A Novel Variational Principle arising from Electromagnetism

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.

💡 Research Summary

The paper addresses a fundamental limitation of classical optimal‑control and variational methods when applied to nonlinear LC circuits. While Pontryagin’s maximum principle and standard Euler‑Lagrange formulations are well‑suited for problems whose extremals are either minima or maxima, many nonlinear LC networks exhibit a richer set of critical points—including saddle points and other non‑extremal stationary solutions—especially when additional integral constraints (such as total charge or energy preservation) are imposed.

To overcome this, the authors revisit an example circuit from reference