# Conformal Anisotropic Mechanics

## TL;DR

This paper introduces a classical mechanics model with anisotropic scale transformations, generalizing conformal mechanics, and explores its symmetries, thermodynamics, and relation to gravity at Lifshitz points.

## Contribution

It develops a classical mechanics framework incorporating anisotropic scaling, extending conformal mechanics, and derives associated symmetries and thermodynamic properties.

## Key findings

- Model consistent with gravity dispersion relations at Lifshitz points
- Constructed dynamical symmetries for arbitrary z
- Derived Boltzmann distribution compatible with anisotropic scaling

## Abstract

In this paper we implement scale anisotropic transformations in the space-time in classical mechanics. The resulting system is consistent with the dispersion relation of gravity at a Lifshitz point recently considered in arXiv:0901.3775. Also, we show that our model is a generalization of the conformal mechanics of Alfaro, Fubini and Furlan. For arbitrary $z$ we construct the dynamical symmetries that correspond to the Schroedinger algebra. Furthermore, we obtain the Boltzman distribution for a gas of free particles compatible with anisotropic scaling transformations and compare our result with the corresponding thermodynamics of the recent anisotropic black branes proposed in arXiv:0907.4755.