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Symmetry transformation group arising from the Laplace-Runge-Lenz vector

Stephen C. Anco, Mahdieh Gol Bashmani Moghadam

TL;DR

The paper derives explicit dynamical symmetry transformations in the Kepler problem arising from the directional LRL vector and from the full LRL vector, staying within physical kinematic variables. It shows that the directional LRL yields an abelian R^3 subalgebra, which, together with rotations, forms the SO(3) ⋊ R^3 group, while the full LRL structure yields energy-dependent isometry groups (SO(4), SO(3,1), or SO(3) ⋊ R^3) depending on the sign of the energy. Poisson-bracket analysis reveals the exact algebraic relations among E, L^i, A^i, and the direction Θ^i, and the paper provides explicit transformation formulas for r, v, and t under these symmetries. These results offer a concrete, solution-space–based realization of LRL-induced dynamical symmetries and lay groundwork for generalizations to other central-force problems and relativistic contexts.

Abstract

The Kepler problem in classical mechanics exhibits a rich structure of conserved quantities, highlighted by the Laplace--Runge--Lenz (LRL) vector. Through Noether's theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The present work derives the infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector, and obtains the explicit form of the symmetry transformations that it generates. When combined with the rotation symmetries,the resulting symmetry group is shown to be the semi-direct product of $SO(3)$ and $R^3$. This stands in contrast to the $SO(4)$ symmetry group generated by the LRL symmetries and the rotations. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quanties is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated.

Symmetry transformation group arising from the Laplace-Runge-Lenz vector

TL;DR

The paper derives explicit dynamical symmetry transformations in the Kepler problem arising from the directional LRL vector and from the full LRL vector, staying within physical kinematic variables. It shows that the directional LRL yields an abelian R^3 subalgebra, which, together with rotations, forms the SO(3) ⋊ R^3 group, while the full LRL structure yields energy-dependent isometry groups (SO(4), SO(3,1), or SO(3) ⋊ R^3) depending on the sign of the energy. Poisson-bracket analysis reveals the exact algebraic relations among E, L^i, A^i, and the direction Θ^i, and the paper provides explicit transformation formulas for r, v, and t under these symmetries. These results offer a concrete, solution-space–based realization of LRL-induced dynamical symmetries and lay groundwork for generalizations to other central-force problems and relativistic contexts.

Abstract

The Kepler problem in classical mechanics exhibits a rich structure of conserved quantities, highlighted by the Laplace--Runge--Lenz (LRL) vector. Through Noether's theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The present work derives the infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector, and obtains the explicit form of the symmetry transformations that it generates. When combined with the rotation symmetries,the resulting symmetry group is shown to be the semi-direct product of and . This stands in contrast to the symmetry group generated by the LRL symmetries and the rotations. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quanties is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated.

Paper Structure

This paper contains 11 sections, 7 theorems, 111 equations, 1 table.

Table of Contents

  1. Introduction
  2. Infinitesimal symmetry generators
  3. Infinitesimal symmetry generators from the constants of motion
  4. Poisson brackets
  5. Symmetry action on constants of motion
  6. Dynamical symmetry algebra
  7. Symmetry group generated by the directional LRL vector
  8. LRL symmetry transformation group
  9. Concluding remarks
  10. Proof of Theorem \ref{['thm:directionLRL.group']}
  11. Proof of Theorem \ref{['thm:LRL.group']}

Key Result

Proposition 1

A function $C(t,{\vec{r}},\dot{{\vec{r}}})$ is a conserved quantity for the Kepler problem, namely $\dot{C} = 0$ for all solutions of the equations of motion eom, if and only if the generator ${\bf X} ={\vec{P}}(t,{\vec{r}},\dot{{\vec{r}}})\cdot\partial_{{\vec{r}}}$ is an infinitesimal dynamical sym

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Proposition 4
  • Theorem 2
  • Theorem 3