A Novel Variational Principle arising from Electromagnetism

TL;DR

This paper introduces a new variational principle tailored for nonlinear LC circuits, extending classical extremal principles and deriving a generalized Euler-Lagrange equation suitable for diverse critical points.

Contribution

A novel variational principle is established that replaces Pontryagin's maximum principle, applicable to all critical points in nonlinear LC circuits with arbitrary topology.

Findings

Derivation of a generalized Euler-Lagrange equation.

Formulation of canonical Hamiltonian systems via Legendre transformation.

Applicable to nonlinear LC circuits with complex topology and constraints.

Abstract

Analyzing one example of LC circuit in [8], show its Lagrange problem only have other type critical points except for minimum type and maximum type under many circumstances. One novel variational principle is established instead of Pontryagin maximum principle or other extremal principles to be suitable for all types of critical points in nonlinear LC circuits. The generalized Euler-Lagrange equation of new form is derived. The canonical Hamiltonian systems of description are also obtained under the Legendre transformation, instead of the generalized type of Hamiltonian systems. This approach is not only very simple in theory but also convenient in applications and applicable for nonlinear LC circuits with arbitrary topology and other additional integral constraints.