A General Variational Principle of Classical Field and Its Application to General relativity I

TL;DR

This paper introduces a comprehensive variational principle for classical fields with higher-order derivatives, deriving Noether's theorem and Hamilton-Jacobi equations, with applications to general relativity and conservation laws in curved spacetime.

Contribution

It presents a unified variational framework for classical fields with derivatives up to order N, extending Noether's theorem and Hamilton-Jacobi theory, and sets the stage for applications in general relativity.

Findings

Derived a general variational principle for classical fields.

Established a generalized Noether's theorem.

Formulated the Hamilton-Jacobi equation for the principal functional.

Abstract

A general variational principle of classical fields with a Lagrangian containing the field quantity and its derivatives of up to the N-th order is presented. Noether's theorem is derived. The generalized Hamilton-Jacobi's equation for the Hamilton's principal functional is obtained. These results are surprisingly in great harmony with each other. They will be applied to the general relativity in the subsequent articles, especially the generalized Noether's theorem will be applied to the problem of conservation and non-conservation in curved spacetime..