Nonlocal Mechanics

We introduce a Hamiltonian framework for nonlocal Lagrangian systems without relying on infinite-derivative expansions. Starting from a (trajectory-based) variational principle and a generalized Noether theorem, we define the canonical momenta and energy. Moreover, we construct a (pre)symplectic form on the kinematic space, and show that its restriction to the phase space (by implementing the constraints) yields a true (pre)symplectic structure encoding the dynamics. Three examples – a finite nonlocal oscillator, the fully nonlocal Pais-Uhlenbeck model, and a delayed harmonic oscillator – demonstrate how phase space and the Hamiltonian emerge without explicitly solving the Euler-Lagrange equations.