Construction of the classical time crystal Lagrangians from Sisyphus dynamics and duality description with the Li\'enard type equation

TL;DR

This paper investigates the relationship between Sisyphus dynamics and Lie9nard equations, deriving classical time crystal Lagrangians from higher-order Lagrangians and exploring velocity-dependent mass functions with potential cosmological implications.

Contribution

It establishes a connection between Sisyphus dynamics and Lie9nard equations, deriving time crystal Lagrangians from higher-order frameworks and analyzing velocity-dependent mass functions.

Findings

Sisyphus dynamics relate closely to Lie9nard-II equations.

Velocity-dependent mass functions appear in Sisyphus dynamics.

Potential links to cosmological time crystals are discussed.

Abstract

We explore the connection between the equations describing Sisyphus dynamics and the generic Li\'{e}nard type or Li\'{e}nard equation from the viewpoint of branched Hamiltonians. The former provides the appropriate setting for classical time crystal being derivable from a higher order Lagrangian. However it appears the equations of Sisyphus dynamics have a close relation with the Li\'{e}nard-II equation when expressed in terms of the `velocity' variable. Another interesting feature of the equations of Sisyphus dynamics is the appearance of velocity dependent "mass function" in contrast to the more commonly encountered position dependent mass. The consequences of such mass functions seem to have connections to cosmological time crystals .