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In the vis-viva equation the mass "m" of the orbiting body (e.g., a spacecraft) is taken to be negligible in comparison to the mass "M" of the central body (e.g., the Earth). The central body and orbiting body are also often referred to as the primary and a particle respectively. In the specific cases of an elliptical or circular orbit, the vis-viva equation may be readily derived from conservation of energy and momentum. |
Specific total energy is constant throughout the orbit. Thus, using the subscripts "a" and "p" to denote apoapsis (apogee) and periapsis (perigee), respectively, |
Recalling that for an elliptical orbit (and hence also a circular orbit) the velocity and radius vectors are perpendicular at apoapsis and periapsis, conservation of angular momentum requires specific angular momentum formula_4, thus formula_5: |
Isolating the kinetic energy at apoapsis and simplifying, |
From the geometry of an ellipse, formula_11 where "a" is the length of the semimajor axis. Thus, |
Substituting this into our original expression for specific orbital energy, |
Thus, formula_14 and the vis-viva equation may be written |
Therefore, the conserved angular momentum L = mh can be derived using formula_17 and formula_18, |
where a is semi-major axis and b is semi-minor axis of the elliptical orbit, as follows - |
Given the total mass and the scalars "r" and "v" at a single point of the orbit, one can compute "r" and "v" at any other point in the orbit. |
Given the total mass and the scalars "r" and "v" at a single point of the orbit, one can compute the specific orbital energy formula_23, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being "suborbital" (a ballistic missile, for example), having enough energy to be "orbital", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and/or go to infinity (as a meteor, for example). |
The formula for escape velocity can be obtained from the Vis-viva equation by taking the limit as formula_24 approaches formula_25: |
In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of "one" spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): |
It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. |
A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is "not" a parity transformation; it is the same as a 180°-rotation. |
In quantum mechanics, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are odd functions. |
Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group. |
Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. The word "projective" refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states. |
The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO(3), are ordinary representations of the special unitary group SU(2) (see Representation theory of SU(2)). Projective representations of the rotation group that are not representations are called spinors and so quantum states may transform not only as tensors but also as spinors. |
If one adds to this a classification by parity, these can be extended, for example, into notions of |
which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing "x"-, "y"-, and "z"-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used. |
Parity forms the abelian group formula_3 due to the relation formula_4. All Abelian groups have only one-dimensional irreducible representations. For formula_3, there are two irreducible representations: one is even under parity, formula_6, the other is odd, formula_7. These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase. |
Newton's equation of motion formula_8 (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity. |
However, angular momentum formula_9 is an axial vector, |
In classical electrodynamics, the charge density formula_11 is a scalar, the electric field, formula_12, and current formula_13 are vectors, but the magnetic field, formula_14 is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector. |
Effect of spatial inversion on some variables of classical physics. |
Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include: |
Classical variables, predominantly vector quantities, which have their sign flipped by spatial inversion include: |
In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, formula_40, is a unitary operator, in general acting on a state formula_41 as follows: formula_42. |
For electronic wavefunctions, even states are usually indicated by a subscript g for "gerade" (German: even) and odd states by a subscript u for "ungerade" (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled formula_56 and the next-closest (higher) energy level is labelled formula_57. |
The wave functions of a particle moving into an external potential, which is centrosymmetric (potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions. |
The law of conservation of parity of particle (not true for the beta decay of nuclei) states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. |
The parity of the states of a particle moving in a spherically symmetric external field is determined by the angular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum. |
When parity generates the Abelian group ℤ2, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number. |
In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if formula_58 commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any scalar potential, i.e., formula_59, hence the potential is spherically symmetric. The following facts can be easily proven: |
Some of the non-degenerate eigenfunctions of formula_70 are unaffected (invariant) by parity formula_58 and the others are merely reversed in sign when the Hamiltonian operator and the parity operator commute: |
where formula_77 is a constant, the eigenvalue of formula_58, |
The overall parity of a many-particle system is the product of the parities of the one-particle states. It is −1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules. |
Atomic orbitals have parity (−1)ℓ, where the exponent ℓ is the azimuthal quantum number. The parity is odd for orbitals p, f, … with ℓ = 1, 3, …, and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol 4So, where the superscript o denotes odd parity. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is 4P (without an o superscript). |
The complete (rotational-vibrational-electronic-nuclear spin) electromagnetic Hamiltonian of any molecule commutes with (or is invariant to) the parity operation P (or E*, in the notation introduced by Longuet-Higgins) and its eigenvalues can be given the parity symmetry label + or - as they are even or odd, respectively. The parity operation involves the inversion of electronic and nuclear spatial coordinates at the molecular center of mass. |
does not commute with the point group inversion operation i because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of g and u vibronic states (called ortho-para mixing) and give rise to ortho-para transitions |
If we can show that the vacuum state is invariant under parity, formula_80, the Hamiltonian is parity invariant formula_81 and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction. |
To show that quantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that canonical quantization is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator: |
where p denotes the momentum of a photon and ± refers to its polarization state. This is equivalent to the statement that the photon has odd intrinsic parity. Similarly all vector bosons can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity. |
A straightforward extension of these arguments to scalar field theories shows that scalars have even parity, since |
This is true even for a complex scalar field. ("Details of spinors are dealt with in the article on the "Dirac equation", where it is shown that fermions and antifermions have opposite intrinsic parity.") |
With fermions, there is a slight complication because there is more than one spin group. |
In the Standard Model of fundamental interactions there are precisely three global internal U(1) symmetry groups available, with charges equal to the baryon number "B", the lepton number "L" and the electric charge "Q". The product of the parity operator with any combination of these rotations is another parity operator. It is conventional to choose one specific combination of these rotations to define a standard parity operator, and other parity operators are related to the standard one by internal rotations. One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges "B", "L" and "Q". In general, one assigns the parity of the most common massive particles, the proton, the neutron and the electron, to be +1. |
Steven Weinberg has shown that if , where "F" is the fermion number operator, then, since the fermion number is the sum of the lepton number plus the baryon number, , for all particles in the Standard Model and since lepton number and baryon number are charges "Q" of continuous symmetries "e""iQ", it is possible to redefine the parity operator so that . However, if there exist Majorana neutrinos, which experimentalists today believe is possible, their fermion number is equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana, and so (−1)"F" would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity ±"i". |
In 1954, a paper by William Chinowsky and Jack Steinberger demonstrated that the pion has negative parity. They studied the decay of an "atom" made from a deuteron () and a negatively charged pion () in a state with zero orbital angular momentum formula_82 into two neutrons (formula_83). |
Although parity is conserved in electromagnetism, strong interactions and gravity, it is violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in charged weak interactions in the Standard Model. This implies that parity is not a symmetry of our universe, unless a hidden mirror sector exists in which parity is violated in the opposite way. |
By the mid-20th century, it had been suggested by several scientists that parity might not be conserved (in different contexts), but without solid evidence these suggestions were not considered important. Then, in 1956, a careful review and analysis by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang went further, showing that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. They were mostly ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards. |
After the fact, it was noted that an obscure 1928 experiment, done by R. T. Cox, G. C. McIlwraith, and B. Kurrelmeyer, had in effect reported parity violation in weak decays, but since the appropriate concepts had not yet been developed, those results had no impact. The discovery of parity violation immediately explained the outstanding τ–θ puzzle in the physics of kaons. |
In 2010, it was reported that physicists working with the Relativistic Heavy Ion Collider (RHIC) had created a short-lived parity symmetry-breaking bubble in quark-gluon plasmas. An experiment conducted by several physicists including Yale's Jack Sandweiss as part of the STAR collaboration, suggested that parity may also be violated in the strong interaction. It is predicted that this local parity violation, which would be analogous to the effect that is induced by fluctuation of the axion field, manifest itself by chiral magnetic effect. |
To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as rho meson decay to pions. |
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. |
In three dimensions, the angular momentum for a point particle is a pseudovector , the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is in Newtonian mechanics. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. |
Angular momentum is an extensive quantity; i.e. the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density (i.e. angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body. |
In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the notion of a quantum particle literally "spinning" about an axis does not exist. Quantum particles "do" possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to actual physical spinning motion. |
Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to mass and linear speed |
angular momentum is proportional to moment of inertia and angular speed measured in radians per second. |
Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, should be referred to as the angular momentum "relative to that center". |
Because formula_3 for a single particle and formula_4 for circular motion, angular momentum can be expanded, formula_5 and reduced to, |
the product of the radius of rotation and the linear momentum of the particle formula_7, where formula_8 in this case is the equivalent linear (tangential) speed at the radius (formula_9). |
This simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, |
where formula_11 is the perpendicular component of the motion. Expanding, formula_12 rearranging, formula_13 and reducing, angular momentum can also be expressed, |
where formula_15 is the length of the "moment arm", a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, to which the term "moment of momentum" refers. |
Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate formula_16 expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass formula_17 constrained to move in a circle of radius formula_18 in the absence of any external force field. The kinetic energy of the system is |
The "generalized momentum" "canonically conjugate to" the coordinate formula_16 is defined by |
To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as: |
This can be expanded, reduced, and by the rules of vector algebra, rearranged: |
which is the cross product of the position vector formula_27 and the linear momentum formula_33 of the particle. By the definition of the cross product, the formula_34 vector is perpendicular to both formula_27 and formula_36. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the formula_34 vector defines the plane in which formula_27 and formula_36 lie. |
By defining a unit vector formula_40 perpendicular to the plane of angular displacement, a scalar angular speed formula_41 results, where |
The two-dimensional scalar equations of the previous section can thus be given direction: |
and formula_46 for circular motion, where all of the motion is perpendicular to the radius formula_47. |
In the spherical coordinate system the angular momentum vector expresses as |
Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape. |
Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product, |
is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the "moment arm". It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a "moment". Hence, the particle's momentum referred to a particular point, |
is the "angular momentum", sometimes called, as here, the "moment of momentum" of the particle versus that particular center point. The equation formula_51 combines a moment (a mass formula_17 turning moment arm formula_47) with a linear (straight-line equivalent) speed formula_8. Linear speed referred to the central point is simply the product of the distance formula_47 and the angular speed formula_41 versus the point: formula_57 another moment. Hence, angular momentum contains a double moment: formula_58 Simplifying slightly, formula_59 the quantity formula_60 is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia. |
Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits. |
For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass. |
For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random. |
In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by, |
Similarly, for a point mass formula_17 the moment of inertia is defined as, |
and for any collection of particles formula_67 as the sum, |
The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the "invariable plane", because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System. |
Newton's second law of motion can be expressed mathematically, |
or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows: |
which means that the torque (i.e. the time derivative of the angular momentum) is |
Because the moment of inertia is formula_72, it follows that formula_73, and formula_74 which, reduces to |
This is the rotational analog of Newton's Second Law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass. |
A rotational analog of Newton's third law of motion might be written, "In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque." Hence, "angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved)." |
Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted by an external influence." Thus "with no external influence to act upon it, the original angular momentum of the system remains constant". |
For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year. |
The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of the mass of her body closer to the axis, she decreases her body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase. |
The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars. Decrease in the size of an object "n" times results in increase of its angular velocity by the factor of "n"2. |
Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved. |
Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved. |
Relation to Newton's second law of motion. |
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