The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time.
For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field.
For one scalar field φ(x, t), the Hamiltonian density is defined from the Lagrangian density by[nb 1] with ∇ the "del" or "nabla" operator, x is the position vector of some point in space, and t is time.
It is the field analogue to the Lagrangian function for a system of discrete particles described by generalized coordinates.
The corresponding dimension is [energy][length]−3, in SI units Joules per metre cubed, J m−3.
is the variational derivative Under the same conditions of vanishing fields on the surface, the following result holds for the time evolution of A (similarly for B): which can be found from the total time derivative of A, integration by parts, and using the above Poisson bracket.
), Taking the partial time derivative of the definition of the Hamiltonian density above, and using the chain rule for implicit differentiation and the definition of the conjugate momentum field, gives the continuity equation: in which the Hamiltonian density can be interpreted as the energy density, and the energy flux, or flow of energy per unit time per unit surface area.
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.
[2] This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum gauge theory.
Covariant Hamiltonian field theory is developed in the Hamilton–De Donder,[4] polysymplectic,[5] multisymplectic[6] and k-symplectic[7] variants.
A phase space of covariant Hamiltonian field theory is a finite-dimensional polysymplectic or multisymplectic manifold.