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Concept
Hamiltonian field theory
Summary
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory.
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. Since each point mass has one or more degrees of freedom, the field formulation has infinitely many degrees of freedom.
The Hamiltonian density is the continuous analogue for fields; it is a function of the fields, the conjugate "momentum" fields, and possibly the space and time coordinates themselves. For one scalar field φ(x, t), the Hamiltonian density is defined from the Lagrangian density by
with ∇ the "del" or "nabla" operator, x is the position vector of some point in space, and t is time. The Lagrangian density is a function of the fields in the system, their space and time derivatives, and possibly the space and time coordinates themselves. It is the field analogue to the Lagrangian function for a system of discrete particles described by generalized coordinates.
As in Hamiltonian mechanics where every generalized coordinate has a corresponding generalized momentum, the field φ(x, t) has a conjugate momentum field π(x, t), defined as the partial derivative of the Lagrangian density with respect to the time derivative of the field,
in which the overdot denotes a partial time derivative ∂/∂t, not a total time derivative d/dt.
For many fields φi(x, t) and their conjugates πi(x, t) the Hamiltonian density is a function of them all:
where each conjugate field is defined with respect to its field,
In general, for any number of fields, the volume integral of the Hamiltonian density gives the Hamiltonian, in three spatial dimensions:
The Hamiltonian density is the Hamiltonian per unit spatial volume.
Official source
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