Conic Pseudo-Finslerian Mechanical Systems

We develop a geometric framework for conservative mechanical systems modeled on conic pseudo-Finslerian manifolds, extending classical Riemannian and semi-Riemannian mechanics to anisotropic geometries. In this setting, we define a Jacobi-type pseudo-Finslerian metric and demonstrate that motions of fixed energy correspond, up to reparametrization, to its geodesics. We establish energy and momentum conservation under suitable symmetry assumptions via a generalized Noether's theorem, and provide new analytical criteria for the completeness of both the Jacobi metric and the mechanical system. Furthermore, for mechanical systems with holonomic and non-holonomic constraints, we prove the existence and uniqueness of reaction forces satisfying d'Alembert's principle. These results provide a unified geometric foundation for constrained and unconstrained anisotropic dynamics.