# Sub-symmetries II. Sub-symmetries and Conservation Laws

## TL;DR

This paper explores the relationship between sub-symmetries and conservation laws in differential systems, demonstrating their significance in generating infinite conservation laws, especially for fluid dynamics equations like the Euler equations.

## Contribution

It establishes a connection between sub-symmetries and conservation laws for non-Lagrangian systems, extending Noether's theorem, and applies these concepts to Euler equations to find new infinite conservation laws.

## Key findings

- Sub-symmetries are crucial in decoupling systems and deforming conservation laws.
- An analogue of Noether's theorem for sub-symmetries is proved.
- Infinite sub-symmetries generate infinite conservation laws for Euler equations.

## Abstract

In our previous paper, the concept of sub-symmetry of a differential system was introduced, and its properties and some applications were studied. It was shown that sub-symmetries are important in decoupling a differential system, and in the deformation of a system's conservation laws, to a greater extent than regular symmetries. In this paper, we study the nature of a correspondence between sub-symmetries and conservation laws of a differential system. We show that for a large class of non-Lagrangian systems, there is a natural association between sub-symmetries and local conservation laws based on the Noether operator identity, and we prove an analogue of the first Noether Theorem for sub-symmetries. We also demonstrate that infinite conservation laws containing arbitrary functions of dependent variables can be generated by infinite sub-symmetries through the Noether operator identity. We discuss the application of sub-symmetries to the incompressible Euler equations of fluid dynamics. Despite the fact that infinite symmetries (with arbitrary functions of dependent variables) are not known for the Euler equations, we show that these equations possess infinite sub-symmetries. We calculate infinite sub-symmetries with arbitrary functions of dependent variables for the two-and three-dimensional Euler equations in the velocity and vorticity formulations with certain constraints. We demonstrate that these sub-symmetries generate known series of infinite conservation laws, and obtain new classes of infinite conservation laws