De Donder Construction for Higher Jets

TL;DR

This paper extends the De Donder construction to higher jets, enabling a generalized approach to boundary forms that is applicable in continuum mechanics and calculus of variations, ensuring invariance of physical laws and conservation laws.

Contribution

It introduces a generalized De Donder construction for higher jets, demonstrating boundary form independence and invariance of conservation laws across different boundary choices.

Findings

Boundary forms depend on the chosen coordinate system.

Splitting of forces is independent of boundary form choice.

Constants of motion are unaffected by boundary form variations.

Abstract

In this paper, we generalize De Donder approach to construct boundary forms that depend on the adapted coordinate system used. In continuum mechanics, use of boundary forms leads to splitting of the total force acting on the body into body force and surface traction. Moreover, this splitting is independent of the choice of the boundary form used. In calculus of variations, use of boundary forms leads to equations in exterior differential forms that are equivalent to the Euler-Lagrange equations. Infinitesimal symmetries of the theory lead to conservation laws valid for any choice of the boundary form used. In an example, we show that the boundary conditions lead to independence of constants of motion of the choice of the boundary form.