A variant of Schwarzian mechanics

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.

Super-Schwarzians via nonlinear realizations

“…As compared to the N = 1 case, the constraints (3.15) turn out to be more stringent and result in a variant of N = 2 super-Schwarzain mechanics in which the super-Schwarzian derivative is equal to a (coupling) constantpḡ −pg gḡ . The latter is an N = 2 analogue of the model studied recently in [11]. As was mentioned in the Introduction, our primarily concern in this work is to understand how the super-Schwarzian derivatives may be obtained within the method of nonlinear realizations.…”

The Schwarzian derivative and Euler--Lagrange equations

“…The derivative appears in the projective and conformal geometry as well as in many other contexts in mathematics and mathematical physics (see e.g. [1,2,10,11,15] and [16] for a survey). In this note we intend to exhibit its variational nature.…”

“…Consequently we interpret (1) as an Euler-Lagrange equation. Note that a variant of the Schwarzian mechanics has been studied in [10] where a Hamiltonian approach is developed. Our results split into two parts.…”

“…Remark. Solution u(t) for S(u) < 0 appeared in [10] as a starting point for the studies of Hamiltonian mechanics related to the Schwarzian.…”