A variant of Schwarzian mechanics

TL;DR

This paper explores a simplified variant of Schwarzian mechanics where the Schwarzian derivative is set to a constant, analyzing its stability, conserved charges, Hamiltonian structure, and embedding into a larger gravitational system.

Contribution

It introduces a new variant of Schwarzian mechanics with a fixed Schwarzian derivative, analyzes its stability, conserved quantities, and embeds it into a broader gravitational framework.

Findings

The system generally exhibits stable evolution.

A fixed point solution is only locally stable.

Conserved charges and Hamiltonian formulation are constructed.

Abstract

The Schwarzian derivative is invariant under SL(2,R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the Schwarzian mechanics for short. In this note, we consider the simplest variant which results from setting the Schwarzian derivative to be equal to a dimensionful coupling constant. It is shown that the corresponding dynamical system in general undergoes stable evolution but for one fixed point solution which is only locally stable. Conserved charges associated with the SL(2,R)-symmetry transformations are constructed and a Hamiltonian formulation reproducing them is proposed. An embedding of the Schwarzian mechanics into a larger dynamical system associated with the geodesics of a Brinkmann-like metric obeying the Einstein equations is constructed.