Moving Frames and Noether's Conservation Laws - the General Case

TL;DR

This paper extends the use of moving frames to derive Noether's conservation laws for variational problems where independent variables are affected by symmetry groups, with applications to fluid dynamics.

Contribution

It generalizes previous methods to cases where independent variables participate in the symmetry action, including explicit examples with SL(2) and SL(3).

Findings

Derived invariantized Euler-Lagrange equations for general cases.

Obtained conservation laws in terms of invariants and the adjoint representation.

Applied results to fluid problems conserving potential vorticity.

Abstract

In recent works, the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.