A direct reformulation of the Hamiltonian formalism in terms of the intrinsic geometry of infinitely prolonged differential equations is obtained. Concepts of spatial equation and spatial-gauge symmetry of a Lagrangian system of equations are introduced. A non-covariant canonical variational principle is proposed and demonstrated using the Maxwell equations as an example. A covariant canonical variational principle is formulated. The results obtained are applicable to any variational equations, including those that do not originate in physics.
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Kostya Druzhkov (2024) studied this question.
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