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In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by |
where is the invariant mass. If (in Minkowski space), then |
Using Einstein's mass-energy equivalence, , this can be rewritten as |
Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy. |
The magnitude of the momentum four-vector is equal to : |
and is invariant across all reference frames. |
The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting it follows that |
In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles. |
The four-momentum of a planar wave can be related to a wave four-vector |
For a particle, the relationship between temporal components, , is the Planck–Einstein relation, and the relation between spatial components, , describes a de Broglie matter wave. |
In Lagrangian mechanics, a Lagrangian is defined as the difference between the kinetic energy and the potential energy : |
If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or Euler–Lagrange equations) are a set of equations: |
If a coordinate is not a Cartesian coordinate, the associated generalized momentum component does not necessarily have the dimensions of linear momentum. Even if is a Cartesian coordinate, will not be the same as the mechanical momentum if the potential depends on velocity. Some sources represent the kinematic momentum by the symbol . |
In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as |
Each component is said to be the "conjugate momentum" for the coordinate . |
Now if a given coordinate does not appear in the Lagrangian (although its time derivative might appear), then |
This is the generalization of the conservation of momentum. |
Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism. |
In Hamiltonian mechanics, the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as |
where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are |
As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved. |
Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem. For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include curved spacetimes in general relativity or time crystals in condensed matter physics. |
In Maxwell's equations, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force ("Lorentz force") on a particle with charge due to a combination of electric field and magnetic field is |
It has an electric potential and magnetic vector potential . |
In the non-relativistic regime, its generalized momentum is |
The quantity formula_60 is sometimes called the "potential momentum". It is the momentum due to the interaction of the particle with the electromagnetic fields. The name is an analogy with the potential energy formula_61, which is the energy due to the interaction of the particle with the electromagnetic fields. These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called hidden-momentum of the electromagnetic fields |
In Newtonian mechanics, the law of conservation of momentum can be derived from the law of action and reaction, which states that every force has a reciprocating equal and opposite force. Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions. Nevertheless, the combined momentum of the particles and the electromagnetic field is conserved. |
The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields. |
In a vacuum, the momentum per unit volume is |
where is the vacuum permeability and is the speed of light. The momentum density is proportional to the Poynting vector which gives the directional rate of energy transfer per unit area: |
If momentum is to be conserved over the volume over a region , changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If is the momentum of all the particles in , and the particles are treated as a continuum, then Newton's second law gives |
and the equation for conservation of each component of the momentum is |
The term on the right is an integral over the surface area of the surface representing momentum flow into and out of the volume, and is a component of the surface normal of . The quantity is called the Maxwell stress tensor, defined as |
The above results are for the "microscopic" Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density is modified to |
where the H-field is related to the B-field and the magnetization by |
The electromagnetic stress tensor depends on the properties of the media. |
In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables. |
For a single particle described in the position basis the momentum operator can be written as |
where is the gradient operator, is the reduced Planck constant, and is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in momentum space the momentum operator is represented as |
where the operator acting on a wave function yields that wave function multiplied by the value , in an analogous fashion to the way that the position operator acting on a wave function yields that wave function multiplied by the value "x". |
For both massive and massless objects, relativistic momentum is related to the phase constant formula_72 by |
Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy). |
In fields such as fluid dynamics and solid mechanics, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a continuum in which there is a particle or fluid parcel at each point that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density and velocity that depend on time and position . The momentum per unit volume is . |
Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is , where is the gravitational acceleration. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure . The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is |
If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative because the fluid in a given volume changes with time. Instead, the material derivative is needed: |
Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to advection as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to . This is equal to the net force on the droplet. |
Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a shear stress , exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or strain rate. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the direction varies with , the tangential force in direction per unit area normal to the direction is |
where is the viscosity. This is also a flux, or flow per unit area, of x-momentum through the surface. |
Including the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are |
These are known as the Navier–Stokes equations. |
The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction and force in direction , there is a stress component . The nine components make up the Cauchy stress tensor , which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation: |
The Cauchy momentum equation is broadly applicable to deformations of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see Types of viscosity). |
A disturbance in a medium gives rise to oscillations, or waves, that propagate away from their source. In a fluid, small changes in pressure can often be described by the acoustic wave equation: |
where is the speed of sound. In a solid, similar equations can be obtained for propagation of pressure (P-waves) and shear (S-waves). |
The flux, or transport per unit area, of a momentum component by a velocity is equal to . In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero. However, nonlinear effects can give rise to a nonzero average. It is possible for momentum flux to occur even though the wave itself does not have a mean momentum. |
The work of Philoponus, and possibly that of Ibn Sīnā, was read and refined by the European philosophers Peter Olivi and Jean Buridan. Buridan, who in about 1350 was made rector of the University of Paris, referred to impetus being proportional to the weight times the speed. Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus. |
René Descartes believed that the total "quantity of motion" () in the universe is conserved, where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more important, he believed that it is speed rather than velocity that is conserved. So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion. Galileo, in his "Two New Sciences", used the Italian word "impeto" to similarly describe Descartes' quantity of motion. |
Leibniz, in his "Discourse on Metaphysics", gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances. He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved. |
Christiaan Huygens concluded quite early that Descartes's laws for the elastic collision of two bodies must be wrong, and he formulated the correct laws. An important step was his recognition of the Galilean invariance of the problems. His views then took many years to be circulated. He passed them on in person to William Brouncker and Christopher Wren in London, in 1661. What Spinoza wrote to Henry Oldenburg about them, in 1666 which was during the Second Anglo-Dutch War, was guarded. Huygens had actually worked them out in a manuscript "De motu corporum ex percussione" in the period 1652–6. The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He published them in the "Journal des sçavans" in 1669. |
In physics, zilch is a conserved quantity of the electromagnetic field. |
Daniel M. Lipkin observed that if he defined the quantities |
which implies that the total "zilch" formula_3 is constant (formula_4 is the "zilch current"). Generalising the result, Lipkin found nine related conservation laws, all unrelated to the stress–energy tensor. He named the quantity zilch because of the apparent lack of physical significance. |
Zilch can also be expressed using the dual electromagnetic tensor formula_5 as |
It was later demonstrated that zilch is part of an infinite number of zilch-like conserved quantities, a general property of free fields. |
Zilch has occasionally been rediscovered. It has been called "optical chirality", since it determines the degree of chiral asymmetry in the rate of excitation of a small chiral molecule by an incident electromagnetic field. A physical interpretation of zilch was offered in 2012; zilch is to the curl or time derivative of the electromagnetic field what helicity, spin and related quantities are to the electromagnetic field itself. The conservation of zilch is not associated with duality transformations, but instead with a more subtle symmetry transformation, which has no special name. |
Magnetic braking is a theory explaining the loss of stellar angular momentum due to material getting captured by the stellar magnetic field and thrown out at great distance from the surface of the star. It plays an important role in the evolution of binary star systems. |
The currently accepted theory of the solar system's evolution states that the Solar System originates from a contracting gas cloud. As the cloud contracts, the angular momentum formula_1 must be conserved. Any small net rotation of the cloud will cause the spin to increase as the cloud collapses, forcing the material into a rotating disk. At the dense center of this disk a protostar forms, which gains heat from the gravitational energy of the collapse. As the collapse continues, the rotation rate can increase to the point where the accreting protostar can break up due to centrifugal force at the equator. |
Thus the rotation rate must be braked during the first 100,000 years of the star's life to avoid this scenario. One possible explanation for the braking is the interaction of the protostar's magnetic field with the stellar wind. In the case of our own Sun, when the planets' angular momenta are compared to the Sun's own, the Sun has less than 1% of its supposed angular momentum. In other words, the Sun has slowed down its spin while the planets have not. |
As ionized material follows the Sun's magnetic field lines, due to the effect of the field lines being frozen in the plasma, the charged particles feel a force formula_2 of the magnitude: |
where formula_4 is the charge, formula_5 is the velocity and formula_6 is the magnetic field vector. This bending action forces the particles to "corkscrew" around the magnetic field lines while held in place by a "magnetic pressure"formula_7, or "energy density", while rotating together with the Sun as a solid body: |
Since magnetic field strength decreases with the cube of the distance there will be a place where the kinetic gas pressure formula_9 of the ionized gas is great enough to break away from the field lines: |
where n is the number of particles, m is the mass of the individual particle and v is the radial velocity away from the Sun, or the speed of the solar wind. |
Due to the high conductivity of the stellar wind, the magnetic field outside the sun declines with radius like the mass density of the wind, i.e. decline as an inverse square law. The magnetic field is therefore given by |
where formula_12 is the magnetic field on the surface of the sun and formula_13 is its radius. The critical distance where the material will break away from the field lines can then be calculated as the distance where the kinetic pressure and the magnetic pressure are equal, i.e. |
If the solar mass loss is omni-directional then the mass loss formula_17; plugging this into the above equation and isolating the critical radius it follows that |
This leads to a critical radius formula_23. This means that the ionized plasma will rotate together with the Sun as a solid body until it reaches a distance of nearly 15 times the radius of the Sun; from there the material will break off and stop affecting the Sun. |
The amount of solar mass needed to be thrown out along the field lines to make the Sun completely stop rotating can then be calculated using the specific angular momentum: |
It has been suggested that the sun lost a comparable amount of material over the course of its lifetime. |
In 2016 scientists at Carnegie Observatories published a research suggesting that stars at a similar stage of life as the Sun were spinning faster than magnetic braking theories predicted. To calculate this they pinpointed the dark spots on the surface of stars and tracked them as they moved with the stars' spin. While this method has been successful for measuring the spin of younger stars, the "weakened" magnetic braking in older stars proved harder to confirm, as the latter notoriously have fewer star spots. In a study published in Nature Astronomy in 2021, researchers at the University of Birmingham used a different approach, namely asteroseismology, to confirm that older stars do appear to rotate faster than expected. |
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. |
is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle "θ" |
the function, after some cancellation of terms, takes exactly the same form |
The rotation of coordinates can be expressed using matrix form using the rotation matrix, |
or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is |
In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. For a real-valued function of three or more real variables, this expression extends easily using appropriate rotation matrices. |
The concept also extends to a vector-valued function f of one or more variables; |
In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself. |
which maps elements from a subset "X" of the real line ℝ to itself, rotational invariance may also mean that the function commutes with rotations of elements in "X". This also applies for an operator that acts on such functions. An example is the two-dimensional Laplace operator |
which acts on a function "f" to obtain another function ∇2"f". This operator is invariant under rotations. |
If "g" is the function "g"("p") = "f"("R"("p")), where "R" is any rotation, then (∇2"g")("p") = (∇2"f" )("R"("p")); that is, rotating a function merely rotates its Laplacian. |
In physics, if a system behaves the same regardless of how it is oriented in space, then its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved. |
In quantum mechanics, rotational invariance is the property that after a rotation the new system still obeys Schrödinger's equation. That is |
for any rotation "R". Since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have ["R", "H"] = 0. |
For infinitesimal rotations (in the "xy"-plane for this example; it may be done likewise for any plane) by an angle "dθ" the (infinitesimal) rotation operator is |
in other words angular momentum is conserved. |
In astrodynamics, the "vis-viva" equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight. |
"Vis viva" (Latin for "living force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the total work of the accelerating forces of a system and that of the retarding forces is equal to one half the "vis viva" accumulated or lost in the system while the work is being done. |
For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the "vis-viva" equation is as follows: |
The product of GM can also be expressed as the standard gravitational parameter using the Greek letter μ. |
Derivation for elliptic orbits (0 ≤ eccentricity < 1). |
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