In mathematical physics, Hamiltonian field theory usually means the symplectic Hamiltonian formalism when applied to classical field theory, that takes the form of the instantaneous Hamiltonian formalism on an infinite-dimensional phase space, and where canonical coordinates are field functions at some instant of time. This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum gauge theory.
The true Hamiltonian counterpart of classical first order Lagrangian field theory is covariant Hamiltonian field theory where canonical momenta pμi correspond to derivatives of fields with respect to all world coordinates xμ. Covariant Hamilton equations are equivalent to the Euler-Lagrange equations in the case of hyperregular Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton–De Donder, polysymplectic, multisymplectic and k-symplectic variants. A phase space of covariant Hamiltonian field theory is a finite-dimensional polysymplectic or multisymplectic manifold.
Hamiltonian non-autonomous mechanics is formulated as covariant Hamiltonian field theory on fiber bundles over the time axis, i.e. the real line ℝ.