On the conservation of energy: Noether's theorem revisited

Abstract

This paper studies the dynamics and conservation of energy. It evaluates the validity of Noether's theorem as a formal argument supporting the law of conservation of energy in physical systems. The analysis examines the role of nonconvexity in energy dynamics. The paper argues that nonconvexity can arise in the presence of catalytic effects or in situations of transitions between multiple regimes. With the introduction of nonconvexity, the analysis relies on a generalized Lagrangian and generalized Hamiltonian. The investigation applies under general conditions, allowing for multiple types of energy with dynamics driven by multiple state variables. Our key result is to show that the conservation of energy (Noether's theorem) holds under convexity but not under nonconvexity. This identifies situations where energy in isolated systems is not necessarily constant over time. By relaxing the law of conservation of energy, our analysis provides new insights into energy dynamics. It offers new directions for scientific inquiries, including improved understanding about the origin of life, the evolution of the early universe and the nature of space and time.