Equivariance, Variational Principles, and the Feynman Integral

TL;DR

This paper proposes replacing traditional variational calculus with equivariance and invariance principles, exploring their implications for deriving physical laws and connecting to Feynman's integral in quantum mechanics.

Contribution

It introduces a novel approach to variational calculus based on equivariance, offering insights into the foundations of Lagrangian theories and their relation to Feynman's integral.

Findings

Equivariance-based methods can replace variational calculus in deriving Euler's equations.

Invariance conditions can derive physical laws without action integrals.

Speculations on the connection between Lagrangian theories and Feynman's integral.

Abstract

We argue that the variational calculus leading to Euler's equations and Noether's theorem can be replaced by equivariance and invariance conditions avoiding the action integral. We also speculate about the origin of Lagrangian theories in physics and their connection to Feynman's integral.