Template:Short description {{#invoke:Hatnote|hatnote}} 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;<ref name="goldstein_1980">Template:Cite book</ref><ref name="taff_1985">Template:Cite book</ref> 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.<ref name="goldstein_1980b">Template:Cite book</ref><ref name="arnold_1989">Template:Cite book</ref><ref name="sommerfeld_1989">Template:Cite book</ref><ref name="lanczos_1970">Template:Cite book</ref>
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,<ref name="pauli_1926">Template:Cite journal</ref><ref name="bohm_1993">Template:Cite book</ref> 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.<ref name="hanca_et_al_2004">Template:Cite journal</ref> 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,<ref name="fock_1935" >Template:Cite journal</ref> so that the whole problem is symmetric under certain rotations of the four-dimensional space.<ref name="bargmann_1936" >Template:Cite journal</ref> 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.<ref name="hamilton_1847_hodograph">Template:Cite journal</ref>
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,<ref name="goldstein_1980c">Template:Cite book</ref><ref name="arnold_1989b">Template:Cite book</ref> the Runge–Lenz vector<ref name="goldstein_1975_1976">Template:Cite journal
Template:Cite journal</ref> and the Lenz vector.<ref name="bohm_1993" /> Ironically, none of those scientists discovered it.<ref name="goldstein_1975_1976" /> The LRL vector has been re-discovered and re-formulated several times;<ref name="goldstein_1975_1976" /> for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics.<ref name="taff_1985" /><ref name="arnold_1989b" /><ref name="hamilton_1847_quaternions">Template:Cite journal</ref> Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.<ref name="landau_lifshitz_1976">Template:Cite book</ref><ref name="fradkin_1967">Template:Cite journal</ref><ref name="yoshida_1987">Template:Cite journal</ref>
ContextEdit
A single particle moving under any conservative central force has at least four constants of motion: the total energy Template:Mvar and the three Cartesian components of the angular momentum vector Template:Math with respect to the center of force.<ref name="goldstein_1980d">Template:Cite book</ref><ref name="symon_1971">Template:Cite book</ref> The particle's orbit is confined to the plane defined by the particle's initial momentum Template:Math (or, equivalently, its velocity Template:Math) and the vector Template:Math between the particle and the center of force<ref name="goldstein_1980d" /><ref name="symon_1971" /> (see Figure 1). This plane of motion is perpendicular to the constant angular momentum vector Template:Math; this may be expressed mathematically by the vector dot product equation Template:Math. Given its mathematical definition below, the Laplace–Runge–Lenz vector (LRL vector) Template:Math is always perpendicular to the constant angular momentum vector Template:Math for all central forces (Template:Math). Therefore, Template:Math always lies in the plane of motion. As shown below, Template:Math points from the center of force to the periapsis of the motion, the point of closest approach, and its length is proportional to the eccentricity of the orbit.<ref name="goldstein_1980" />
The LRL vector Template:Math is constant in length and direction, but only for an inverse-square central force.<ref name="goldstein_1980" /> For other central forces, the vector Template:Math is not constant, but changes in both length and direction. If the central force is approximately an inverse-square law, the vector Template:Math is approximately constant in length, but slowly rotates its direction.<ref name="arnold_1989b" /> A generalized conserved LRL vector <math>\mathcal{A}</math> can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form.<ref name="fradkin_1967" /><ref name="yoshida_1987" />
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 rediscoveryEdit
The LRL vector Template:Math 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.<ref name="goldstein_1975_1976" />
Jakob Hermann was the first to show that Template:Math is conserved for a special case of the inverse-square central force,<ref>Template:Cite journal
Template:Cite journal</ref> 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.<ref>Template:Cite journal</ref> At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of Template:Math, deriving it analytically, rather than geometrically.<ref>Template:Cite book</ref> In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below,<ref name="hamilton_1847_quaternions" /> using it to show that the momentum vector Template:Math moves on a circle for motion under an inverse-square central force (Figure 3).<ref name="hamilton_1847_hodograph" />
At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis.<ref>Template:Cite book</ref> Gibbs' derivation was used as an example by Carl Runge in a popular German textbook on vectors,<ref>Template:Cite book</ref> which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom.<ref>Template:Cite journal</ref> 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,<ref name="pauli_1926" /> after which it became known mainly as the Runge–Lenz vector.<ref name="goldstein_1975_1976" />
DefinitionEdit
An inverse-square central force acting on a single particle is described by the equation <math display="block"> \mathbf{F}(r)=-\frac{k}{r^{2}}\mathbf{\hat{r}}; </math> The corresponding potential energy is given by <math>V(r) = - k / r</math>. The constant parameter Template:Mvar describes the strength of the central force; it is equal to Template:Math for gravitational and Template:Math for electrostatic forces. The force is attractive if Template:Math and repulsive if Template:Math.
The LRL vector Template:Math is defined mathematically by the formula<ref name="goldstein_1980" /> Template:Equation box 1,</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F9FFF7}} where
- Template:Mvar is the mass of the point particle moving under the central force,
- Template:Math is its momentum vector,
- Template:Math is its angular momentum vector,
- Template:Math is the position vector of the particle (Figure 1),
- <math>\mathbf{\hat{r}}</math> is the corresponding unit vector, i.e., <math>\mathbf{\hat{r}} = \frac{\mathbf{r}}{r}</math>, and
- Template:Mvar is the magnitude of Template:Math, 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 Template:Math and Template:Math are kg⋅m/s and J⋅s, respectively. This agrees with the units of Template:Mvar (kg) and of Template:Mvar (N⋅m2).
This definition of the LRL vector Template:Math pertains to a single point particle of mass Template:Mvar 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 Template:Mvar as the reduced mass of the two bodies and Template:Math as the vector between the two bodies.
Since the assumed force is conservative, the total energy Template:Mvar is a constant of motion, <math display="block"> E = \frac{p^{2}}{2m} - \frac{k}{r} = \frac{1}{2} mv^{2} - \frac{k}{r}. </math>
The assumed force is also a central force. Hence, the angular momentum vector Template:Math is also conserved and defines the plane in which the particle travels. The LRL vector Template:Math is perpendicular to the angular momentum vector Template:Math because both Template:Math and Template:Math are perpendicular to Template:Math. It follows that Template:Math 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 Template:Math, the force parameter Template:Math or the angular momentum Template:Math.<ref name="goldstein_1975_1976" /> The most common variant is to divide Template:Math by Template:Math, which yields the eccentricity vector,<ref name="taff_1985" /><ref name="hamilton_1847_quaternions" /> a dimensionless vector along the semi-major axis whose modulus equals the eccentricity of the conic: <math display="block"> \mathbf{e} = \frac{\mathbf{A}}{m k} = \frac{1}{m k}(\mathbf{p} \times \mathbf{L}) - \mathbf{\hat{r}}. </math> An equivalent formulation<ref name="arnold_1989b" /> multiplies this eccentricity vector by the major semiaxis Template:Mvar, giving the resulting vector the units of length. Yet another formulation<ref name="symon_1971b">Template:Cite book</ref> divides Template:Math 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 <math display="block"> u \equiv \frac{1}{r} = \frac{km}{L^2} + \frac{A}{L^2} \cos\theta </math> where <math>\theta</math> is the angle between Template:Math and the position vector Template:Math. Further alternative formulations are given below.
Derivation of the Kepler orbitsEdit
The shape and orientation of the orbits can be determined from the LRL vector as follows.<ref name="goldstein_1980" /> Taking the dot product of Template:Math with the position vector Template:Math gives the equation <math display="block"> \mathbf{A} \cdot \mathbf{r} = A \cdot r \cdot \cos\theta = \mathbf{r} \cdot \left( \mathbf{p} \times \mathbf{L} \right) - mkr, </math> where Template:Mvar is the angle between Template:Math and Template:Math (Figure 2). Permuting the scalar triple product yields <math display="block"> \mathbf{r} \cdot\left(\mathbf{p}\times \mathbf{L}\right) = \left(\mathbf{r} \times \mathbf{p}\right)\cdot\mathbf{L} = \mathbf{L}\cdot\mathbf{L}=L^2 </math>
Rearranging yields the solution for the Kepler equation Template:Equation box 1 \cos\theta</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F9FFF7}}
This corresponds to the formula for a conic section of eccentricity e <math display="block"> \frac{1}{r} = C \cdot \left( 1 + e \cdot \cos\theta \right) </math> where the eccentricity <math>e = \frac{A}{\left| mk \right|} \geq 0</math> and Template:Mvar is a constant.<ref name="goldstein_1980" />
Taking the dot product of Template:Math with itself yields an equation involving the total energy Template:Mvar,<ref name="goldstein_1980" /> <math display="block"> A^2 = m^2 k^2 + 2 m E L^2, </math> which may be rewritten in terms of the eccentricity,<ref name="goldstein_1980" /> <math display="block"> e^{2} = 1 + \frac{2L^2}{mk^2}E. </math>
Thus, if the energy Template:Mvar 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"<ref name="goldstein_1980" />), the eccentricity is greater than one and the orbit is a hyperbola.<ref name="goldstein_1980" /> Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola.<ref name="goldstein_1980" /> In all cases, the direction of Template:Math lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.<ref name="goldstein_1980" />
Circular momentum hodographsEdit
The conservation of the LRL vector Template:Math and angular momentum vector Template:Math is useful in showing that the momentum vector Template:Math moves on a circle under an inverse-square central force.<ref name="hamilton_1847_hodograph" /><ref name="goldstein_1975_1976" />
Taking the dot product of <math display="block"> mk \hat{\mathbf{r}} = \mathbf{p} \times \mathbf{L} - \mathbf{A} </math> with itself yields <math display="block"> (mk)^2= A^2+ p^2 L^2 + 2 \mathbf{L} \cdot (\mathbf{p} \times \mathbf{A}). </math>
Further choosing Template:Math along the Template:Mvar-axis, and the major semiaxis as the Template:Mvar-axis, yields the locus equation for Template:Math, Template:Equation box 1
In other words, the momentum vector Template:Math is confined to a circle of radius Template:Math centered on Template:Math.<ref>The conserved binormal Hamilton vector <math>\mathbf{B}\equiv \mathbf{L} \times \mathbf{A} / L^2</math> on this momentum plane (pink) has a simpler geometrical significance, and may actually supplant it, as <math>\mathbf{A} =\mathbf {B}\times \mathbf {L}</math>, see Patera, R. P. (1981). "Momentum-space derivation of the Runge-Lenz vector", Am. J. Phys 49 593–594. It has length Template:Math and is discussed in section #Alternative scalings, symbols and formulations.</ref> For bounded orbits, the eccentricity Template:Mvar corresponds to the cosine of the angle Template:Mvar 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 Template:Math, the circle centers at the origin Template:Math. 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 superintegrabilityEdit
The seven scalar quantities Template:Mvar, Template:Math and Template:Math (being vectors, the latter two contribute three conserved quantities each) are related by two equations, Template:Math and Template:Math, giving five independent constants of motion. (Since the magnitude of Template:Math, hence the eccentricity Template:Mvar of the orbit, can be determined from the total angular momentum Template:Mvar and the energy Template:Mvar, only the direction of Template:Math is conserved independently; moreover, since Template:Math must be perpendicular to Template:Math, 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 Template:Mvar degrees of freedom can have at most Template:Math constants of motion, since there are Template:Math initial conditions and the initial time cannot be determined by a constant of motion. A system with more than Template:Mvar constants of motion is called superintegrable and a system with Template:Math constants is called maximally superintegrable.<ref>Template:Cite journal</ref> Since the solution of the Hamilton–Jacobi equation in one coordinate system can yield only Template:Mvar constants of motion, superintegrable systems must be separable in more than one coordinate system.<ref>Template:Cite book</ref>Template:Irrelevant citation The Kepler problem is maximally superintegrable, since it has three degrees of freedom (Template:Math) and five independent constant of motion; its Hamilton–Jacobi equation is separable in both spherical coordinates and parabolic coordinates,<ref name="landau_lifshitz_1976" /> as described below.
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,<ref>Template:Cite journal</ref> alternatively, and equivalently, to Hamiltonian mechanics.
Maximally superintegrable systems can be quantized using commutation relations, as illustrated below.<ref>Template:Cite journal</ref> Nevertheless, equivalently, they are also quantized in the Nambu framework, such as this classical Kepler problem into the quantum hydrogen atom.<ref>Template:Cite journal</ref>
Evolution under perturbed potentialsEdit
The Laplace–Runge–Lenz vector Template:Math 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 Template:Math. 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 Template:Math is a conservative central force, which implies that the total energy Template:Mvar and angular momentum vector Template:Math are conserved. Thus, the motion still lies in a plane perpendicular to Template:Math and the magnitude Template:Mvar is conserved, from the equation Template:Math. The perturbation potential Template:Math 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 Template:Math. Using canonical perturbation theory and action-angle coordinates, it is straightforward to show<ref name="goldstein_1980" /> that Template:Math rotates at a rate of, <math display="block">\begin{align} \frac{\partial}{\partial L} \langle h(r) \rangle & = \frac{\partial}{\partial L} \left\{ \frac{1}{T} \int_0^T h(r) \, dt \right\} \\[1em] & = \frac{\partial}{\partial L} \left\{ \frac{m}{L^{2}} \int_0^{2\pi} r^2 h(r) \, d\theta \right\}, \end{align}</math> where Template:Mvar is the orbital period, and the identity Template:Math was used to convert the time integral into an angular integral (Figure 5). The expression in angular brackets, Template:Math, represents the perturbing potential, but averaged over one full period; that is, averaged over one full passage of the body around its 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 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,<ref name="einstein_1915">Template:Cite journal</ref> <math display="block"> 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 <math display="block"> \frac{1}{r} = \frac{mk}{L^2} \left( 1 + \frac{A}{mk} \cos\theta \right) </math> to express Template:Mvar in terms of Template:Mvar, the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be<ref name="einstein_1915" /> <math display="block"> \frac{6 \pi k^2}{T L^2 c^2}, </math> which closely matches the observed anomalous precession of Mercury<ref>Template:Cite journal</ref> and binary pulsars.<ref>Template:Cite book</ref> This agreement with experiment is strong evidence for general relativity.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>
Poisson bracketsEdit
Unscaled functionsEdit
The algebraic structure of the problem is, as explained in later sections, Template:Math.<ref name="bargmann_1936" /> The three components Li of the angular momentum vector Template:Math have the Poisson brackets<ref name="goldstein_1980" /> <math display="block"> \{ L_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s, </math> where Template:Mvar=1,2,3 and Template:Math is the fully antisymmetric tensor, i.e., the Levi-Civita symbol; the summation index Template:Mvar is used here to avoid confusion with the force parameter Template:Mvar defined above. Then since the LRL vector Template:Math transforms like a vector, we have the following Poisson bracket relations between Template:Math and Template:Math:<ref>Template:Harvnb Proposition 17.25.</ref> <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 Template:Math are as follows:<ref>Template:Harvnb Proposition 18.7; note that Hall uses a different normalization of the LRL vector.</ref> <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 Template:Math and the components of Template:Math 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 Template:Math and Template:Math are constants of motion, we have <math display="block">\{A_i, H\} = \{L_i, H\} = 0.</math>
The Poisson brackets will be extended to quantum mechanical commutation relations in the next section and to Lie brackets in a following section.
Scaled functionsEdit
As noted below, a scaled Laplace–Runge–Lenz vector Template:Math may be defined with the same units as angular momentum by dividing Template:Math by <math display="inline">p_0 = \sqrt{2m|H|}</math>. Since Template:Math still transforms like a vector, the Poisson brackets of Template:Math with the angular momentum vector Template:Math can then be written in a similar form<ref name="bargmann_1936" /><ref name="bohm_1993" /> <math display="block"> \{ D_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} D_s. </math>
The Poisson brackets of Template:Math with itself depend on the sign of Template:Mvar, i.e., on whether the energy is negative (producing closed, elliptical orbits under an inverse-square central force) or positive (producing open, hyperbolic orbits under an inverse-square central force). For negative energies—i.e., for bound systems—the Poisson brackets are<ref name="Hall 2013">Template:Harvnb Theorem 18.9.</ref> <math display="block"> \{ D_i, D_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s. </math> We may now appreciate the motivation for the chosen scaling of Template:Math: 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 Template:Math and the three components of Template:Math forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to Template:Math, the Lie algebra of the 4-dimensional rotation group Template:Math.<ref name="ReferenceA">Template:Harvnb</ref>
By contrast, for positive energy, the Poisson brackets have the opposite sign, <math display="block"> \{ D_i, D_j\} = -\sum_{s=1}^3 \varepsilon_{ijs} L_s. </math> In this case, the Lie algebra is isomorphic to Template:Math.
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 spaceEdit
Scaled Laplace-Runge-Lenz operator in the momentum space was found in 2022 .<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> 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 levelsEdit
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 Template:Math and Template:Math, <math display="block"> \{ 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 Template:Mvar, Template:Mvar and Template:Mvar. 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.<ref name="bohm_1993" /><ref name="ReferenceA" /> This derivation is discussed in detail in the next section.
Quantum mechanics of the hydrogen atomEdit
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 Template:Math.<ref>Template:Cite book</ref>
By carrying out this quantization and calculating the eigenvalues of the Template:Mvar1 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.<ref name="pauli_1926" /> This elegant 1926 derivation was obtained before the development of the Schrödinger equation.<ref>Template:Cite journal</ref>
A subtlety of the quantum mechanical operator for the LRL vector Template:Math is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of Template:Math and Template:Math must be defined carefully.<ref name="bohm_1993" /> Typically, the operators for the Cartesian components Template:Math are defined using a symmetrized (Hermitian) product, <math display="block"> A_s = - m k \hat{r}_s + \frac{1}{2} \sum_{i=1}^3 \sum_{j=1}^3 \varepsilon_{sij} (p_i \ell_j + \ell_j p_i), </math> Once this is done, one can show that the quantum LRL operators satisfy commutations relations exactly analogous to the Poisson bracket relations in the previous section—just replacing the Poisson bracket with <math>1/(i\hbar)</math> times the commutator.<ref>Template:Harvnb Proposition 18.12.</ref><ref>Template:Cite book</ref>
From these operators, additional ladder operators for Template:Math 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 Template:Math, so different spin multiplets, among themselves.
A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined, <math display="block">C_1 = - \frac{m k^2}{2 \hbar^{2}} H^{-1} - I,</math> where Template:Math is the inverse of the Hamiltonian energy operator, and Template:Mvar is the identity operator.
Applying these ladder operators to the eigenstates |ℓTemplate:Math〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, Template:Mvar1, are seen to be quantized, Template:Math. Importantly, by dint of the vanishing of C2, they are independent of the ℓ and Template:Mvar quantum numbers, making the energy levels degenerate.<ref name="bohm_1993" />
Hence, the energy levels are given by <math display="block"> E_n = - \frac{m k^2}{2\hbar^{2} n^2}, </math> which coincides with the Rydberg formula for hydrogen-like atoms (Figure 6). The additional symmetry operators Template:Math have connected the different ℓ multiplets among themselves, for a given energy (and C1), dictating Template:Math states at each level. In effect, they have enlarged the angular momentum group Template:Math to Template:Math.<ref>Template:Harvnb Theorem 18.14.</ref>
Conservation and symmetryEdit
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.<ref name="goldstein_1980" /> For example, every central force is symmetric under the rotation group SO(3), leading to the conservation of the angular momentum Template:Math. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the spherical harmonics of the same quantum number Template:Mvar without changing the energy.
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 Template:Math and the LRL vector Template:Math (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers Template:Mvar and Template:Mvar. 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".<ref name="prince_eliezer_1981" />
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 Template:Mvar and Template:Mvar quantum numbers, such as the Template:Mvar(Template:Math) and Template:Mvar(Template:Math) 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 Template:Math, which preserves the length of four-dimensional vectors <math display="block"> |\mathbf{e}|^2 = e_1^2 + e_2^2 + e_3^2 + e_4^2. </math>
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.<ref name="fock_1935" /> 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. Rotation of the sphere and re-projection results in a continuous mapping of the elliptical orbits without changing the energy, an Template:Math symmetry sometimes known as Fock symmetry;<ref>Template:Cite journal</ref> quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number Template:Mvar. Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector Template:Math and the scaled LRL vector Template:Math formed the Lie algebra for Template:Math.<ref name="bargmann_1936" /><ref name="Hall 2013"/> Simply put, the six quantities Template:Math and Template:Math correspond to the six conserved angular momenta in four dimensions, associated with the six possible simple rotations in that space (there are six ways of choosing two axes from four). 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.
For positive energies – i.e., for unbound, "scattered" systems – the higher symmetry group is Template:Math, which preserves the Minkowski length of 4-vectors <math display="block"> ds^2 = e_1^2 + e_2^2 + e_3^2 - e_4^2. </math>
Both the negative- and positive-energy cases were considered by Fock<ref name="fock_1935" /> and Bargmann<ref name="bargmann_1936" /> and have been reviewed encyclopedically by Bander and Itzykson.<ref name="bander_itzykson_1966">Template:Cite journal</ref><ref>Template:Cite journal</ref>
The orbits of central-force systems – and those of the Kepler problem in particular – are also symmetric under reflection. Therefore, the Template:Math, Template:Math and Template:Math groups cited above are not the full symmetry groups of their orbits; the full groups are [[orthogonal group|Template:Math]], Template:Math, and O(3,1), respectively. Nevertheless, only the connected subgroups, Template:Math, Template:Math, and Template:Math, 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 dimensionsEdit
The connection between the Kepler problem and four-dimensional rotational symmetry Template:Math can be readily visualized.<ref name="bander_itzykson_1966" /><ref name="rogers_1973">Template:Cite journal</ref><ref>Template:Cite book</ref> Let the four-dimensional Cartesian coordinates be denoted Template:Math where Template:Math represent the Cartesian coordinates of the normal position vector Template:Math. The three-dimensional momentum vector Template:Math 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 Template:Mvar axis. The transformation mapping Template:Math to Template:Math can be uniquely inverted; for example, the Template:Mvar component of the momentum equals <math display="block"> p_x = p_0 \frac{\eta_x}{1 - \eta_w}, </math> and similarly for Template:Math and Template:Math. In other words, the three-dimensional vector Template:Math is a stereographic projection of the four-dimensional <math>\boldsymbol\eta</math> vector, scaled by Template:Math (Figure 8).
Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the Template:Mvar axis is aligned with the angular momentum vector Template:Math and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the Template:Mvar axis. Since the motion is planar, and Template:Math and Template:Math are perpendicular, Template:Math and attention may be restricted to the three-dimensional vector Template:Nowrap 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 Template:Math axis at the two foci Template:Math, corresponding to the momentum hodograph foci at Template:Math. These great circles are related by a simple rotation about the Template:Math-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 Template:Math. 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 Template:Math<ref>Template:Cite journal</ref> <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 Template:Math, Template:Math and Template:Math are Jacobi's elliptic functions.
Generalizations to other potentials and relativityEdit
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 Template:Math, the generalized Laplace–Runge–Lenz vector <math>\mathcal{A}</math> is<ref name="landau_lifshitz_1976" /><ref>Template:Cite journal</ref> <math display="block"> \mathcal{A} = \mathbf{A} + \frac{mq}{2} \left[ \left( \mathbf{r} \times \mathbf{E} \right) \times \mathbf{r} \right], </math> where Template:Mvar 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<ref name="fradkin_1967" /> <math display="block"> \mathcal{A} = \left( \frac{\partial \xi}{\partial u} \right) \left(\mathbf{p} \times \mathbf{L}\right) + \left[ \xi - u \left( \frac{\partial \xi}{\partial u} \right)\right] L^{2} \mathbf{\hat{r}}, </math> where Template:Math and Template:Math, with the angle Template:Mvar defined by <math display="block"> \theta = L \int^u \frac{du}{\sqrt{m^2 c^2 (\gamma^2 - 1) - L^2 u^{2}}}, </math> and Template:Mvar is the Lorentz factor. As before, we may obtain a conserved binormal vector Template:Math by taking the cross product with the conserved angular momentum vector <math display="block"> \mathcal{B} = \mathbf{L} \times \mathcal{A}. </math>
These two vectors may likewise be combined into a conserved dyadic tensor Template:Math, <math display="block"> \mathcal{W} = \alpha \mathcal{A} \otimes \mathcal{A} + \beta \, \mathcal{B} \otimes \mathcal{B}. </math>
In illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated.<ref name="fradkin_1967" /> Since the force is central, <math display="block"> \mathbf{F}(r)= -k \mathbf{r}, </math> the angular momentum vector is conserved and the motion lies in a plane.
The conserved dyadic tensor can be written in a simple form <math display="block"> \mathcal{W} = \frac{1}{2m} \mathbf{p} \otimes \mathbf{p} + \frac{k}{2} \, \mathbf{r} \otimes \mathbf{r}, </math> although Template:Math and Template:Math are not necessarily perpendicular.
The corresponding Runge–Lenz vector is more complicated, <math display="block"> \mathcal{A} = \frac{1}{\sqrt{mr^2 \omega_0 A - mr^2 E + L^2}} \left\{ \left( \mathbf{p} \times \mathbf{L} \right) + \left(mr\omega_0 A - mrE \right) \mathbf{\hat{r}} \right\}, </math> where <math display="block">\omega_0 = \sqrt{\frac{k}{m}}</math> is the natural oscillation frequency, and <math display="block">A = (E^2-\omega^2 L^2)^{1/2} / \omega.</math>
Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problemsEdit
The following are arguments showing that the LRL vector is conserved under central forces that obey an inverse-square law.
Direct proof of conservationEdit
A central force <math>\mathbf{F}</math> acting on the particle is <math display="block"> \mathbf{F} = \frac{d\mathbf{p}}{dt} = f(r) \frac{\mathbf{r}}{r} = f(r) \mathbf{\hat{r}} </math> for some function <math>f(r)</math> of the radius <math>r</math>. Since the angular momentum <math>\mathbf{L} = \mathbf{r} \times \mathbf{p}</math> is conserved under central forces, <math display="inline">\frac{d}{dt}\mathbf{L} = 0</math> and <math display="block"> \frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = \frac{d\mathbf{p}}{dt} \times \mathbf{L} = f(r) \mathbf{\hat{r}} \times \left( \mathbf{r} \times m \frac{d\mathbf{r}}{dt} \right) = f(r) \frac{m}{r} \left[ \mathbf{r} \left(\mathbf{r} \cdot \frac{d\mathbf{r}}{dt} \right) - r^2 \frac{d\mathbf{r}}{dt} \right], </math> where the momentum <math display="inline">\mathbf{p} = m \frac{d\mathbf{r}}{dt}</math> and where the triple cross product has been simplified using Lagrange's formula <math display="block"> \mathbf{r} \times \left( \mathbf{r} \times \frac{d\mathbf{r}}{dt} \right) = \mathbf{r} \left(\mathbf{r} \cdot \frac{d\mathbf{r}}{dt} \right) - r^2 \frac{d\mathbf{r}}{dt}. </math>
The identity <math display="block"> \frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = 2 \mathbf{r} \cdot \frac{d\mathbf{r}}{dt} = \frac{d}{dt} (r^2) = 2r\frac{dr}{dt} </math> yields the equation <math display="block"> \frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = -m f(r) r^2 \left[ \frac{1}{r} \frac{d\mathbf{r}}{dt} - \frac{\mathbf{r}}{r^2} \frac{dr}{dt}\right] = -m f(r) r^2 \frac{d}{dt} \left( \frac{\mathbf{r}}{r}\right). </math>
For the special case of an inverse-square central force <math display="inline">f(r)=\frac{-k}{r^{2}}</math>, this equals <math display="block"> \frac{d}{dt} \left( \mathbf{p} \times \mathbf{L} \right) = m k \frac{d}{dt} \left( \frac{\mathbf{r}}{r}\right) = \frac{d}{dt} (mk\mathbf{\hat{r}}). </math>
Therefore, Template:Math is conserved for inverse-square central forces<ref>Template:Harvnb Proposition 2.34.</ref> <math display="block"> \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 <math display="block">
\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, Template:Math is seen to be a difference of two vectors with equal time derivatives.
As described elsewhere in this article, this LRL vector Template:Math is a special case of a general conserved vector <math>\mathcal{A}</math> that can be defined for all central forces.<ref name="fradkin_1967" /><ref name="yoshida_1987" /> However, since most central forces do not produce closed orbits (see Bertrand's theorem), the analogous vector <math>\mathcal{A}</math> rarely has a simple definition and is generally a multivalued function of the angle Template:Mvar between Template:Math and <math>\mathcal{A}</math>.
Hamilton–Jacobi equation in parabolic coordinatesEdit
The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates Template:Math, which are defined by the equations <math display="block">\begin{align} \xi &= r + x, \\ \eta &= r - x, \end{align}</math> where Template:Mvar represents the radius in the plane of the orbit <math display="block">r = \sqrt{x^2 + y^2}.</math>
The inversion of these coordinates is <math display="block">\begin{align} x &= \tfrac{1}{2} (\xi - \eta), \\ y &= \sqrt{\xi\eta}, \end{align}</math>
Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations<ref name="landau_lifshitz_1976" /><ref>Template:Cite journal</ref>
<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 Template:Math is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta Template:Math and Template:Math shows that Template:Math is equivalent to the LRL vector <math display="block"> \Gamma = p_y (x p_y - y p_x) - mk\frac{x}{r} = A_x. </math>
Noether's theoremEdit
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 <math display="block"> \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 <math display="block"> \delta L = \varepsilon \frac{d}{dt} G(\mathbf{q}, t) </math> corresponds to a conserved quantity Template:Math <math display="block"> \Gamma = -G + \sum_i g_i \left( \frac{\partial L}{\partial \dot{q}_i}\right). </math>
In particular, the conserved LRL vector component Template:Math corresponds to the variation in the coordinates<ref>Template:Cite journal</ref> <math display="block"> \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 Template:Mvar equals 1, 2 and 3, with Template:Math and Template:Math being the Template:Mvar-th components of the position and momentum vectors Template:Math and Template:Math, respectively; as usual, Template:Math represents the Kronecker delta. The resulting first-order change in the Lagrangian is <math display="block"> \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 Template:Math yields the conserved component Template:Math of the LRL vector, <math display="block"> 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 transformationEdit
Noether's theorem derivation of the conservation of the LRL vector Template:Math is elegant, but has one drawback: the coordinate variation Template:Math involves not only the position Template:Math, but also the momentum Template:Math or, equivalently, the velocity Template:Math.<ref>Template:Cite journal</ref> This drawback may be eliminated by instead deriving the conservation of Template:Math using an approach pioneered by Sophus Lie.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> Specifically, one may define a Lie transformation<ref name="prince_eliezer_1981" >Template:Cite journal</ref> in which the coordinates Template:Math and the time Template:Mvar are scaled by different powers of a parameter λ (Figure 9), <math display="block"> 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 Template:Mvar and energy Template:Mvar, <math display="block"> L \rightarrow \lambda L, \qquad E \rightarrow \frac{1}{\lambda^{2}} E, </math> but preserves their product EL2. Therefore, the eccentricity Template:Mvar and the magnitude Template:Mvar are preserved, as may be seen from the [[#Derivation of the Kepler orbits|equation for Template:Math]] <math display="block">A^2 = m^2 k^2 e^{2} = m^2 k^2 + 2 m E L^2.</math>
The direction of Template:Math 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 Template:Mvar and the period Template:Mvar form a constant Template:Math.
Alternative scalings, symbols and formulationsEdit
Unlike the momentum and angular momentum vectors Template:Math and Template:Math, 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 Template:Math to obtain a dimensionless conserved eccentricity vector <math display="block"> \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 Template:Math is the velocity vector. This scaled vector Template:Math has the same direction as Template:Math 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 Template:Math by Template:Mvar alone <math display="block"> \mathbf{M} = \mathbf{v} \times \mathbf{L} - k\mathbf{\hat{r}}, </math> or by Template:Math <math display="block"> \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 Template:Math.
In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by Template:Math. Other common symbols for the LRL vector include Template:Math, Template:Math, Template:Math, Template:Math and Template:Math. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.
An alternative conserved vector is the binormal vector Template:Math studied by William Rowan Hamilton,<ref name="hamilton_1847_quaternions" /> Template:Equation box 1 which is conserved and points along the minor semiaxis of the ellipse. (It is not defined for vanishing eccentricity.)
The LRL vector Template:Math is the cross product of Template:Math and Template:Math (Figure 4). On the momentum hodograph in the relevant section above, Template:Math is readily seen to connect the origin of momenta with the center of the circular hodograph, and to possess magnitude Template:Math. At perihelion, it points in the direction of the momentum.
The vector Template:Math is denoted as "binormal" since it is perpendicular to both Template:Math and Template:Math. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.
The two conserved vectors, Template:Math and Template:Math can be combined to form a conserved dyadic tensor Template:Math,<ref name="fradkin_1967" /> <math display="block"> \mathbf{W} = \alpha \mathbf{A} \otimes \mathbf{A} + \beta \, \mathbf{B} \otimes \mathbf{B}, </math> where Template:Mvar and Template:Mvar are arbitrary scaling constants and <math>\otimes</math> represents the tensor product (which is not related to the vector cross product, despite their similar symbol). Written in explicit components, this equation reads <math display="block"> W_{ij} = \alpha A_i A_j + \beta B_i B_j. </math>
Being perpendicular to each another, the vectors Template:Math and Template:Math can be viewed as the principal axes of the conserved tensor Template:Math, i.e., its scaled eigenvectors. Template:Math is perpendicular to Template:Math , <math display="block"> \mathbf{L} \cdot \mathbf{W} = \alpha \left( \mathbf{L} \cdot \mathbf{A} \right) \mathbf{A} + \beta \left( \mathbf{L} \cdot \mathbf{B} \right) \mathbf{B} = 0, </math> since Template:Math and Template:Math are both perpendicular to Template:Math as well, Template:Math.
More directly, this equation reads, in explicit components, <math display="block"> \left( \mathbf{L} \cdot \mathbf{W} \right)_j = \alpha \left( \sum_{i=1}^3 L_i A_i \right) A_j + \beta \left( \sum_{i=1}^3 L_i B_i \right) B_j = 0. </math>
See alsoEdit
ReferencesEdit
Further readingEdit
- {{#invoke:citation/CS1|citation
|CitationClass=web }}
- {{#invoke:citation/CS1|citation
|CitationClass=web }}
- {{#invoke:citation/CS1|citation
|CitationClass=web }} Updated version of previous source.