Laplace–Runge–Lenz vector

In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.

Thus the hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today.

In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system. 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, so that the whole problem is symmetric under certain rotations of the four-dimensional space. 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.

The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector, the Runge–Lenz vector and the Lenz vector. Ironically, none of those scientists discovered it. The LRL vector has been re-discovered and re-formulated several times; for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics. Its first use seems to go back at least to Jakob Hermann. Various generalisations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.

Context

The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector. Thus, the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum.

History of rediscovery

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.

Hermann's work was generalized to its modern form by Johann Bernoulli in 1710.

At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis. Gibbs' derivation was used as an example by Carl Runge in a popular German textbook on vectors, which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom. In 1926, Wolfgang Pauli used the LRL vector to derive the energy levels of the hydrogen atom using the matrix mechanics formulation of quantum mechanics, after which it became known mainly as the Runge–Lenz vector.

Definition

where

The SI units of the LRL vector are joule-kilogram-meter (J⋅kg⋅m).

The assumed force is also a central force.

Further alternative formulations are given below.

Derivation of the Kepler orbits

The shape and orientation of the orbits can be determined from the LRL vector as follows.

Rearranging yields the solution for the Kepler equation

Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"), the eccentricity is greater than one and the orbit is a hyperbola. Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola.

Circular momentum hodographs

This circular hodograph is useful in illustrating the symmetry of the Kepler problem.

Constants of motion and superintegrability

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.

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, alternatively, and equivalently, to Hamiltonian mechanics.

Maximally superintegrable systems can be quantized using commutation relations, as illustrated below. Nevertheless, equivalently, they are also quantized in the Nambu framework, such as this classical Kepler problem into the quantum hydrogen atom.

Evolution under perturbed potentials

In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow apsidal precession of the orbit.

Mathematically, this time average corresponds to the following quantity in curly braces. This averaging helps to suppress fluctuations in the rate of rotation.

This agreement with experiment is strong evidence for general relativity.

Poisson brackets

Unscaled functions

The Poisson brackets will be extended to quantum mechanical commutation relations in the next section and to Lie brackets in a following section.

Scaled functions

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.

Laplace-Runge-Lenz operator for the hydrogen atom in momentum space

Scaled Laplace-Runge-Lenz operator in the momentum space was found in 2022. The formula for the operator is simpler than in position space:

Casimir invariants and the energy levels

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. This derivation is discussed in detail in the next section.

Quantum mechanics of the hydrogen atom

This elegant 1926 derivation was obtained before the development of the Schrödinger equation.

Conservation and symmetry

The conservation of the LRL vector corresponds to a subtle symmetry of the system. In classical mechanics, symmetries are continuous operations that map one orbit onto another without changing the energy of the system; in quantum mechanics, symmetries are continuous operations that "mix" electronic orbitals of the same energy, i.e., degenerate energy levels. A conserved quantity is usually associated with such symmetries.

The symmetry for the inverse-square central force is higher and more subtle. 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".

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. Such mixing cannot be done with ordinary three-dimensional translations or rotations, but is equivalent to a rotation in a higher dimension.

In 1935, Vladimir Fock showed that the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a three-dimensional unit sphere in four-dimensional space. Specifically, Fock showed that the Schrödinger wavefunction in the momentum space for the Kepler problem was the stereographic projection of the spherical harmonics on the sphere. This conclusion does not imply that our universe is a three-dimensional sphere; it merely means that this particular physics problem (the two-body problem for inverse-square central forces) is mathematically equivalent to a free particle on a three-dimensional sphere.

Both the negative- and positive-energy cases were considered by Fock and Bargmann and have been reviewed encyclopedically by Bander and Itzykson.

The orbits of central-force systems – and those of the Kepler problem in particular – are also symmetric under reflection.

Rotational symmetry in four dimensions

This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.

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 illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated.

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

Hamilton–Jacobi equation in parabolic coordinates

Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations

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 Noether's theorem.

Lie transformation

Alternative scalings, symbols and formulations

However, the choice of scaling and symbol for the LRL vector do not affect its conservation.

which is conserved and points along the minor semiaxis of the ellipse. (It is not defined for vanishing eccentricity.)

At perihelion, it points in the direction of the momentum.

Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.