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Some new aspects of first integrals and symmetries for central force dynamics

### Abstract

For the general central force equations of motion in n > 1 dimensions, a complete set of 2n first integrals is derived in an explicit algorithmic way without the use of dynamical symmetries or Noether’s theorem. The derivation uses the polar formulation of the equations of motion and yields energy, angular momentum, a generalized Laplace-Runge-Lenz vector, and a temporal quantity involving the time variable explicitly. A variant of the general Laplace-Runge-Lenz vector, which generalizes Hamilton’s eccentricity vector, is also obtained. The physical meaning of the general Laplace-Runge-Lenz vector, its variant, and the temporal quantity is discussed for general central forces. Their properties are compared for precessing bounded trajectories versus non-precessing bounded trajectories, as well as unbounded trajectories, by considering an inverse-square force (Kepler problem) and a cubically perturbed inverse-square force (Newtonian revolving orbit problem).

Published by AIP Publishing.

Received 28 August 2015
Accepted 14 May 2016
Published online 03 June 2016

Acknowledgments:
S.C.A. is supported by an NSERC research Grant. Georgios Papadopoulos is thanked for stimulating discussions on this work.

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
III. DERIVATION AND PROPERTIES OF POLAR FIRST INTEGRALS
A. Normalization (“zero-point” values)
B. Evaluation using turning points
C. Piecewise property (“multiplicities”) and trajectory shapes
IV. EXAMPLES
A. Inverse-square force
1. Turning points for the Kepler problem
2. Inertial points for the Kepler problem
B. Inverse square force with cubic corrections
1. Turning points for the Newtonian revolving orbit problem
2. Inertial points for the Newtonian revolving orbit problem
V. SYMMETRY FORMULATION
A. Transformation of first integrals under dynamical symmetries
B. Method of extended point symmetries for finding first integrals and hidden dynamical symmetries
VI. FIRST INTEGRALS IN *n* DIMENSIONS
A. General Laplace-Runge-Lenz vector
VII. EXAMPLES OF *n*-DIMENSIONAL GENERAL LAPLACE-RUNGE-LENZ VECTOR
A. Inverse-square force
B. Inverse-square force with cubic corrections
VIII. CONCLUDING REMARKS

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2016-06-03

2016-10-24

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