Dynamics-dependent symmetries in Newtonian mechanics

TL;DR

This paper identifies two novel symmetries in one-dimensional Newtonian mechanics involving complex, potential-dependent time transformations, and explores their connection to conserved quantities and quantum mechanics.

Contribution

It introduces two new symmetries in Newtonian mechanics that relate solutions through nonlinear, potential-dependent time transformations, and analyzes their conserved quantities.

Findings

Invariant-potential symmetry is linked to energy conservation.

The symmetries involve nonlinear transformations of time.

The invariant-potential symmetry is not a symmetry of the Schrödinger equation.

Abstract

We exhibit two symmetries of one-dimensional Newtonian mechanics whereby a solution is built from the history of another solution via a generally nonlinear and complex potential-dependent transformation of the time. One symmetry intertwines the square roots of the kinetic and potential energies and connects solutions of the same dynamical problem (the potential is an invariant function). The other symmetry connects solutions of different dynamical problems (the potential is a scalar function). The existence of corresponding conserved quantities is examined using Noethers theorem and it is shown that the invariant-potential symmetry is correlated with energy conservation. In the Hamilton-Jacobi picture the invariant-potential transformation provides an example of a field-dependent symmetry in point mechanics. It is shown that this transformation is not a symmetry of the Schroedinger equation.