Laplace–Runge–Lenz vector
In
The
. However, this approach is rarely used today.In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system.[9] The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere,[10] so that the whole problem is symmetric under certain rotations of the four-dimensional space.[11] This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.[12]
The Laplace–Runge–Lenz vector is named after
Context
A single particle moving under any
The LRL vector A is constant in length and direction, but only for an inverse-square central force.[1] For other central forces, the vector A is not constant, but changes in both length and direction. If the central force is approximately an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction.[14] A generalized conserved LRL vector
The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding
History of rediscovery
The LRL vector A is a constant of motion of the Kepler problem, and is useful in describing astronomical orbits, such as the motion of planets and binary stars. Nevertheless, it has never been well known among physicists, possibly because it is less intuitive than momentum and angular momentum. Consequently, it has been rediscovered independently several times over the last three centuries.[15]
Jakob Hermann was the first to show that A is conserved for a special case of the inverse-square central force,[22] and worked out its connection to the eccentricity of the orbital ellipse. Hermann's work was generalized to its modern form by Johann Bernoulli in 1710.[23] At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of A, deriving it analytically, rather than geometrically.[24] In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below,[16] using it to show that the momentum vector p moves on a circle for motion under an inverse-square central force (Figure 3).[12]
At the beginning of the twentieth century,
Mathematical definition
An inverse-square central force acting on a single particle is described by the equation
The LRL vector A is defined mathematically by the formula[1]
where
- m is the mass of the point particle moving under the central force,
- p is its momentum vector,
- L = r × p is its angular momentum vector,
- r is the position vector of the particle (Figure 1),
- is the corresponding unit vector, i.e., , and
- r is the magnitude of r, the distance of the mass from the center of force.
The SI units of the LRL vector are joule-kilogram-meter (J⋅kg⋅m). This follows because the units of p and L are kg⋅m/s and J⋅s, respectively. This agrees with the units of m (kg) and of k (N⋅m2).
This definition of the LRL vector A pertains to a single point particle of mass m moving under the action of a fixed force. However, the same definition may be extended to two-body problems such as the Kepler problem, by taking m as the reduced mass of the two bodies and r as the vector between the two bodies.
Since the assumed force is conservative, the total energy E is a constant of motion,
The assumed force is also a central force. Hence, the angular momentum vector L is also conserved and defines the plane in which the particle travels. The LRL vector A is perpendicular to the angular momentum vector L because both p × L and r are perpendicular to L. It follows that A lies in the plane of motion.
Alternative formulations for the same constant of motion may be defined, typically by scaling the vector with constants, such as the mass m, the force parameter k or the angular momentum L.[15] The most common variant is to divide A by mk, which yields the eccentricity vector,[2][16] a dimensionless vector along the semi-major axis whose modulus equals the eccentricity of the conic:
Derivation of the Kepler orbits
The shape and orientation of the orbits can be determined from the LRL vector as follows.[1] Taking the dot product of A with the position vector r gives the equation
Rearranging yields the solution for the Kepler equation
This corresponds to the formula for a conic section of
Taking the dot product of A with itself yields an equation involving the total energy E,[1]
Thus, if the energy E is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"[1]), the eccentricity is greater than one and the orbit is a hyperbola.[1] Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola.[1] In all cases, the direction of A lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.[1]
Circular momentum hodographs
The conservation of the LRL vector A and angular momentum vector L is useful in showing that the momentum vector p moves on a circle under an inverse-square central force.[12][15]
Taking the dot product of
Further choosing L along the z-axis, and the major semiaxis as the x-axis, yields the locus equation for p,
In other words, the momentum vector p is confined to a circle of radius mk/L = L/ℓ centered on (0, A/L).[29] For bounded orbits, the eccentricity e corresponds to the cosine of the angle η shown in Figure 3. For unbounded orbits, we have and so the circle does not intersect the -axis.
In the degenerate limit of circular orbits, and thus vanishing A, the circle centers at the origin (0,0). For brevity, it is also useful to introduce the variable .
This circular hodograph is useful in illustrating the symmetry of the Kepler problem.
Constants of motion and superintegrability
The seven scalar quantities E, A and L (being vectors, the latter two contribute three conserved quantities each) are related by two equations, A ⋅ L = 0 and A2 = m2k2 + 2 mEL2, giving five independent
This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since the initial time is not determined by a constant of motion. The resulting 1-dimensional orbit in 6-dimensional phase space is thus completely specified.
A mechanical system with d degrees of freedom can have at most 2d − 1 constants of motion, since there are 2d initial conditions and the initial time cannot be determined by a constant of motion. A system with more than d constants of motion is called superintegrable and a system with 2d − 1 constants is called .
Maximally superintegrable systems follow closed, one-dimensional orbits in phase space, since the orbit is the intersection of the phase-space isosurfaces of their constants of motion. Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross products of all of these gradients. As a result, all superintegrable systems are automatically describable by Nambu mechanics,[32] alternatively, and equivalently, to Hamiltonian mechanics.
Maximally superintegrable systems can be
Evolution under perturbed potentials
The Laplace–Runge–Lenz vector A is conserved only for a perfect inverse-square central force. In most practical problems such as planetary motion, however, the interaction potential energy between two bodies is not exactly an inverse square law, but may include an additional central force, a so-called perturbation described by a potential energy h(r). In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow apsidal precession of the orbit.
By assumption, the perturbing potential h(r) is a conservative central force, which implies that the total energy E and angular momentum vector L are conserved. Thus, the motion still lies in a plane perpendicular to L and the magnitude A is conserved, from the equation A2 = m2k2 + 2mEL2. The perturbation potential h(r) may be any sort of function, but should be significantly weaker than the main inverse-square force between the two bodies.
The rate at which the LRL vector rotates provides information about the perturbing potential h(r). Using canonical perturbation theory and action-angle coordinates, it is straightforward to show[1] that A rotates at a rate of,
This approach was used to help verify Einstein's theory of general relativity, which adds a small effective inverse-cubic perturbation to the normal Newtonian gravitational potential,[35]
Inserting this function into the integral and using the equation
Poisson brackets
The unscaled functions
The algebraic structure of the problem is, as explained in later sections, SO(4)/Z2 ~ SO(3) × SO(3).[11] The three components Li of the angular momentum vector L have the Poisson brackets[1]
Finally, since both L and A are constants of motion, we have
The Poisson brackets will be extended to quantum mechanical commutation relations in the next section and to Lie brackets in a following section.
The scaled functions
As noted below, a scaled Laplace–Runge–Lenz vector D may be defined with the same units as angular momentum by dividing A by . Since D still transforms like a vector, the Poisson brackets of D with the angular momentum vector L can then be written in a similar form[11][8]
The Poisson brackets of D with itself depend on the
By contrast, for positive energy, the Poisson brackets have the opposite sign,
The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the square root of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where ) and negative values (where ).
Laplace-Runge-Lenz operator for the hydrogen atom in momentum space
Scaled Laplace-Runge-Lenz operator in the momentum space was found in 2022 .[44][45] The formula for the operator is simpler than in position space:
where the "degree operator"
multiplies a homogeneous polynomial by its degree.
Casimir invariants and the energy levels
The
and have vanishing Poisson brackets with all components of D and L,
However, the other invariant, C1, is non-trivial and depends only on m, k and E. Upon canonical quantization, this invariant allows the energy levels of hydrogen-like atoms to be derived using only quantum mechanical canonical commutation relations, instead of the conventional solution of the Schrödinger equation.[8][43] This derivation is discussed in detail in the next section.
Quantum mechanics of the hydrogen atom
Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical operators is specified by the Poisson bracket of the corresponding classical variables, multiplied by iħ.[46]
By carrying out this quantization and calculating the eigenvalues of the C1
A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of p and L must be defined carefully.[8] Typically, the operators for the Cartesian components As are defined using a symmetrized (Hermitian) product,
From these operators, additional
A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined,
Applying these ladder operators to the
Hence, the energy levels are given by
Conservation and symmetry
The conservation of the LRL vector corresponds to a subtle symmetry of the system. In
The symmetry for the inverse-square central force is higher and more subtle. The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers ℓ and m. The symmetry is more subtle, however, because the symmetry operation must take place in a higher-dimensional space; such symmetries are often called "hidden symmetries".[51]
Classically, the higher symmetry of the Kepler problem allows for continuous alterations of the orbits that preserve energy but not angular momentum; expressed another way, orbits of the same energy but different angular momentum (eccentricity) can be transformed continuously into one another. Quantum mechanically, this corresponds to mixing orbitals that differ in the ℓ and m quantum numbers, such as the s(ℓ = 0) and p(ℓ = 1) atomic orbitals. Such mixing cannot be done with ordinary three-dimensional translations or rotations, but is equivalent to a rotation in a higher dimension.
For negative energies – i.e., for bound systems – the higher symmetry group is SO(4), which preserves the length of four-dimensional vectors
In 1935,
For positive energies – i.e., for unbound, "scattered" systems – the higher symmetry group is
Both the negative- and positive-energy cases were considered by Fock[10] and Bargmann[11] and have been reviewed encyclopedically by Bander and Itzykson.[53][54]
The orbits of central-force systems – and those of the Kepler problem in particular – are also symmetric under
, SO(3), SO(4), and SO+(3,1), are needed to demonstrate the conservation of the angular momentum and LRL vectors; the reflection symmetry is irrelevant for conservation, which may be derived from the Lie algebra of the group.Rotational symmetry in four dimensions
The connection between the Kepler problem and four-dimensional rotational symmetry SO(4) can be readily visualized.[53][55][56] Let the four-dimensional Cartesian coordinates be denoted (w, x, y, z) where (x, y, z) represent the Cartesian coordinates of the normal position vector r. The three-dimensional momentum vector p is associated with a four-dimensional vector on a three-dimensional unit sphere
where is the unit vector along the new w axis. The transformation mapping p to η can be uniquely inverted; for example, the x component of the momentum equals
Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the z axis is aligned with the angular momentum vector L and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the y axis. Since the motion is planar, and p and L are perpendicular, pz = ηz = 0 and attention may be restricted to the three-dimensional vector . The family of Apollonian circles of momentum hodographs (Figure 7) correspond to a family of great circles on the three-dimensional sphere, all of which intersect the ηx axis at the two foci ηx = ±1, corresponding to the momentum hodograph foci at px = ±p0. These great circles are related by a simple rotation about the ηx-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; however, such a rotation is orthogonal to the usual three-dimensional rotations, since it transforms the fourth dimension ηw. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.
An elegant
Generalizations to other potentials and relativity
The Laplace–Runge–Lenz vector can also be generalized to identify conserved quantities that apply to other situations.
In the presence of a uniform electric field E, the generalized Laplace–Runge–Lenz vector is[17][58]
Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as[18]
where u = 1/r and ξ = cos θ, with the angle θ defined by
and γ is the Lorentz factor. As before, we may obtain a conserved binormal vector B by taking the cross product with the conserved angular momentum vector
These two vectors may likewise be combined into a conserved
In illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated.[18] Since the force is central,
The conserved dyadic tensor can be written in a simple form
The corresponding Runge–Lenz vector is more complicated,
Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problems
The following are arguments showing that the LRL vector is conserved under central forces that obey an inverse-square law.
Direct proof of conservation
A central force acting on the particle is
for some function of the radius . Since the angular momentum is conserved under central forces, and
where the momentum and where the triple cross product has been simplified using
The identity
yields the equation
For the special case of an inverse-square central force , this equals
Therefore, A is conserved for inverse-square central forces[59]
A shorter proof is obtained by using the relation of angular momentum to angular velocity, , which holds for a particle traveling in a plane perpendicular to . Specifying to inverse-square central forces, the time derivative of is
As described elsewhere in this article, this LRL vector A is a special case of a general conserved vector that can be defined for all central forces.[18][19] However, since most central forces do not produce closed orbits (see Bertrand's theorem), the analogous vector rarely has a simple definition and is generally a multivalued function of the angle θ between r and .
Hamilton–Jacobi equation in parabolic coordinates
The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates (ξ, η), which are defined by the equations
The inversion of these coordinates is
Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations[17][60]
Noether's theorem
The connection between the rotational symmetry described above and the conservation of the LRL vector can be made quantitative by way of
that causes the Lagrangian to vary to first order by a total time derivative
corresponds to a conserved quantity Γ
In particular, the conserved LRL vector component As corresponds to the variation in the coordinates[61]
where i equals 1, 2 and 3, with xi and pi being the i-th components of the position and momentum vectors r and p, respectively; as usual, δis represents the Kronecker delta. The resulting first-order change in the Lagrangian is
Substitution into the general formula for the conserved quantity Γ yields the conserved component As of the LRL vector,
Lie transformation
Noether's theorem derivation of the conservation of the LRL vector A is elegant, but has one drawback: the coordinate variation δxi involves not only the position r, but also the momentum p or, equivalently, the velocity v.[62] This drawback may be eliminated by instead deriving the conservation of A using an approach pioneered by Sophus Lie.[63][64] Specifically, one may define a Lie transformation[51] in which the coordinates r and the time t are scaled by different powers of a parameter λ (Figure 9),
This transformation changes the total angular momentum L and energy E,
The direction of A is preserved as well, since the semiaxes are not altered by a global scaling. This transformation also preserves
Alternative scalings, symbols and formulations
Unlike the momentum and angular momentum vectors p and L, there is no universally accepted definition of the Laplace–Runge–Lenz vector; several different scaling factors and symbols are used in the scientific literature. The most common definition is given above, but another common alternative is to divide by the quantity mk to obtain a dimensionless conserved eccentricity vector
where v is the velocity vector. This scaled vector e has the same direction as A and its magnitude equals the eccentricity of the orbit, and thus vanishes for circular orbits.
Other scaled versions are also possible, e.g., by dividing A by m alone
In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by −1. Other common symbols for the LRL vector include a, R, F, J and V. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.
An alternative conserved vector is the
which is conserved and points along the minor semiaxis of the ellipse. (It is not defined for vanishing eccentricity.)
The LRL vector A = B × L is the cross product of B and L (Figure 4). On the momentum hodograph in the relevant section above, B is readily seen to connect the origin of momenta with the center of the circular hodograph, and to possess magnitude A/L. At perihelion, it points in the direction of the momentum.
The vector B is denoted as "binormal" since it is perpendicular to both A and L. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.
The two conserved vectors, A and B can be combined to form a conserved dyadic tensor W,[18]
Being perpendicular to each another, the vectors A and B can be viewed as the
More directly, this equation reads, in explicit components,
See also
- Astrodynamics
- Bertrand's theorem
- Binet equation
- Two-body problem
References
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Further reading
- Baez, John (2008). "The Kepler Problem Revisited: The Laplace–Runge–Lenz Vector"(PDF). Retrieved 2021-05-31.
- Baez, John (2003). "Mysteries of the gravitational 2-body problem". Archived from the originalon 2008-10-21. Retrieved 2004-12-11.
- Baez, John (2018). "Mysteries of the gravitational 2-body problem". Retrieved 2021-05-31. Updated version of previous source.
- D'Eliseo, M. M. (2007). "The first-order orbital equation". American Journal of Physics. 75 (4): 352–355. .
- Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158.
- Leach, P. G. L.; G. P. Flessas (2003). "Generalisations of the Laplace–Runge–Lenz vector". J. Nonlinear Math. Phys. 10 (3): 340–423. S2CID 73707398.