Analytic Mechanics of Locally Conservative Physical Systems

The analysis of the dynamics of a material point perfectly constrained to a submanifold of the three-dimensional euclidean space and subjected to a locally conservative force’s field, namely a force’s field corresponding to a closed but not necessarily exact differential form on such a submanifold, requires a generalization of the Lagrangian and the Hamiltonian formalism that is here developed.

💡 Research Summary

The paper introduces a systematic extension of classical Lagrangian and Hamiltonian mechanics to handle “locally conservative” force fields—forces that are described by closed but not necessarily exact differential forms on a submanifold (M\subset\mathbb{R}^3). In the usual formulation, a conservative force admits a global scalar potential (V(q)) such that (\mathbf{F}=-\nabla V); mathematically this means the associated 1‑form (\omega = \mathbf{F}\cdot d\mathbf{q}) is exact ((\omega = dV)). The author points out that on manifolds with non‑trivial first cohomology (e.g., a torus, a cylinder, or any multiply‑connected surface) a closed 1‑form need not be exact. Consequently, a global potential may not exist, yet the force remains “conservative” in the sense that the work around any contractible loop vanishes.

To treat such forces, the paper proceeds in several steps:

1. Mathematical Setting – The force field is encoded by a closed 1‑form (\omega) with (d\omega=0). Its cohomology class (