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. 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

A single particle moving under any conservative central force has at least four constants of motion: the total energy and the three Cartesian components of the angular momentum vector with respect to the center of force. The particle's orbit is confined to the plane defined by the particle's initial momentum (or, equivalently, its velocity ) and the vector between the particle and the center of force 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. At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of , deriving it analytically, rather than geometrically. In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below, 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, divides by <math>L^2</math>, yielding an equivalent conserved quantity with units of inverse length, a quantity that appears in the solution of the Kepler problem

u \equiv \frac{1}{r} = \frac{km}{L^2} + \frac{A}{L^2} \cos\theta

</math>

where <math>\theta</math> is the angle between and the position vector . Further alternative formulations are given below.

Derivation of the Kepler orbits

thumb|right|350px|Figure 2: Simplified version of Figure 1, defining the angle between and at one point of the orbit.

The shape and orientation of the orbits can be determined from the LRL vector as follows. For bounded orbits, the eccentricity corresponds to the cosine of the angle shown in Figure 3. For unbounded orbits, we have <math> A > m k</math> and so the circle does not intersect the <math>p_x</math>-axis.

In the degenerate limit of circular orbits, and thus vanishing , the circle centers at the origin .

For brevity, it is also useful to introduce the variable <math display="inline">p_0 = \sqrt{2m|E|}</math>.

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

Constants of motion and superintegrability

The seven scalar quantities , and (being vectors, the latter two contribute three conserved quantities each) are related by two equations, and , giving five independent constants of motion. (Since the magnitude of , hence the eccentricity of the orbit, can be determined from the total angular momentum and the energy , only the direction of is conserved independently; moreover, since must be perpendicular to , it contributes only one additional conserved quantity.)

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 degrees of freedom can have at most constants of motion, since there are initial conditions and the initial time cannot be determined by a constant of motion. A system with more than constants of motion is called superintegrable and a system with constants is called maximally superintegrable. Since the solution of the Hamilton–Jacobi equation in one coordinate system can yield only constants of motion, superintegrable systems must be separable in more than one coordinate system. The Kepler problem is maximally superintegrable, since it has three degrees of freedom () and five independent constant of motion; its Hamilton–Jacobi equation is separable in both spherical coordinates and parabolic coordinates, 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

right|thumb|250px|Figure 5: Gradually precessing elliptical orbit, with an eccentricity e = 0.667. Such precession arises in the Kepler problem if the attractive central force deviates slightly from an inverse-square law. The rate of precession can be calculated using the formulae in the text.

The Laplace–Runge–Lenz vector 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 . 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 is a conservative central force, which implies that the total energy and angular momentum vector are conserved. Thus, the motion still lies in a plane perpendicular to and the magnitude is conserved, from the equation . The perturbation potential 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 . Using canonical perturbation theory and action-angle coordinates, it is straightforward to show

h(r) = \frac{kL^{2{m^{2}c^{2 \left( \frac{1}{r^{3 \right).

</math>

Inserting this function into the integral and using the equation

\frac{1}{r} = \frac{mk}{L^2} \left( 1 + \frac{A}{mk} \cos\theta \right)

</math>

to express in terms of , the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be and binary pulsars. This agreement with experiment is strong evidence for general relativity.

Poisson brackets

Unscaled functions

The algebraic structure of the problem is, as explained in later sections, .

<math display="block">\{A_i,L_j\}=\sum_{s=1}^3\varepsilon_{ijs}A_s.</math>

Finally, the Poisson bracket relations between the different components of are as follows:

<math display="block">\{A_i,A_j\}=-2mH\sum_{s=1}^3\varepsilon_{ijs}L_s,</math>

where <math>H</math> is the Hamiltonian. Note that the span of the components of and the components of is not closed under Poisson brackets, because of the factor of <math>H</math> on the right-hand side of this last relation.

Finally, since both and 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.

Scaled functions

As noted below, a scaled Laplace–Runge–Lenz vector may be defined with the same units as angular momentum by dividing by <math display="inline">p_0 = \sqrt{2m|H|}</math>. Since still transforms like a vector, the Poisson brackets of with the angular momentum vector can then be written in a similar form

\{ D_i, D_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s.

</math>

We may now appreciate the motivation for the chosen scaling of : With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of and the three components of forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to , the Lie algebra of the 4-dimensional rotation group .

By contrast, for positive energy, the Poisson brackets have the opposite sign,

\{ D_i, D_j\} = -\sum_{s=1}^3 \varepsilon_{ijs} L_s.

</math>

In this case, the Lie algebra is isomorphic to .

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 <math>|H| = H</math>) and negative values (where <math>|H| = -H</math>).

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:

: <math > \hat \mathbf{A}_{\mathbf p}=\imath(\hat l_{\mathbf p}+1 )\mathbf p -\frac{(p^2+1)}{2}\imath\mathbf\nabla_{\mathbf p } , </math>

where the "degree operator"

: <math > \hat l_{\mathbf p }=(\mathbf p \mathbf \nabla_{\mathbf p} ) </math>

multiplies a homogeneous polynomial by its degree.

Casimir invariants and the energy levels

The Casimir invariants for negative energies are

<math display="block"> \begin{align}

C_1 &= \mathbf{D} \cdot \mathbf{D} + \mathbf{L} \cdot \mathbf{L} = \frac{mk^2}{2|E|}, \\

C_2 &= \mathbf{D} \cdot \mathbf{L} = 0,

\end{align}</math>

and have vanishing Poisson brackets with all components of and ,

\{ C_1, L_i \} = \{ C_1, D_i\} =

\{ C_2, L_i \} = \{ C_2, D_i \} = 0.

</math>

C2 is trivially zero, since the two vectors are always perpendicular.

However, the other invariant, C1, is non-trivial and depends only on , and . 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.

By carrying out this quantization and calculating the eigenvalues of the 1 Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum.

A subtlety of the quantum mechanical operator for the LRL vector is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of and must be defined carefully.

From these operators, additional ladder operators for can be defined,

<math display="block">\begin{align}

J_0 &= A_3, \\

J_{\pm 1} &= \mp \tfrac{1}{\sqrt{2 \left( A_1 \pm i A_2 \right).

\end{align}</math>

These further connect different eigenstates of , so different spin multiplets, among themselves.

A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined,

where is the inverse of the Hamiltonian energy operator, and is the identity operator.

Applying these ladder operators to the eigenstates |ℓ〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, 1, are seen to be quantized, . Importantly, by dint of the vanishing of C2, they are independent of the ℓ and quantum numbers, making the energy levels degenerate.

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. quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number . Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector and the scaled LRL vector formed the Lie algebra for .

The orbits of central-force systems – and those of the Kepler problem in particular – are also symmetric under reflection. Therefore, the , and groups cited above are not the full symmetry groups of their orbits; the full groups are orthogonal group|, , and O(3,1), respectively. Nevertheless, only the connected subgroups, , , and , 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

thumb|right|300px|Figure 8: The momentum hodographs of Figure 7 correspond to stereographic projections of [[great circles on the three-dimensional unit sphere. All of the great circles intersect the axis, which is perpendicular to the page; the projection is from the North pole (the unit vector) to the − plane, as shown here for the magenta hodograph by the dashed black lines. The great circle at a latitude corresponds to an eccentricity . The colors of the great circles shown here correspond to their matching hodographs in Figure 7.]]

The connection between the Kepler problem and four-dimensional rotational symmetry can be readily visualized. Let the four-dimensional Cartesian coordinates be denoted where represent the Cartesian coordinates of the normal position vector . The three-dimensional momentum vector is associated with a four-dimensional vector <math>\boldsymbol\eta</math> on a three-dimensional unit sphere

<math display="block">\begin{align}

\boldsymbol\eta & = \frac{p^2 - p_0^2}{p^2 + p_0^2} \mathbf{\hat{w + \frac{2 p_0}{p^2 + p_0^2} \mathbf{p} \\[1em]

& = \frac{mk - r p_0^2}{mk} \mathbf{\hat{w + \frac{rp_0}{mk} \mathbf{p},

\end{align}</math>

where <math>\mathbf{\hat{w</math> is the unit vector along the new axis. The transformation mapping to can be uniquely inverted; for example, the component of the momentum equals

and similarly for and . In other words, the three-dimensional vector is a stereographic projection of the four-dimensional <math>\boldsymbol\eta</math> vector, scaled by (Figure 8).

Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the axis is aligned with the angular momentum vector and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the axis. Since the motion is planar, and and are perpendicular, 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 <math>\boldsymbol\eta</math> sphere, all of which intersect the axis at the two foci , corresponding to the momentum hodograph foci at . These great circles are related by a simple rotation about the -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 . This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.

An elegant action-angle variables solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates <math>\boldsymbol\eta</math> in favor of elliptic cylindrical coordinates

<math display="block">\begin{align}

\eta_w &= \operatorname{cn} \chi \operatorname{cn} \psi, \\[1ex]

\eta_x &= \operatorname{sn} \chi \operatorname{dn} \psi \cos \phi, \\[1ex]

\eta_y &= \operatorname{sn} \chi \operatorname{dn} \psi \sin \phi, \\[1ex]

\eta_z &= \operatorname{dn} \chi \operatorname{sn} \psi,

\end{align}</math>

where , and are Jacobi's elliptic functions.

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 , the generalized Laplace–Runge–Lenz vector <math>\mathcal{A}</math> is

\mathcal{A} = \mathbf{A} + \frac{mq}{2} \left[ \left( \mathbf{r} \times \mathbf{E} \right) \times \mathbf{r} \right],

</math>

where is the charge of the orbiting particle. Although <math>\mathcal{A}</math> is not conserved, it gives rise to a conserved quantity, namely <math>\mathcal{A} \cdot \mathbf{E}</math>.

Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as

\frac{d}{dt} \mathbf{A} = \frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) - \frac{d}{dt} \left( mk\mathbf{\hat{r \right) = \mathbf{0}.

</math>

A shorter proof is obtained by using the relation of angular momentum to angular velocity, <math> \mathbf{L} = m r^2 \boldsymbol{\omega}</math>, which holds for a particle traveling in a plane perpendicular to <math> \mathbf{L}</math>. Specifying to inverse-square central forces, the time derivative of <math>\mathbf{p} \times \mathbf{L}</math> is

\frac{d}{dt} \mathbf{p} \times \mathbf{L} = \left( \frac{-k}{r^2} \mathbf{\hat{r \right) \times \left(m r^2 \boldsymbol{\omega}\right)

= m k \, \boldsymbol{\omega} \times \mathbf{\hat{r = m k \,\frac{d}{dt}\mathbf{\hat{r,

</math>

where the last equality holds because a unit vector can only change by rotation, and <math>\boldsymbol{\omega}\times\mathbf{\hat{r</math> is the orbital velocity of the rotating vector. Thus, is seen to be a difference of two vectors with equal time derivatives.

As described elsewhere in this article, this LRL vector is a special case of a general conserved vector <math>\mathcal{A}</math> that can be defined for all central forces.

<math display="block"> \begin{align}

2\xi p_\xi^2 - mk - mE\xi &= -\Gamma, \\

2\eta p_\eta^2 - mk - mE\eta &= \Gamma,

\end{align}</math>

where is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta and shows that is equivalent to the LRL vector

\Gamma = p_y (x p_y - y p_x) - mk\frac{x}{r} = A_x.

</math>

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. This theorem, which is used for finding constants of motion, states that any infinitesimal variation of the generalized coordinates of a physical system

\delta q_i = \varepsilon g_i(\mathbf{q}, \mathbf{\dot{q, t)

</math>

that causes the Lagrangian to vary to first order by a total time derivative

\delta L = \varepsilon \frac{d}{dt} G(\mathbf{q}, t)

</math>

corresponds to a conserved quantity

\Gamma = -G + \sum_i g_i \left( \frac{\partial L}{\partial \dot{q}_i}\right).

</math>

In particular, the conserved LRL vector component corresponds to the variation in the coordinates

\delta_s x_i = \frac{\varepsilon}{2} \left[ 2 p_i x_s - x_i p_s - \delta_{is} \left( \mathbf{r} \cdot \mathbf{p} \right) \right],

</math>

where equals 1, 2 and 3, with and being the -th components of the position and momentum vectors and , respectively; as usual, represents the Kronecker delta. The resulting first-order change in the Lagrangian is

\delta L = \frac{1}{2}\varepsilon mk\frac{d}{dt} \left( \frac{x_s}{r} \right).

</math>

Substitution into the general formula for the conserved quantity yields the conserved component of the LRL vector,

A_s = \left[ p^2 x_s - p_s \ \left(\mathbf{r} \cdot \mathbf{p}\right) \right] - mk \left( \frac{x_s}{r} \right) =

\left[ \mathbf{p} \times \left( \mathbf{r} \times \mathbf{p} \right) \right]_s - mk \left( \frac{x_s}{r} \right).

</math>

Lie transformation

thumb|right|350px|Figure 9: The Lie transformation from which the conservation of the LRL vector is derived. As the scaling parameter varies, the energy and angular momentum changes, but the eccentricity and the magnitude and direction of do not.

Noether's theorem derivation of the conservation of the LRL vector is elegant, but has one drawback: the coordinate variation involves not only the position , but also the momentum or, equivalently, the velocity . This drawback may be eliminated by instead deriving the conservation of using an approach pioneered by Sophus Lie. Specifically, one may define a Lie transformation in which the coordinates and the time are scaled by different powers of a parameter λ (Figure 9),

t \rightarrow \lambda^{3}t , \qquad \mathbf{r} \rightarrow \lambda^{2}\mathbf{r} , \qquad\mathbf{p} \rightarrow \frac{1}{\lambda}\mathbf{p}.

</math>

This transformation changes the total angular momentum and energy ,

L \rightarrow \lambda L, \qquad E \rightarrow \frac{1}{\lambda^{2 E,

</math>

but preserves their product EL2. Therefore, the eccentricity and the magnitude are preserved, as may be seen from the equation for

The direction of is preserved as well, since the semiaxes are not altered by a global scaling. This transformation also preserves Kepler's third law, namely, that the semiaxis and the period form a constant .

Alternative scalings, symbols and formulations

Unlike the momentum and angular momentum vectors and , 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 to obtain a dimensionless conserved eccentricity vector

\mathbf{e} =

\frac{1}{mk} \left(\mathbf{p} \times \mathbf{L} \right) - \mathbf{\hat{r =

\frac{m}{k} \left(\mathbf{v} \times \left( \mathbf{r} \times \mathbf{v} \right) \right) - \mathbf{\hat{r,

</math>

where is the velocity vector. This scaled vector has the same direction as 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 by alone

\mathbf{M} = \mathbf{v} \times \mathbf{L} - k\mathbf{\hat{r,

</math>

or by

\mathbf{D} = \frac{\mathbf{A{p_{0 =

\frac{1}{\sqrt{2m|E|

\left( \mathbf{p} \times \mathbf{L} - m k \mathbf{\hat{r \right),

</math>

which has the same units as the angular momentum vector .

In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by . Other common symbols for the LRL vector include , , , and . However, the choice of scaling and symbol for the LRL vector do not affect its conservation.

thumb|right|250px|Figure 4: The angular momentum vector , the LRL vector and Hamilton's vector, the binormal , are mutually perpendicular; and point along the major and minor axes, respectively, of an elliptical orbit of the Kepler problem.

An alternative conserved vector is the binormal vector studied by William Rowan Hamilton,