ) ) _ ( r ) ) where ê and ê are the polar unit vectors . Differentiating with respect to time gives the velocity v = e ^ r d r d t + r ω e ^ θ ( \ displaystyle \ mathbf ( v ) = \ mathbf ( \ hat ( e ) ) _ ( r ) ( \ frac ( dr ) ( dt ) ) + r \ omega \ mathbf ( \ hat ( e ) ) _ ( \ theta ) ) with radial component dr / dt and an additional component rω due to the rotation . Differentiating with respect to time again obtains the acceleration a = ( d 2 r d t 2 − r ω 2 ) e ^ r + ( r α + 2 ω d r d t ) e ^ θ ( \ displaystyle \ mathbf ( a ) = \ left ( ( \ frac ( d ^ ( 2 ) r ) ( dt ^ ( 2 ) ) ) - r \ omega ^ ( 2 ) \ right ) \ mathbf ( \ hat ( e ) ) _ ( r ) + \ left ( r \ alpha + 2 \ omega ( \ frac ( dr ) ( dt ) ) \ right ) \ mathbf ( \ hat ( e ) ) _ ( \ theta ) ) which breaks into the radial acceleration d r / dt , centripetal acceleration -- rω , Coriolis acceleration 2ω dr / dt , and angular acceleration rα . Special cases of motion described be these equations are summarized qualitatively in the table below . Two have already been discussed above , in the cases that either the radial components or the angular components are zero , and the non-zero component of motion describes uniform acceleration . State of motion Constant r r linear in t r quadratic in t r non-linear in t Constant θ Stationary Uniform translation ( constant translational velocity ) Uniform translational acceleration Non-uniform translation θ linear in t Uniform angular motion in a circle ( constant angular velocity ) Uniform angular motion in a spiral , constant radial velocity Angular motion in a spiral , constant radial acceleration Angular motion in a spiral , varying radial acceleration θ quadratic in t Uniform angular acceleration in a circle Uniform angular acceleration in a spiral , constant radial velocity Uniform angular acceleration in a spiral , constant radial acceleration Uniform angular acceleration in a spiral , varying radial acceleration θ non-linear in t Non-uniform angular acceleration in a circle Non-uniform angular acceleration in a spiral , constant radial velocity Non-uniform angular acceleration in a spiral , constant radial acceleration Non-uniform angular acceleration in a spiral , varying radial acceleration General 3D motion ( edit ) Main article : Spherical coordinate system In 3D space , the equations in spherical coordinates ( r , θ , φ ) with corresponding unit vectors ê , ê and ê , the position , velocity , and acceleration generalize respectively to r = r ( t ) = r e ^ r v = v e ^ r + r d θ d t e ^ θ + r d φ d t sin  θ e ^ φ a = ( a − r ( d θ d t ) 2 − r ( d φ d t ) 2 sin 2  θ ) e ^ r + ( r d 2 θ d t 2 + 2 v d θ d t − r ( d φ d t ) 2 sin  θ cos  θ ) e ^ θ + ( r d 2 φ d t 2 sin  θ + 2 v d φ d t sin  θ + 2 r d θ d t d φ d t cos  θ ) e ^ φ ( \ displaystyle ( \ begin ( aligned ) \ mathbf ( r ) & = \ mathbf ( r ) \ left ( t \ right ) = r \ mathbf ( \ hat ( e ) ) _ ( r ) \ \ \ mathbf ( v ) & = v \ mathbf ( \ hat ( e ) ) _ ( r ) + r \ , ( \ frac ( d \ theta ) ( dt ) ) \ mathbf ( \ hat ( e ) ) _ ( \ theta ) + r \ , ( \ frac ( d \ varphi ) ( dt ) ) \ , \ sin \ theta \ mathbf ( \ hat ( e ) ) _ ( \ varphi ) \ \ \ mathbf ( a ) & = \ left ( a-r \ left ( ( \ frac ( d \ theta ) ( dt ) ) \ right ) ^ ( 2 ) - r \ left ( ( \ frac ( d \ varphi ) ( dt ) ) \ right ) ^ ( 2 ) \ sin ^ ( 2 ) \ theta \ right ) \ mathbf ( \ hat ( e ) ) _ ( r ) \ \ & + \ left ( r ( \ frac ( d ^ ( 2 ) \ theta ) ( dt ^ ( 2 ) ) ) + 2v ( \ frac ( d \ theta ) ( dt ) ) - r \ left ( ( \ frac ( d \ varphi ) ( dt ) ) \ right ) ^ ( 2 ) \ sin \ theta \ cos \ theta \ right ) \ mathbf ( \ hat ( e ) ) _ ( \ theta ) \ \ & + \ left ( r ( \ frac ( d ^ ( 2 ) \ varphi ) ( dt ^ ( 2 ) ) ) \ , \ sin \ theta + 2v \ , ( \ frac ( d \ varphi ) ( dt ) ) \ , \ sin \ theta + 2r \ , ( \ frac ( d \ theta ) ( dt ) ) \ , ( \ frac ( d \ varphi ) ( dt ) ) \ , \ cos \ theta \ right ) \ mathbf ( \ hat ( e ) ) _ ( \ varphi ) \ end ( aligned ) ) \ , \ ! ) In the case of a constant φ this reduces to the planar equations above . Dynamic equations of motion ( edit ) Newtonian mechanics ( edit ) Main article : Newtonian mechanics The first general equation of motion developed was Newton 's second law of motion , in its most general form states the rate of change of momentum p = p ( t ) = mv ( t ) of an object equals the force F = F ( x ( t ) , v ( t ) , t ) acting on it , F = d p d t ( \ displaystyle \ mathbf ( F ) = ( \ frac ( d \ mathbf ( p ) ) ( dt ) ) ) The force in the equation is not the force the object exerts . Replacing momentum by mass times velocity , the law is also written more famously as F = m a ( \ displaystyle \ mathbf ( F ) = m \ mathbf ( a ) ) since m is a constant in Newtonian mechanics . Newton 's second law applies to point - like particles , and to all points in a rigid body . They also apply to each point in a mass continua , like deformable solids or fluids , but the motion of the system must be accounted for , see material derivative . In the case the mass is not constant , it is not sufficient to use the product rule for the time derivative on the mass and velocity , and Newton 's second law requires some modification consistent with conservation of momentum , see variable - mass system . It may be simple to write down the equations of motion in vector form using Newton 's laws of motion , but the components may vary in complicated ways with spatial coordinates and time , and solving them is not easy . Often there is an excess of variables to solve for the problem completely , so Newton 's laws are not always the most efficient way to determine the motion of a system . In simple cases of rectangular geometry , Newton 's laws work fine in Cartesian coordinates , but in other coordinate systems can become dramatically complex . The momentum form is preferable since this is readily generalized to more complex systems , generalizes to special and general relativity ( see four - momentum ) . It can also be used with the momentum conservation . However , Newton 's laws are not more fundamental than momentum conservation , because Newton 's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum , while a resultant force implies the momentum is not constant . Momentum conservation is always true for an isolated system not subject to resultant forces . For a number of particles ( see many body problem ) , the equation of motion for one particle i influenced by other particles is d p i d t = F E + ∑ i ≠ j F i j ( \ displaystyle ( \ frac ( d \ mathbf ( p ) _ ( i ) ) ( dt ) ) = \ mathbf ( F ) _ ( E ) + \ sum _ ( i \ neq j ) \ mathbf ( F ) _ ( ij ) \ , \ ! ) where p is the momentum of particle i , F is the force on particle i by particle j , and F is the resultant external force due to any agent not part of system . Particle i does not exert a force on itself . Euler 's laws of motion are similar to Newton 's laws , but they are applied specifically to the motion of rigid bodies . The Newton -- Euler equations combine the forces and torques acting on a rigid body into a single equation . Newton 's second law for rotation takes a similar form to the translational case , τ = d L d t , ( \ displaystyle ( \ boldsymbol ( \ tau ) ) = ( \ frac ( d \ mathbf ( L ) ) ( dt ) ) \ , , ) by equating the torque acting on the body to the rate of change of its angular momentum L. Analogous to mass times acceleration , the moment of inertia tensor I depends on the distribution of mass about the axis of rotation , and the angular acceleration is the rate of change of angular velocity , τ = I ⋅ α . ( \ displaystyle ( \ boldsymbol ( \ tau ) ) = \ mathbf ( I ) \ cdot ( \ boldsymbol ( \ alpha ) ) . ) Again , these equations apply to point like particles , or at each point of a rigid body . Likewise , for a number of particles , the equation of motion for one particle i is d L i d t = τ E + ∑ i ≠ j τ i j , ( \ displaystyle ( \ frac ( d \ mathbf ( L ) _ ( i ) ) ( dt ) ) = ( \ boldsymbol ( \ tau ) ) _ ( E ) + \ sum _ ( i \ neq j ) ( \ boldsymbol ( \ tau ) ) _ ( ij ) \ , , ) where L is the angular momentum of particle i , τ the torque on particle i by particle j , and τ is resultant external torque ( due to any agent not part of system ) . Particle i does not exert a torque on itself . Applications ( edit ) Some examples of Newton 's law include describing the motion of a simple pendulum , − m g sin  θ = m d 2 ( l θ ) d t 2 ⇒ d 2 θ d t 2 = − g l sin  θ , ( \ displaystyle - mg \ sin \ theta = m ( \ frac ( d ^ ( 2 ) ( l \ theta ) ) ( dt ^ ( 2 ) ) ) \ quad \ Rightarrow \ quad ( \ frac ( d ^ ( 2 ) \ theta ) ( dt ^ ( 2 ) ) ) = - ( \ frac ( g ) ( l ) ) \ sin \ theta \ , , ) and a damped , sinusoidally driven harmonic oscillator , F 0 sin  ( ω t ) = m ( d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x ) . ( \ displaystyle F_ ( 0 ) \ sin ( \ omega t ) = m \ left ( ( \ frac ( d ^ ( 2 ) x ) ( dt ^ ( 2 ) ) ) + 2 \ zeta \ omega _ ( 0 ) ( \ frac ( dx ) ( dt ) ) + \ omega _ ( 0 ) ^ ( 2 ) x \ right ) \ , . ) For describing the motion of masses due to gravity , Newton 's law of gravity can be combined with Newton 's second law . For two examples , a ball of mass m thrown in the air , in air currents ( such as wind ) described by a vector field of resistive forces R = R ( r , t ) , − G m M r 2 e ^ r + R = m d 2 r d t 2 + 0 ⇒ d 2 r d t 2 = − G M r 2 e ^ r + A ( \ displaystyle - ( \ frac ( GmM ) ( \ mathbf ( r ) ^ ( 2 ) ) ) \ mathbf ( \ hat ( e ) ) _ ( r ) + \ mathbf ( R ) = m ( \ frac ( d ^ ( 2 ) \ mathbf ( r ) ) ( dt ^ ( 2 ) ) ) + 0 \ quad \ Rightarrow \ quad ( \ frac ( d ^ ( 2 ) \ mathbf ( r ) ) ( dt ^ ( 2 ) ) ) = - ( \ frac ( GM ) ( \ mathbf ( r ) ^ ( 2 ) ) ) \ mathbf ( \ hat ( e ) ) _ ( r ) + \ mathbf ( A ) \ , \ ! ) where G is the gravitational constant , M the mass of the Earth , and A = R / m is the acceleration of the projectile due to the air currents at position r and time t . The classical N - body problem for N particles each interacting with each other due to gravity is a set of N nonlinear coupled second order ODEs , d 2 r i d t 2 = G ∑ i ≠ j m i m j r j − r i 3 ( r j − r i ) ( \ displaystyle ( \ frac ( d ^ ( 2 ) \ mathbf ( r ) _ ( i ) ) ( dt ^ ( 2 ) ) ) = G \ sum _ ( i \ neq j ) ( \ frac ( m_ ( i ) m_ ( j ) ) ( \ mathbf ( r ) _ ( j ) - \ mathbf ( r ) _ ( i ) ^ ( 3 ) ) ) ( \ mathbf ( r ) _ ( j ) - \ mathbf ( r ) _ ( i ) ) ) where i = 1 , 2 , ... , N labels the quantities ( mass , position , etc . ) associated with each particle . Analytical mechanics ( edit ) Main articles : Analytical mechanics , Lagrangian mechanics and Hamiltonian mechanics As the system evolves , q traces a path through configuration space ( only some are shown ) . The path taken by the system ( red ) has a stationary action ( δS = 0 ) under small changes in the configuration of the system ( δq ) . Using all three coordinates of 3D space is unnecessary if there are constraints on the system . If the system has N degrees of freedom , then one can use a set of N generalized coordinates q ( t ) = ( q ( t ) , q ( t ) ... q ( t ) ) , to define the configuration of the system . They can be in the form of arc lengths or angles . They are a considerable simplification to describe motion , since they take advantage of the intrinsic constraints that limit the system 's motion , and the number of coordinates is reduced to a minimum . The time derivatives of the generalized coordinates are the generalized velocities q ̇ = d q d t . ( \ displaystyle \ mathbf ( \ dot ( q ) ) = ( \ frac ( d \ mathbf ( q ) ) ( dt ) ) \ , . ) The Euler -- Lagrange equations are d d t ( ∂ L ∂ q ̇ ) = ∂ L ∂ q , ( \ displaystyle ( \ frac ( d ) ( dt ) ) \ left ( ( \ frac ( \ partial L ) ( \ partial \ mathbf ( \ dot ( q ) ) ) ) \ right ) = ( \ frac ( \ partial L ) ( \ partial \ mathbf ( q ) ) ) \ , , ) where the Lagrangian is a function of the configuration q and its time rate of change dq / dt ( and possibly time t ) L = L ( q ( t ) , q ̇ ( t ) , t ) . ( \ displaystyle L = L \ left ( \ mathbf ( q ) ( t ) , \ mathbf ( \ dot ( q ) ) ( t ) , t \ right ) \ , . ) Setting up the Lagrangian of the system , then substituting into the equations and evaluating the partial derivatives and simplifying , a set of coupled N second order ODEs in the coordinates are obtained . Hamilton 's equations are p ̇ = − ∂ H ∂ q , q ̇ = + ∂ H ∂ p , ( \ displaystyle \ mathbf ( \ dot ( p ) ) = - ( \ frac ( \ partial H ) ( \ partial \ mathbf ( q ) ) ) \ , , \ quad \ mathbf ( \ dot ( q ) ) = + ( \ frac ( \ partial H ) ( \ partial \ mathbf ( p ) ) ) \ , , ) where the Hamiltonian H = H ( q ( t ) , p ( t ) , t ) , ( \ displaystyle H = H \ left ( \ mathbf ( q ) ( t ) , \ mathbf ( p ) ( t ) , t \ right ) \ , , ) is a function of the configuration q and conjugate `` generalized '' momenta p = ∂ L ∂ q ̇ , ( \ displaystyle \ mathbf ( p ) = ( \ frac ( \ partial L ) ( \ partial \ mathbf ( \ dot ( q ) ) ) ) \ , , ) in which ∂ / ∂ q = ( ∂ / ∂ q , ∂ / ∂ q , ... , ∂ / ∂ q ) is a shorthand notation for a vector of partial derivatives with respect to the indicated variables ( see for example matrix calculus for this denominator notation ) , and possibly time t , Setting up the Hamiltonian of the system , then substituting into the equations and evaluating the partial derivatives and simplifying , a set of coupled 2N first order ODEs in the coordinates q and momenta p are obtained . The Hamilton -- Jacobi equation is − ∂ S ( q , t ) ∂ t = H ( q , p , t ) . ( \ displaystyle - ( \ frac ( \ partial S ( \ mathbf ( q ) , t ) ) ( \ partial t ) ) = H \ left ( \ mathbf ( q ) , \ mathbf ( p ) , t \ right ) \ , . ) where S ( q , t ) = ∫ t 1 t 2 L ( q , q ̇ , t ) d t , ( \ displaystyle S ( \ mathbf ( q ) , t ) = \ int _ ( t_ ( 1 ) ) ^ ( t_ ( 2 ) ) L ( \ mathbf ( q ) , \ mathbf ( \ dot ( q ) ) , t ) \ , dt \ , , ) is Hamilton 's principal function , also called the classical action is a functional of L. In this case , the momenta are given by p = ∂ S ∂ q . ( \ displaystyle \ mathbf ( p ) = ( \ frac ( \ partial S ) ( \ partial \ mathbf ( q ) ) ) \ , . ) Although the equation has a simple general form , for a given Hamiltonian it is actually a single first order non-linear PDE , in N + 1 variables . The action S allows identification of conserved quantities for mechanical systems , even when the mechanical problem itself can not be solved fully , because any differentiable symmetry of the action of a physical system has a corresponding conservation law , a theorem due to Emmy Noether . All classical equations of motion can be derived from the variational principle known as Hamilton 's principle of least action δ S = 0 , ( \ displaystyle \ delta S = 0 \ , , ) stating the path the system takes through the configuration space is the one with the least action S . Electrodynamics ( edit ) Lorentz force f on a charged particle ( of charge q ) in motion ( instantaneous velocity v ) . The E field and B field vary in space and time . In electrodynamics , the force on a charged particle of charge q is the Lorentz force : F = q ( E + v × B ) ( \ displaystyle \ mathbf ( F ) = q \ left ( \ mathbf ( E ) + \ mathbf ( v ) \ times \ mathbf ( B ) \ right ) \ , \ ! ) Combining with Newton 's second law gives a first order differential equation of motion , in terms of position of the particle : m d 2 r d t 2 = q ( E + d r d t × B ) ( \ displaystyle m ( \ frac ( d ^ ( 2 ) \ mathbf ( r ) ) ( dt ^ ( 2 ) ) ) = q \ left ( \ mathbf ( E ) + ( \ frac ( d \ mathbf ( r ) ) ( dt ) ) \ times \ mathbf ( B ) \ right ) \ , \ ! ) or its momentum : d p d t = q ( E + p × B m ) ( \ displaystyle ( \ frac ( d \ mathbf ( p ) ) ( dt ) ) = q \ left ( \ mathbf ( E ) + ( \ frac ( \ mathbf ( p ) \ times \ mathbf ( B ) ) ( m ) ) \ right ) \ , \ ! ) The same equation can be obtained using the Lagrangian ( and applying Lagrange 's equations above ) for a charged particle of mass m and charge q : L = 1 2 m r ̇ ⋅ r ̇ + q A ⋅ r ̇ − q φ ( \ displaystyle L = ( \ tfrac ( 1 ) ( 2 ) ) m \ mathbf ( \ dot ( r ) ) \ cdot \ mathbf ( \ dot ( r ) ) + q \ mathbf ( A ) \ cdot ( \ dot ( \ mathbf ( r ) ) ) - q \ phi ) where A and φ are the electromagnetic scalar and vector potential fields . The Lagrangian indicates an additional detail : the canonical momentum in Lagrangian mechanics is given by : P = ∂ L ∂ r ̇ = m r ̇ + q A ( \ displaystyle \ mathbf ( P ) = ( \ frac ( \ partial L ) ( \ partial ( \ dot ( \ mathbf ( r ) ) ) ) ) = m ( \ dot ( \ mathbf ( r ) ) ) + q \ mathbf ( A ) ) instead of just mv , implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle . The Lagrangian expression was first used to derive the force equation . Alternatively the Hamiltonian ( and substituting into the equations ) : H = ( P − q A ) 2 2 m + q φ ( \ displaystyle H = ( \ frac ( \ left ( \ mathbf ( P ) - q \ mathbf ( A ) \ right ) ^ ( 2 ) ) ( 2m ) ) + q \ phi \ , \ ! ) can derive the Lorentz force equation . General relativity ( edit ) Geodesic equation of motion ( edit ) Geodesics on a sphere are arcs of great circles ( yellow curve ) . On a 2D -- manifold ( such as the sphere shown ) , the direction of the accelerating geodesic is uniquely fixed if the separation vector ξ is orthogonal to the `` fiducial geodesic '' ( green curve ) . As the separation vector ξ changes to ξ after a distance s , the geodesics are not parallel ( geodesic deviation ) . Main articles : Geodesics in general relativity and Geodesic equation The above equations are valid in flat spacetime . In curved space spacetime , things become mathematically more complicated since there is no straight line ; this is generalized and replaced by a geodesic of the curved spacetime ( the shortest length of curve between two points ) . For curved manifolds with a metric tensor g , the metric provides the notion of arc length ( see line element for details ) , the differential arc length is given by : d s = g α β d x α d x β ( \ displaystyle ds = ( \ sqrt ( g_ ( \ alpha \ beta ) dx ^ ( \ alpha ) dx ^ ( \ beta ) ) ) ) and the geodesic equation is a second - order differential equation in the coordinates , the general solution is a family of geodesics : d 2 x μ d s 2 = − Γ μ α β d x α d s d x β d s ( \ displaystyle ( \ frac ( d ^ ( 2 ) x ^ ( \ mu ) ) ( ds ^ ( 2 ) ) ) = - \ Gamma ^ ( \ mu ) ( ) _ ( \ alpha \ beta ) ( \ frac ( dx ^ ( \ alpha ) ) ( ds ) ) ( \ frac ( dx ^ ( \ beta ) ) ( ds ) ) ) where Γ is a Christoffel symbol of the second kind , which contains the metric ( with respect to the coordinate system ) . Given the mass - energy distribution provided by the stress -- energy tensor T , the Einstein field equations are a set of non-linear second - order partial differential equations in the metric , and imply the curvature of space time is equivalent to a gravitational field ( see principle of equivalence ) . Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force . The relative acceleration of one geodesic to another in curved spacetime is given by the geodesic deviation equation : D 2 ξ α d s 2 = − R α β γ δ d x α d s ξ γ d x δ d s ( \ displaystyle ( \ frac ( D ^ ( 2 ) \ xi ^ ( \ alpha ) ) ( ds ^ ( 2 ) ) ) = - R ^ ( \ alpha ) ( ) _ ( \ beta \ gamma \ delta ) ( \ frac ( dx ^ ( \ alpha ) ) ( ds ) ) \ xi ^ ( \ gamma ) ( \ frac ( dx ^ ( \ delta ) ) ( ds ) ) ) where ξ = x − x is the separation vector between two geodesics , D / ds ( not just d / ds ) is the covariant derivative , and R is the Riemann curvature tensor , containing the Christoffel symbols . In other words , the geodesic deviation equation is the equation of motion for masses in curved spacetime , analogous to the Lorentz force equation for charges in an electromagnetic field . For flat spacetime , the metric is a constant tensor so the Christoffel symbols vanish , and the geodesic equation has the solutions of straight lines . This is also the limiting case when masses move according to Newton 's law of gravity . Spinning objects ( edit ) In general relativity , rotational motion is described by the relativistic angular momentum tensor , including the spin tensor , which enter the equations of motion under covariant derivatives with respect to proper time . The Mathisson -- Papapetrou -- Dixon equations describe the motion of spinning objects moving in a gravitational field . Analogues for waves and fields ( edit ) Unlike the equations of motion for describing particle mechanics , which are systems of coupled ordinary differential equations , the analogous equations governing the dynamics of waves and fields are always partial differential equations , since the waves or fields are functions of space and time . For a particular solution , boundary conditions along with initial conditions need to be specified . Sometimes in the following contexts , the wave or field equations are also called `` equations of motion '' . Field equations ( edit ) Equations that describe the spatial dependence and time evolution of fields are called field equations . These include Maxwell 's equations for the electromagnetic field , Poisson 's equation for Newtonian gravitational or electrostatic field potentials , the Einstein field equation for gravitation ( Newton 's law of gravity is a special case for weak gravitational fields and low velocities of particles ) . This terminology is not universal : for example although the Navier -- Stokes equations govern the velocity field of a fluid , they are not usually called `` field equations '' , since in this context they represent the momentum of the fluid and are called the `` momentum equations '' instead . Wave equations ( edit ) Equations of wave motion are called wave equations . The solutions to a wave equation give the time - evolution and spatial dependence of the amplitude . Boundary conditions determine if the solutions describe traveling waves or standing waves . From classical equations of motion and field equations ; mechanical , gravitational wave , and electromagnetic wave equations can be derived . The general linear wave equation in 3D is : 1 v 2 ∂ 2 X ∂ t 2 = ∇ 2 X ( \ displaystyle ( \ frac ( 1 ) ( v ^ ( 2 ) ) ) ( \ frac ( \ partial ^ ( 2 ) X ) ( \ partial t ^ ( 2 ) ) ) = \ nabla ^ ( 2 ) X ) where X = X ( r , t ) is any mechanical or electromagnetic field amplitude , say : the transverse or longitudinal displacement of a vibrating rod , wire , cable , membrane etc. , the fluctuating pressure of a medium , sound pressure , the electric fields E or D , or the magnetic fields B or H , the voltage V or current I in an alternating current circuit , and v is the phase velocity . Nonlinear equations model the dependence of phase velocity on amplitude , replacing v by v ( X ) . There are other linear and nonlinear wave equations for very specific applications , see for example the Korteweg -- de Vries equation . Quantum theory ( edit ) In quantum theory , the wave and field concepts both appear . In quantum mechanics , in which particles also have wave - like properties according to wave -- particle duality , the analogue of the classical equations of motion ( Newton 's law , Euler -- Lagrange equation , Hamilton -- Jacobi equation , etc . ) is the Schrödinger equation in its most general form : i ħ ∂ Ψ ∂ t = H ^ Ψ , ( \ displaystyle i \ hbar ( \ frac ( \ partial \ Psi ) ( \ partial t ) ) = ( \ hat ( H ) ) \ Psi \ , , ) where Ψ is the wavefunction of the system , Ĥ is the quantum Hamiltonian operator , rather than a function as in classical mechanics , and ħ is the Planck constant divided by 2π . Setting up the Hamiltonian and inserting it into the equation results in a wave equation , the solution is the wavefunction as a function of space and time . The Schrödinger equation itself reduces to the Hamilton -- Jacobi equation when one considers the correspondence principle , in the limit that ħ becomes zero . Throughout all aspects of quantum theory , relativistic or non-relativistic , there are various formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system , for instance : the Heisenberg equation of motion resembles the time evolution of classical observables as functions of position , momentum , and time , if one replaces dynamical observables by their quantum operators and the classical Poisson bracket by the commutator , the phase space formulation closely follows classical Hamiltonian mechanics , placing position and momentum on equal footing , the Feynman path integral formulation extends the principle of least action to quantum mechanics and field theory , placing emphasis on the use of a Lagrangians rather than Hamiltonians . See also ( edit ) Scalar ( physics ) Vector Distance Displacement Speed Velocity Acceleration Angular displacement Angular speed Angular velocity Angular acceleration Equations for a falling body Parabolic trajectory Curvilinear coordinates Orthogonal coordinates Newton 's laws of motion Torricelli 's equation Euler -- Lagrange equation Generalized forces Defining equation ( physics ) Newton -- Euler laws of motion for a rigid body References ( edit ) Jump up ^ Encyclopaedia of Physics ( second Edition ) , R.G. Lerner , G.L. Trigg , VHC Publishers , 1991 , ISBN ( Verlagsgesellschaft ) 3 - 527 - 26954 - 1 ( VHC Inc . ) 0 - 89573 - 752 - 3 ^ Jump up to : Analytical Mechanics , L.N. Hand , J.D. Finch , Cambridge University Press , 2008 , ISBN 978 - 0 - 521 - 57572 - 0 Jump up ^ See History of Mathematics Jump up ^ The Britannica Guide to History of Mathematics , ed . 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ISBN 0 - 679 - 77631 - 1 . ^ Jump up to : Classical Mechanics ( second edition ) , T.W.B. Kibble , European Physics Series , 1973 , ISBN 0 - 07 - 084018 - 0 Jump up ^ Electromagnetism ( second edition ) , I.S. Grant , W.R. Phillips , Manchester Physics Series , 2008 ISBN 0 - 471 - 92712 - 0 Jump up ^ Classical Mechanics ( second Edition ) , T.W.B. Kibble , European Physics Series , Mc Graw Hill ( UK ) , 1973 , ISBN 0 - 07 - 084018 - 0 . Jump up ^ Misner , Thorne , Wheeler , Gravitation Jump up ^ C.B. Parker ( 1994 ) . McGraw Hill Encyclopaedia of Physics ( second ed . ) . p. 1199 . ISBN 0 - 07 - 051400 - 3 . Jump up ^ C.B. Parker ( 1994 ) . McGraw Hill Encyclopaedia of Physics ( second ed . ) . p. 1200 . ISBN 0 - 07 - 051400 - 3 . Jump up ^ J.A. Wheeler ; C. Misner ; K.S. Thorne ( 1973 ) . Gravitation . W.H. Freeman & Co. pp. 34 -- 35 . ISBN 0 - 7167 - 0344 - 0 . Jump up ^ H.D. Young ; R.A. Freedman ( 2008 ) . University Physics ( 12th ed . ) . Addison - Wesley ( Pearson International ) . ISBN 0 - 321 - 50130 - 6 . Retrieved from `` https://en.wikipedia.org/w/index.php?title=Equations_of_motion&oldid=810175763 '' Categories : Classical mechanics Equations of physics Talk Contents About Wikipedia Afrikaans Català Čeština Cymraeg Dansk Deutsch Eesti Español فارسی Français 한국어 हिन्दी Hrvatski Italiano עברית Қазақша Latina Lëtzebuergesch Nederlands 日本 語 Norsk Norsk nynorsk Polski Português Română Русский Shqip Srpskohrvatski / српскохрватски தமிழ் Türkçe Українська Tiếng Việt 中文 Edit links This page was last edited on 13 November 2017 , at 19 : 29 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . By using this site , you agree to the Terms of Use and Privacy Policy . Wikipedia ® is a registered trademark of the Wikimedia Foundation , Inc. , a non-profit organization . About Wikipedia


Hickey (surname) Hickey is a common surname of Irish origin . The original form is Ó hÍceadha , which is still used in Ireland . Notable people with the surname include : Contents ( hide ) 1 Arts and entertainment 2 Sport 3 Politics and law 4 History and current affairs 5 Religion 6 Science and academia 7 Fictional characters 8 See also Arts and entertainment ( edit ) Cheryl Hickey , born 1976 , entertainment reporter for the Global Television Network Dale Hickey born 1937 , Australian artist Dave Hickey , American art critic , author of Air Guitar : Essays on Art & Democracy ( 1998 ) Eddie Hickey , US sportsman Emily Henrietta Hickey , Irish author and translator Ersel Hickey , rockabilly singer James Harden - Hickey , 19th - century American writer Kenny Hickey , US musician Michael Hickey , US screenwriter Rich Hickey , Creator of the Clojure programming language Tom Hickey , Irish actor , famous for his role in The Riordans Thomas Hickey , 1741 -- 1842 , Irish painter William Hickey ( actor ) William Hickey ( memoirist ) May Wynn , actress , born Donna Lee Hickey before taking the stage name `` May Wynn , '' after playing a character by that name in The Caine Mutiny Sport ( edit ) Ambrose Hickey ( 1945 -- 2016 ) , Irish Gaelic footballer Anthony Hickey , born 1992 , American basketball player Charlie Hickey , American college baseball coach Colin Hickey , Australian speed skater Denis Hickie , Irish international Rugby Union Jack Hickey ( rugby ) , Australian Dual internationalist Rugby Union , Rugby league Jarrad Hickey , born 1985 , Australian Rugby League player Jim Hickey ( American football ) Jim Hickey ( baseball coach ) Miriam Hickey , US Soccer player Noah Hickey , New Zealand football player Noel Hickey , Irish sportsman Pat Hickey , born 1953 , Canadian Ice Hockey player Pat Hickey ( sports administrator ) , born 1945 , Irish sports administrator Red Hickey , 1917 -- 2006 , athlete , played American Football Reg Hickey , player and coach for Geelong in Australian rules football Thomas Hickey , professional ice hockey player , L.A. Kings Politics and law ( edit ) Bonnie Hickey , Canadian politician Eileen Hickey ( New York ) ( 1945 -- 1999 ) , New York politician Eileen M. Hickey ( 1886 -- 1960 ) , Northern Irish politician James Hickey ( Irish politician ) Jimmy Hickey Jr. , American politician John J. Hickey , American politician Maggie Hickey , Australian politician Theodore M. Hickey , American politician Vivian Hickey ( 1916 - 2016 ) , American politician and educator William J. Hickey , New York politician and judge History and current affairs ( edit ) Colonel James Hickey ( soldier ) , in charge of the US Special forces team which captured Saddam Hussein Elizabeth Hickey historian and writer Jim Hickey ( weather presenter ) Thomas Hickey ( soldier ) , executed for mutiny during the American Revolutionary War Religion ( edit ) Antony Hickey , born 1586 , Irish Franciscan theologian David Francis Hickey ( 1882 -- 1973 ) , American - born Catholic bishop in Belize Rev. William Hickey ( 1787 - 1875 ) , Irish priest , writer and philanthropist Science and academia ( edit ) Joseph Hickey ( 1907 - 1993 ) , American ornithologist Raymond Hickey , Irish linguist Fictional characters ( edit ) Earl Hickey , fictional character in the NBC show My Name Is Earl Randy Hickey , fictional character in the NBC show My Name Is Earl See also ( edit ) Hickey ( disambiguation ) Ó hÍceadha Irish medical families This page lists people with the surname Hickey . If an internal link intending to refer to a specific person led you to this page , you may wish to change that link by adding the person 's given name ( s ) to the link . Retrieved from `` https://en.wikipedia.org/w/index.php?title=Hickey_(surname)&oldid=813665472 '' Categories : Surnames Anglicised Irish - language surnames Hidden categories : All set index articles Talk Contents About Wikipedia Русский Edit links This page was last edited on 4 December 2017 , at 16 : 46 . About Wikipedia


Maggie Pierce Margaret `` Maggie '' Pierce , M.D. is a fictional character from the American television medical drama Grey 's Anatomy , which airs on the American Broadcasting Company ( ABC ) in the United States . The character was created by series producer Shonda Rhimes and is portrayed by actress Kelly McCreary from the tenth season 's penultimate episode onwards . It is revealed in the `` season ten finale '' , Maggie 's biological parents are Richard Webber ( James Pickens Jr . ) and Ellis Grey ( Kate Burton ) , making her Meredith 's half - sister . McCreary was upgraded to a series regular in the eleventh episode of the eleventh season . Pierce has been described as `` perky '' and `` chatty '' , as well as focusing mostly on her academic career rather than social life , causing her to be socially inept at times and childlike . She graduated from high school and medical school early , and becomes the head of her department at the young age of 31 years old . McCreary was well received upon her introduction . Contents ( hide ) 1 Storylines 2 Development 2.1 Casting and creation 2.2 Characterization 2.3 Introduction 3 Reception 4 References 5 External links Storylines ( edit ) This section does not cite any sources . Please help improve this section by adding citations to reliable sources . Unsourced material may be challenged and removed . ( February 2017 ) ( Learn how and when to remove this template message ) Maggie is first introduced in the penultimate episode of the tenth season . Cristina Yang ( Sandra Oh ) interviews various different candidates to replace her as the head of her department in Cardiothoracic surgery . Maggie , having finished high school and medical school early , becomes head of her department at 31 years old . It is revealed in the season 10 finale `` Fear ( of the Unknown ) '' , Maggie 's biological mother was Ellis Grey . Richard Webber is shocked by this , knowing he is her biological father . Throughout the eleventh season , Webber struggles to deal with the revelation and keeps it from Maggie . Maggie explains that 2 years prior , she looked for her birth mom and needed to go to the court for the records . She found out that she was dead , but got her name , and discovered that she was an amazing surgeon that worked at the Seattle Grace Hospital . She stated that that 's not why she took the job , but it 's `` a little bit '' why she took the interview . When Maggie identifies Meredith Grey , her half - sister , she attempts to bond with her . However , Meredith attempts to take charge and conflicts with her on a case . After several arguments , Maggie tells Meredith that she is her half - sister . Meredith , in disbelief , points out that she would know that her mother would be pregnant when she was five years old , as Maggie is five years younger than her . Meredith works with Alex Karev ( Justin Chambers ) to decide whether Maggie is lying or if she is wrong . They discover she is correct . Meredith gets flashes back to the time of her mother 's crisis at five years old , and remembers being at a hospital and hearing a screaming baby . After several weeks , Webber admits to Maggie that he is her father . Maggie is angry at him . Later , when he comes to apologize for not being there as a father , Maggie corrects him and says he is not her father , as she has adoptive parents she loves , and that she is only angry because of his dishonesty and making her look foolish and naive for weeks . Derek and Meredith decide to have Maggie come over to their house for dinner , to get to know Maggie better . After Maggie says yes , Derek decides it is a good idea to ask Richard as well . Richard reluctantly agrees . The night of the dinner , Maggie and Richard show up , but no - one answers . Maggie leaves him after that . Maggie develops a relationship with radiologist , Ethan Boyd . She admits her awkwardness in relationships , having ended her previous engagement to Dean , who she was unable to sleep in the same bed with . Meredith ends up in a bind with no one to watch her kids , so Maggie offers to watch them . Richard and Maggie eventually come to good terms , and Maggie 's helping with Meredith 's kids help her integrate into her family home more . Maggie learns about the histories of Meredith , Alex and Callie Torres ( Sara Ramirez ) as they commemorate their dead friend , George O'Malley ( TR Knight ) . After Derek 's death , Meredith abandons Seattle for a year , leaving Maggie a note . Maggie is shocked by this , and admits she felt abandoned by Meredith right when they were starting to become a family . She invites Webber to a Christmas party , noticing his loneliness . Meredith eventually returns home , hugging Maggie and re-integrating into her old family home . While Maggie and her colleagues are busy dealing with Keith Gardner after a tunnel collapse , Maggie gets a phone call that her parents are getting a divorce . Her mother confesses to her that she 'd been having an affair with their car serviceman for eleven years and they had only stayed together for her sake , but were getting a divorce now that she was out on her own . Maggie is emotional about this , but she does n't want to bother her colleagues about it . Later , she confides in Meredith over the issue , but feels it is silly in comparison to her husband 's death . Meredith , however , says she can tell her anything and she should come to her whenever she is having a problem . After Meredith buys her house back from Alex , she invites Maggie and Amelia Shepherd ( Caterina Scorsone ) to move in with her . They assist Meredith in caring for her kids . Maggie also often mediates between Meredith and Amelia , who frequently get into fights over little things . She often helps with Meredith 's three kids , with each `` sister '' in charge of getting one kid ready for the day . In the twelfth season , Maggie develops a relationship with intern Andrew DeLuca . Maggie sleeps with DeLuca after a short conversation and a couple of drinks at Joe 's . They agree it is a one - time thing , before they have sex again . Maggie and DeLuca keep their relationship private to avoid difficulties at work . DeLuca grows frustrated with their secrecy , so Maggie abruptly decides to go public . When this happens , DeLuca becomes uncomfortable with the way he is viewed by others because he is dating an attending , and does n't want others to think he is getting special treatment . He also feels intimidated by her authority . When she confronts him , he ends things with her . Maggie attempts to move on and develops an interest in Nathan Riggs ( Martin Henderson ) . However , she is unaware Meredith has a sexual relationship with him . She confides in Meredith about this at Amelia 's wedding to Owen Hunt ( Kevin McKidd ) . In the thirteenth season , Maggie is personally hurt by Meredith lying to her about Alex 's actions assaulting DeLuca . She tells her to never lie to her again , but she does so immediately afterwards about Riggs . Riggs tells Meredith she needs to tell Maggie . Maggie works up the courage to ask Riggs out , but he rejects her . Development ( edit ) Casting and creation ( edit ) McCreary on the role When I auditioned , all they told me was that the character might recur a little bit into Season 11 ; they did not tell me it was Meredith 's sister and there was long - term potential . I had no idea the scope of it in terms of how ( she ) fit into the mythology of the series . Her character ties into a lot of things that have already occurred in these people 's lives . `` '' Benjamin Lindsay , Backstage 2015 On April 5 , 2014 , TVLine reported that Kelly McCreary best known for her series regular role on The CW 's short lived medical drama Emily Owens , M.D. and recent guest appearances on Scandal had been cast in a guest star role . McCeary was slated to debut on the May 8 , 2015 episode . On October 23 , 2014 , it was announced that McCreary was promoted to series regular after being credited as guest - starring until the eleventh episode . It was a couple of months after McCreary wrapped her stint on Scandal that she was invited to audition for Grey 's . McCeary revealed that she originally auditioned for the character of `` Claudette '' with dummy sides , unbeknownst to her . The only information McCeary had about the character was `` that she was adopted , that she would recur , and she was going to be a very important storyline . '' Just before the script reading for the season 10 finale , Rhimes summoned McCeary to inform her of Maggie 's true identity . `` I was really psyched '' McCeary revealed when Rhimes informed her of Maggie 's lineage . `` I felt really honored to be the one to accept the challenge . '' However , it was n't until McCeary experienced the reactions to Maggie 's existence from her fellow cast mates that McCeary realized just how big of a deal the character would be . The character 's potential existence was first referenced in the press in February 2009 when Michael Ausiello hinted at the possibility of Meredith having a half - sibling , the product of her mother 's affair with Richard . Initial speculation pointed to Jesse Williams ' Jackson Avery as the child , but the actor refuted those rumors . Shonda Rhimes revealed that the character had always been a part of the original story plans , but the character 's existence was not set in stone until season 4 . Characterization ( edit ) Kelly McCeary expressed her excitement about how special Maggie in an interview with BuddyTV . `` I just feel like it 's a rare opportunity to be able to play somebody that is so complex and fully realized and smart and fierce and compassionate and awkward and the whole thing . '' Maggie is very good at her job so she can find a place in a professional capacity , but in her personal life , she `` gets more than she bargained for '' McCreary said . `` I think she 's a really good doctor '' McCeary said of Maggie . `` She 's obviously super-bright . She 's very young to be as high in her field as she is . '' McCeary further described Maggie as a `` really great problem - solver . '' Maggie is `` very compassionate '' and at the same time can `` keep a clear head and do the right thing medically all the time . '' McCeary further described Maggie as `` so dynamic . '' While Maggie initially `` came across as all business , '' Kelly McCeary relishes in playing `` Maggie 's neurotic , existential , meltdown side . '' `` Maggie also wo n't really back down from a fight ; she 's very decisive . '' Introduction ( edit ) Further information : Puzzle With a Piece Missing and Got to Be Real ( Grey 's Anatomy ) Kate Aurthur noted that until Maggie 's introduction , Grey 's was not `` known for dropping in stories that it planted seasons before -- it 's never been filled with twisty mythology . '' Rhimes insisted that the timing was n't right for Maggie 's introduction in previous seasons . In season 3 , the series introduced Meredith 's paternal half - sister Lexie Grey ( Chyler Leigh ) and it was at that time that Maggie became a serious topic of discussion . Though Lexie was killed off in the season 8 finale , `` Flight , '' Rhimes felt it was still too soon for Maggie 's arrival . The scribe even thought the show might end without the character ever being introduced . `` Then we hit this moment in time and I remember walking into the writers ' room and going , ' You guys , it 's time . ' '' When Maggie makes the sudden revelation that she is Ellis ' child to Richard , viewers are left to wonder about her history . The second episode of the season puts Maggie front and center . The episode appropriately titled `` Puzzle With a Piece Missing , '' focuses on Maggie 's struggle to fit in throughout her life , even with her own adoptive parents whom she is very different from . Though she is very close with her parents , and they get along very well , she 's grown up `` still feeling like something 's missing . '' Kelly McCreary said `` That 's what 's really compelling to me about Maggie : that sort of dark underbelly of this person who seems to really have it all together . '' Maggie struggles to build relationships with her colleagues . She unknowingly shares confidential information about Alex Karev ( Justin Chambers ) and she gets she also insults the hospital board when she tries to prop up Miranda Bailey ( Chandra Wilson ) . Though she 's got the `` best intentions , '' things backfire . Maggie also serves as a `` mentor '' to Jo Wilson ( Camilla Luddington ) . As Maggie discovers she has several things in common with Richard , the stage is set for Maggie to learn that Richard is her father . `` I think people will relate to the awkwardness of being the new kid at school '' McCeary said of the episode 's comedic tone . Despite so many different interactions , the episode highlights `` Maggie 's loneliness . '' The Maggie - centric episode also featured Kelly McCreary stepping into the coveted role of narrator for the episode , which was usually reserved for Ellen Pompeo 's Meredith , and on occasion other veteran cast members . `` It was overwhelming and very surprising '' McCeary said of being the center of attention so early into her Grey 's tenure . She continued , `` I felt terrified and also really grateful that they trusted me with the institution about of Grey 's Anatomy for a whole episode . '' McCeary relished in finally learning the character 's history . `` ( It ) was a huge gift to have 65 pages worth of material telling me exactly who she is . It was great ! '' Reception ( edit ) This section needs expansion . You can help by adding to it . ( February 2017 ) Critics responded positively to the development of Pierce in her centric episode `` Puzzle With a Piece Missing '' . Fempop gave a largely positive review writing , `` Puzzle With a Piece Missing is a funny and insightful look both into this latest interloper and the established characters of Grey 's . '' praising the new addition , `` Thankfully she 's ( Pierce ) her own person , confident and happy to bring the thunder and put everyone , from interns to the chief of surgery , into their places . '' Entertainment Weekly lauded the episode stating , `` Coming into the second week without Cristina Yang , Grey 's was smart to focus all of its attention on the newcomer . With most other main characters only making occasional cameos , it made it more difficult to feel Yang 's absence . And by showing Maggie 's side of the story , it made her more relatable . It was a win - win . '' adding on McCreary 's character , `` this episode is about how Maggie is her own , very likable person . '' On her character TV Fanatic wrote , Overall , we got to know Maggie a little better and it 's nice to see that she 's made some friends . It was a fair episode that showed Maggie 's basically a work in progress . Maybe she 'll grow on us eventually . '' For the thirteenth season , Vulture writer , Maggie Fremont spoke positively about the character : `` It 's never easy for a long - running series to introduce a new character , especially when that character is meant to immediately fit in with the rest of the ensemble . But when Maggie Pierce arrived in season ten , that 's exactly what Grey 's Anatomy did . It could 've been easy to dislike her -- she showed up to take over cardio as Cristina Yang was leaving , for chrissakes ! She was yet another one of Meredith 's long - lost sisters . The cards were really stacked against Maggie Pierce . Yet , thanks to the writers and Kelly McCreary 's terrific performance , Maggie was immediately endearing . She is weird and neurotic and cheerful . She is nothing like Cristina , but she fills a void that Meredith needs filled in order to function . Can you really imagine Grey 's without Maggie Pierce ? '' References ( edit ) Jump up ^ http://www.hollywoodreporter.com/live-feed/greys-anatomy-11-big-developments-735827 Jump up ^ Writer : William Harper ; Director : Tony Phelan ( May 15 , 2014 ) . `` Fear ( of the Unknown ) '' . Grey 's Anatomy . Season 10 . Episode 24 . American Broadcasting Company . access - date = requires url = ( help ) ^ Jump up to : Lindsay , Benjamin ( December 8 , 2015 ) . `` How Kelly McCreary of ' Grey 's Anatomy ' Finds Catharsis in Emotional Scenes '' . Backstage . Backstage , LLC . Retrieved September 24 , 2016 . Jump up ^ Ausiello , Michael ( April 5 , 2014 ) . `` Exclusive : Ill - Fated Scandal Guest Star Resurrected on Grey 's Anatomy '' . TVLine . Penske Media Corporation . Retrieved September 27 , 2014 . Jump up ^ Raftery , Liz ( October 23 , 2014 ) . `` Grey 's Anatomy : Kelly McCreary Upped to Series Regular '' . TV Guide . CBS Interactive . Retrieved October 23 , 2014 . Jump up ^ Goldberg , Lesley ( October 23 , 2014 ) . `` ' Grey 's Anatomy 's ' Kelly McCreary Upped to Series Regular '' . The Hollywood Reporter . Prometheus Global Media . Retrieved October 24 , 2014 . Jump up ^ `` ' Grey 's Anatomy ' Promotes Kelly McCreary to Series Regular '' . Variety . Penske Media Corporation . October 23 , 2014 . Retrieved October 24 , 2014 . Jump up ^ `` ' Grey 's Anatomy ' promotes Kelly McCreary to series regular '' . Entertainment Weekly . Time Inc . October 23 , 2014 . Retrieved October 24 , 2014 . ^ Jump up to : Greco , Patti ( October 9 , 2014 ) . `` '' Grey 's Anatomy '' Newbie Kelly McCreary on What 's Next for Meredith and Maggie `` . Cosmopolitan . Hearst . Retrieved September 24 , 2016 . ^ Jump up to : Aurthur , Kate ( January 29 , 2015 ) . `` Everything You Need To Know About The Woman Who Plays Meredith Grey 's Sister On `` Grey 's Anatomy '' `` . BuzzFeed . BuzzFeed Inc . Retrieved September 24 , 2016 . ^ Jump up to : Nelli , Victoria ( October 16 , 2014 ) . `` Maggie 's Anatomy : Kelly McCreary Dishes on Joining GREY 'S ANATOMY and Previews Tonight 's Flashback Fun '' . The TV Addict . Retrieved September 24 , 2016 . ^ Jump up to : Goldberg , Lesley ( October 2 , 2014 ) . `` ' Grey 's Anatomy 's Kelly McCreary : There 's a Lot Maggie Does n't Know '' . The Hollywood Reporter . Prometheus Global Media . Retrieved September 25 , 2016 . Jump up ^ Ausiello , Michael ( February 25 , 2009 ) . `` ' Grey 's ' exclusive : Does Mer have another MIA sibling ? '' . Entertainment Weekly . Time Inc . Retrieved December 8 , 2013 . Jump up ^ Webb Mitovich , Matt ( December 8 , 2009 ) . `` Grey 's Anatomy 's Jesse Williams Addresses Meredith 's Brother Rumor '' . Archived from the original on December 12 , 2009 . Retrieved December 8 , 2013 . ^ Jump up to : Abrams , Natalie ( July 15 , 2014 ) . `` Grey 's Anatomy 's Shonda Rhimes Teases Half - Sister Twist , Possible New Romance '' . TV Guide . CBS Interactive . Retrieved October 3 , 2014 . ^ Jump up to : Jacobs , Meredith ( October 1 , 2014 ) . `` ' Grey 's Anatomy ' Exclusive Interview : Kelly McCreary on ' Clashing ' with Meredith and a ' Shift ' with Richard '' . BuddyTV . Retrieved September 24 , 2016 . Jump up ^ Goldberg , Lesley ( September 25 , 2014 ) . `` ' Grey 's Anatomy ' Stars Tease Season 11 : New Love , Board Drama and Meddling Meredith '' . The Hollywood Reporter . Prometheus Global Media . Retrieved October 3 , 2014 . Jump up ^ Goldberg , Lesley ( July 16 , 2014 ) . `` ' Grey 's Anatomy 's ' Shonda Rhimes : Season 11 to Focus on Meredith , Explore Callie and Arizona 's ' Band - Aid ' '' . The Hollywood Reporter . Prometheus Global Media . Retrieved September 24 , 2016 . ^ Jump up to : Abrams , Natalie ( September 30 , 2014 ) . `` ' Grey 's Anatomy ' sneak peek : Things get even worse between Meredith and Maggie '' . Entertainment Weekly . Time Inc . Retrieved September 24 , 2016 . ^ Jump up to : Raftery , Liz ( October 1 , 2014 ) . `` Grey 's Anatomy Newcomer Kelly McCreary : Maggie and Meredith Are Like `` Oil and Water '' `` . TV Guide . CBS Interactive . Retrieved September 24 , 2016 . ^ Jump up to : Bucksbaum , Sydney ( October 2 , 2014 ) . `` Maggie 's Grey 's Anatomy Secret Wo n't Stay Hidden for Long ! Kelly McCreary Gives the Scoop '' . E !. NBCUniversal . Retrieved September 25 , 2016 . Jump up ^ Goldberg , Lesley ( October 2 , 2014 ) . `` ' Grey 's Anatomy ' Postmortem : ' Meredith Will Delve Deep Into the Past ' to Uncover the Truth '' . The Hollywood Reporter . Prometheus Global Media . Retrieved September 25 , 2016 . Jump up ^ Samson , Janalen ( October 2 , 2014 ) . `` ' Grey 's Anatomy ' Recap : Maggie Opens Pandora 's Box '' . BuddyTV . Retrieved September 25 , 2016 . Jump up ^ Alex Cranz . `` Review : Grey 's Anatomy - ' Puzzle With a Missing Piece ' '' . FemPop Magazine . Retrieved August 14 , 2015 . Jump up ^ `` Grey 's Anatomy recap : ' Puzzle With A Piece Missing ' '' . Entertainment Weekly . Retrieved August 14 , 2015 . Jump up ^ Tran , Christina . `` Grey 's Anatomy '' . TV Fanatic . Retrieved August 14 , 2015 . Jump up ^ Fremont , Maggie . `` Grey 's Anatomy Recap : Oh , Mother '' . Vulture . Retrieved February 10 , 2017 . External links ( edit ) Grey 's Anatomy portal Fictional characters portal Television portal Maggie Piece at ABC.com Maggie Pierce on IMDb ( hide ) Grey 's Anatomy Episodes Season 1 5 6 7 8 9 10 11 12 13 14 Characters ( cast members ) Meredith Grey Cristina Yang Izzie Stevens Alex Karev George O'Malley Miranda Bailey Richard Webber Derek Shepherd Preston Burke Addison Montgomery Mark Sloan Callie Torres Lexie Grey Erica Hahn Owen Hunt Sadie Harris Arizona Robbins Teddy Altman Jackson Avery April Kepner Amelia Shepherd Stephanie Edwards Maggie Pierce Other Awards and nominations Soundtrack Video game Private Practice A Corazón Abierto ( Mexican telenovela ) A Corazón Abierto ( Colombian telenovela ) Portal Retrieved from `` https://en.wikipedia.org/w/index.php?title=Maggie_Pierce&oldid=799923131 '' Categories : Grey 's Anatomy characters Fictional African - American people Fictional characters from Washington ( state ) Fictional storytellers Fictional characters introduced in 2014 Fictional surgeons Fictional female doctors Hidden categories : Pages using citations with accessdate and no URL Pages using infobox character with unknown parameters Articles needing additional references from February 2017 All articles needing additional references Articles to be expanded from February 2017 All articles to be expanded Articles using small message boxes Talk Contents About Wikipedia Français Italiano Edit links This page was last edited on 10 September 2017 , at 16 : 48 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . By using this site , you agree to the Terms of Use and Privacy Policy . Wikipedia ® is a registered trademark of the Wikimedia Foundation , Inc. , a non-profit organization . About Wikipedia


I'll Be There for You (The Rembrandts song) `` I 'll Be There for You '' is a song recorded by the American duo The Rembrandts . It is best known as the theme song to the American sitcom Friends , which premiered during 1994 and ended in 2004 . The song was also released as the first single from the group 's third studio album LP . Contents ( hide ) 1 Song information 2 Music video 3 Track listing 4 Chart performance 5 Charts 6 Compilation usage 7 Cover versions 8 In popular culture 9 See also 10 References 11 External links Song information ( edit ) `` I 'll Be There for You '' was co-written by Friends producers David Crane and Marta Kauffman , Kauffman 's husband , composer Michael Skloff , and songwriter Allee Willis , along with Phil Sōlem and Danny Wilde , both of the Rembrandts . It is strongly influenced by The Beatles , especially reminiscent of the `` I Feel Fine '' guitar riff , and is also highly reminiscent of The Monkees ' `` Pleasant Valley Sunday . '' The original theme , which is under one minute long , was later re-recorded as a three - minute pop song . After Nashville program director Charlie Quinn , along with radio announcer and music director Tom Peace , looped the original short version into a full - length track and broadcast it on radio station WYHY , it became so popular that they had to re-record it . `` Our record label said we had to finish the song and record it . There was no way to get out of it , '' lead singer Phil Sōlem said . Music video ( edit ) The music video of the song featured the six main stars of Friends , taking the singing and dancing roles of the band after , amusingly , attacking them all and bemusing everyone else on the set with their daft behaviour . The video was included as a bonus feature on several of the show 's Complete Season DVD Boxsets . Track listing ( edit ) European CD maxi - single `` I 'll Be There for You '' -- 3 : 09 `` Fixin ' to Blow '' -- 5 : 03 `` Just the Way It Is Baby '' -- 4 : 06 `` Snippets Medley '' -- 6 : 46 Chart performance ( edit ) In the United States , the song topped the Billboard Hot 100 Airplay chart for eight weeks . When the single was released 1995 , it peaked at # 3 on the UK Singles Chart in the United Kingdom and at # 17 on the Billboard Hot 100 in the United States as the flipside to `` This House Is Not a Home '' . It also peaked at # 1 on the U.S. Top 40 Mainstream and Hot Adult Contemporary Tracks charts . The song was the most successful in Canada where it peaked at number 1 for 5 consecutive weeks . In April 2004 , the song was ranked at # 15 on Blender magazine 's list of the `` 50 Worst Songs Ever '' . Charts ( edit ) Chart ( 1995 -- 1997 ) Peak position Australia ( ARIA ) Belgium ( Ultratop 50 Flanders ) 10 Canada ( RPM ) Netherlands ( Single Top 100 ) 20 France ( SNEP ) 25 Germany ( Official German Charts ) 77 Ireland ( IRMA ) New Zealand ( Recorded Music NZ ) Norway ( VG - lista ) 5 Sweden ( Sverigetopplistan ) 34 UK Singles ( Official Charts Company ) U.S. Billboard Hot 100 17 U.S. Billboard Hot 100 Airplay U.S. Billboard Hot Adult Contemporary Tracks U.S. Billboard Hot Adult Top 40 Tracks 7 U.S. Billboard Hot Modern Rock Tracks 23 U.S. Billboard Top 40 Mainstream Compilation usage ( edit ) `` I 'll Be There for You '' has been included on the following compilation albums : Friends : Music From The TV Series ( 1994 ) -- for the American series Friends ' opening song ; Serial TV : Les Plus Grands Hits Des Séries TV Happy Songs Billboard Top Hits : 1995 -- for Billboard 's 1995 CD . Cover versions ( edit ) American pop rock band The Goo Goo Dolls recorded their own version of this song which contained slightly altered lyrics , and a more upbeat tempo , rockier sound , as well as a glockenspiel in the rhythm track . Yakko , Wakko , and Dot of Animaniacs parodied the theme song as `` We Wo n't Ever Leave '' during a 1997 episode entitled `` Acquaintances '' ( which is itself a spoof of Friends ) . American punk rock band Pink Lincolns covered this song under the title `` Friends '' for a 1997 compilation album titled Show & Tell . Fictional American band Alvin and the Chipmunks covered this song for their 2007 video game Alvin and the Chipmunks . `` Weird Al '' Yankovic had planned to record a parody for his Bad Hair Day album called `` Theme for Home Improvement or I 'll Repair For You '' . Although the Rembrandts were fine with the parody idea , the producers of Friends were not , fearing that the show 's theme would be overexposed . `` Theme for Home Improvement '' is nevertheless a concert favorite . In popular culture ( edit ) The 2013 comedy film We 're the Millers , which stars Jennifer Aniston , features a blooper during the credits where Jason Sudeikis flips through radio stations and ends up on `` I 'll Be There for You '' . Sudeikis , Emma Roberts , and Will Poulter begin singing the song as Aniston covers her face laughing . See also ( edit ) List of number - one adult contemporary singles of 1995 ( U.S. ) References ( edit ) Jump up ^ `` The Rembrandts : ' So no one told me it was going to be this way ' '' . Independent . London . April 27 , 2004 . Retrieved 2010 - 08 - 20 . Jump up ^ Etkin , Jaimie . `` The Amazing Story Behind `` I 'll Be There For You , '' According To The Rembrandts `` . Buzzfeed News . Retrieved 16 February 2016 . Jump up ^ `` Nashville DJ helped turn `` Friends '' song into radio hit `` . Web.archive.org . Archived from the original on 2004 - 12 - 21 . Retrieved 2015 - 09 - 14 . Jump up ^ Whitburn , Joel ( 2002 ) . Top Adult Contemporary : 1961 -- 2001 . Record Research . p. 202 . Jump up ^ `` Item Display -- RPM -- Library and Archives Canada '' . Collectionscanada.gc.ca . Retrieved 2015 - 09 - 14 . Jump up ^ ( 1 ) Archived December 31 , 2009 , at the Wayback Machine . Jump up ^ `` Australian-charts.com -- The Rembrandts -- I 'll Be There For You '' . ARIA Top 50 Singles . Retrieved 4 April 2011 . Jump up ^ `` Ultratop.be -- The Rembrandts -- I 'll Be There For You '' ( in Dutch ) . Ultratop 50 . Retrieved 4 April 2011 . Jump up ^ `` Item Display -- RPM -- Library and Archives Canada '' . Collectionscanada.gc.ca . Retrieved 2015 - 09 - 14 . Jump up ^ `` Dutchcharts.nl -- The Rembrandts -- I 'll Be There For You '' ( in Dutch ) . Single Top 100 . Retrieved 4 April 2011 . Jump up ^ `` Lescharts.com -- The Rembrandts -- I 'll Be There For You '' ( in French ) . Les classement single . Retrieved 4 April 2011 . Jump up ^ `` Musicline.de -- The Rembrandts Single - Chartverfolgung '' ( in German ) . Media Control Charts . PhonoNet GmbH . Retrieved 4 April 2011 . Jump up ^ Jaclyn Ward . `` The Irish Charts -- All there is to know '' . Irishcharts.ie . Archived from the original on 2009 - 06 - 03 . Retrieved 2015 - 09 - 14 . Jump up ^ `` Charts.org.nz -- The Rembrandts -- I 'll Be There For You '' . Top 40 Singles . Retrieved 4 April 2011 . Jump up ^ `` Norwegiancharts.com -- The Rembrandts -- I 'll Be There For You '' . VG - lista . Retrieved 4 April 2011 . Jump up ^ `` Swedishcharts.com -- The Rembrandts -- I 'll Be There For You '' . Singles Top 100 . Retrieved 4 April 2011 . Jump up ^ `` Archive Chart : 1995 '' UK Singles Chart . Retrieved 4 April 2011 . ^ Jump up to : The Rembrandts . `` The Rembrandts Biography , Albums , & Streaming Radio '' . AllMusic . Retrieved 2015 - 09 - 14 . External links ( edit ) Lyrics of this song at MetroLyrics The Rembrandts Phil Solem Danny Wilde Mark Karan Studio albums The Rembrandts ( 1990 ) Untitled ( 1992 ) L.P. ( 1995 ) Spin This ( 1998 ) Lost Together ( 2001 ) Compilation albums Choice Picks ( 2005 ) Greatest Hits ( 2006 ) Singles `` Just the Way It Is , Baby '' `` I 'll Be There for You '' Friends Characters Chandler Bing Phoebe Buffay Monica Geller Ross Geller Rachel Green Joey Tribbiani Episodes Season 1 `` The Pilot '' ( 1.01 ) `` The One with the Sonogram at the End '' ( 1.02 ) `` The One with George Stephanopoulos '' ( 1.04 ) Season 2 `` The One with the Lesbian Wedding '' ( 2.11 ) `` The One After the Superbowl '' ( 2.12 -- 2.13 ) `` The One with the Prom Video '' ( 2.14 ) `` The One Where Dr. Ramoray Dies '' ( 2.18 ) Season 3 `` The One Where No One 's Ready '' ( 3.02 ) Season 4 `` The One with the Embryos '' ( 4.12 ) `` The One with Ross 's Wedding '' ( 4.23 -- 4.24 ) Season 5 `` The One After Ross Says Rachel '' ( 5.01 ) `` The One Hundredth '' ( 5.03 ) `` The One with All the Thanksgivings '' ( 5.08 ) `` The One with the Girl Who Hits Joey '' ( 5.15 ) `` The One Where Rachel Smokes '' ( 5.18 ) `` The One Where Ross Ca n't Flirt '' ( 5.19 ) `` The One with the Ride - Along '' ( 5.20 ) Season 6 `` The One After Vegas '' ( 6.01 ) `` The One with the Apothecary Table '' ( 6.11 ) `` The One with the Proposal '' ( 6.24 -- 6.25 ) Season 7 `` The One with Monica and Chandler 's Wedding '' ( 7.23 -- 7.24 ) Season 8 `` The One with Rachel 's Date '' ( 8.05 ) `` The One with the Rumor '' ( 8.09 ) Season 9 -- Season 10 `` The Last One '' ( 10.17 -- 10.18 ) Related articles Music `` I 'll Be There for You '' Joey Mad About You Awards and nominations Friends ... ' Til the End Retrieved from `` https://en.wikipedia.org/w/index.php?title=I%27ll_Be_There_for_You_(The_Rembrandts_song)&oldid=801058880 '' Categories : 1995 singles The Rembrandts songs Billboard Adult Contemporary number - one singles Billboard Mainstream Top 40 ( Pop Songs ) number - one singles RPM Top Singles number - one singles Songs written by Allee Willis Friends Television theme songs Hidden categories : Webarchive template wayback links Articles that may contain original research from September 2015 All articles that may contain original research Articles with hAudio microformats All articles with unsourced statements Articles with unsourced statements from August 2017 Singlechart usages for Australia Singlechart usages for Belgium ( Flanders ) Singlechart usages for Dutch100 Singlechart usages for France Singlechart usages for German Singlechart usages for New Zealand Singlechart usages for Norway Singlechart usages for Sweden Singlechart usages for UK Talk Contents About Wikipedia Čeština Ελληνικά Español Euskara Français Galego Italiano Nederlands 日本 語 Português Română Русский Suomi Türkçe 中文 Edit links This page was last edited on 17 September 2017 , at 11 : 55 . 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