# Dynamical Similarity

## TL;DR

This paper explores the concept of dynamical similarities in Lagrangian systems, establishing how invariants form autonomous contact systems that simplify the dynamics, with applications to cosmology and classical mechanics.

## Contribution

It introduces a framework linking symmetries of dynamical systems to invariant contact systems, providing new methods to analyze and reduce complex Lagrangian dynamics.

## Key findings

- Invariant subalgebra of phase space exists for systems with dynamical similarities.
- The invariants evolve as a contact system with a friction-like term.
- Contact Hamiltonian and one-form are invariants that reproduce invariant dynamics.

## Abstract

We examine "dynamical similarities" in the Lagrangian framework. These are symmetries of an intrinsically determined physical system under which observables remain unaffected, but the extraneous information is changed. We establish three central results in this context: i) Given a system with such a symmetry there exists a system of invariants which form a subalgebra of phase space, whose evolution is autonomous; ii) this subalgebra of autonomous observables evolves as a contact system, in which the friction-like term describes evolution along the direction of similarity; iii) the contact Hamiltonian and one-form are invariants, and reproduce the dynamics of the invariants. As the subalgebra of invariants is smaller than phase space, dynamics is determined only in terms of this smaller space. We show how to obtain the contact system from the symplectic system, and the embedding which inverts the process. These results are then illustrated in the case of homogeneous Lagrangians, including flat cosmologies minimally coupled to matter; the n-body problem and homogeneous, anisotropic cosmology.