A description of Lagrangian and Hamiltonian formalisms strictly obtained from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used, in order to formulate the canonical symplectic structure, are borrowed from the Cartan's theory of differential systems on associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler-Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems.
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