On the connections between symmetries and conservation rules of dynamical systems

TL;DR

This paper explores the deep relationship between symmetries and conservation laws in dynamical systems, highlighting how symmetries can be used to identify conserved quantities and how weaker symmetries influence conservation in Hamiltonian systems.

Contribution

It provides new insights into how different types of symmetries, including weaker forms like bb- and bLambda-symmetries, relate to conserved quantities in dynamical systems.

Findings

Symmetries can be used to derive conserved quantities in dynamical systems.

Weaker symmetries like bb- and bLambda-symmetries control deviations from conservation.

Examples demonstrate the application of symmetry-conservation relationships.

Abstract

The strict connection between Lie point-symmetries of a dynamical system and its constants of motion is discussed and emphasized, through old and new results. It is shown in particular how the knowledge of a symmetry of a dynamical system can allow to obtain conserved quantities which are invariant under the symmetry. In the case of Hamiltonian dynamical systems it is shown that, if the system admits a symmetry of "weaker" type (specifically, a \lambda\ or a \Lambda-symmetry), then the generating function of the symmetry is not a conserved quantity, but the deviation from the exact conservation is "controlled" in a well defined way. Several examples illustrate the various aspects.