The Riemannian geometry of physical systems of curves

Summary The properties of a physical system S_k where k ≠−1, of ∞²ⁿ⁻¹ trajectories C. in a Riemannian space V_n are developed. The intrinsic differential equations and the equations of Lagrange, of a physical system S_k, are derived. The Lagrangian function L and the Hamiltonian function H, are studied in the conservative case. Also included are systems of the type (G), curvature trajectories, and natural families. The Appell transformation T of a dynamical system S ₀ in a Riemannian space V_n, is obtained. Finally, contact transformations and the transformation theory of a physical system S_k where k ≠−1, are considered in detail. To Enrico Bompiani on his scientific Jubilee Kasner,Differential geometric aspecte of dynamics, The Princeton Colloquium Lectures, 1909. Published by the ? American Mathematical Society, Providence, Rhode Island, 1913, and reprinted 1934.