The Herglotz variational problem: Noether Theorems, Hamilton-Jacobi theory

Within the super-lagrangian context developed in Massa and Pagani [J. Math. Phys. 64, 102902 (2023)], a gauge-invariant definition of geometric symmetry of the Herglotz functional and a consequent formulation of Noether’s first theorem are proposed.Noether’s second theorem for Herglotz functionals admitting an infinite-dimensionalsymmetry group is analysed. The underdeterminacy of the resulting Euler-Lagrange equations is highlighted. A possible adaptation of the Hamilton-Jacobi theory to the study of the Herglotz variational problem is proposed. Keywords: Variational principles in Physics, Herglotz variational problem, Gauge structure of Lagrangian Mechanics, Noether theorems, Hamilton-Jacobi theory. 1