Extensions of Noether's Second Theorem: from continuous to discrete systems

Abstract

A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler--Lagrange equations of any variational problem whose symmetries depend upon a set of free or partly-constrained functions. Our approach extends further to deal with finite difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations.

Cited by in corpus (15)

• Structure-Preserving Geometric Particle-in-Cell Methods for Vlasov-Maxwell Systems
• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics I: Lie Dragging Approach
• Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs
• Potential Vorticity in Magnetohydrodynamics
• The geometric theory of charge conservation in particle-in-cell simulations
• Second Noether theorem for quasi-Noether systems
• Vorticity and Symplecticity in Multi-Symplectic Lagrangian Gas Dynamics
• Symmetries, conservation laws and Noether's theorem for differential-difference equations
• Multi-Symplectic Magnetohydrodynamics
• Moving Frames and Noether's Finite Difference Conservation Laws I
• Invariant Discretization of Partial Differential Equations Admitting Infinite-Dimensional Symmetry Groups
• Action Principles and Conservation Laws for Chew-Goldberger-Low Anisotropic Plasmas
• Transformations, symmetries and Noether theorems for differential-difference equations
• Characteristics of conservation laws for difference equations
• The second Noether theorem on time scale