De Donder-Weyl theory

In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theory in the way that a field is represented as a system that varies both in space and in time. This generalization is different from the canonical Hamiltonian formalism in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.

The De Donder–Weyl theory is based on a change of variables known as Legendre transformation. Let xⁱ be spacetime coordinates, for i = 1 to n (with n = 4 representing 3 + 1 dimensions of space and time), and y^a field variables, for a = 1 to m, and L the Lagrangian density

With the polymomenta pⁱ_a defined as

and the De Donder–Weyl Hamiltonian function H defined as

the De Donder–Weyl equations are:

This De Donder-Weyl Hamiltonian form of field equations is covariant and it is equivalent to the Euler-Lagrange equations when the Legendre transformation to the variables pⁱ_a and H is not singular. The theory is a formulation of a covariant Hamiltonian field theory which is different from the canonical Hamiltonian formalism and for n = 1 it reduces to Hamiltonian mechanics (see also action principle in the calculus of variations).

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