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.
Let xi be spacetime coordinates, for i = 1 to n (with n = 4 representing 3 + 1 dimensions of space and time), and ya field variables, for a = 1 to m, and L the Lagrangian density With the polymomenta pia defined as and the De Donder–Weyl Hamiltonian function H defined as the De Donder–Weyl equations are:[1] 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 pia and H is not singular.
The work of De Donder on the other hand started from the theory of integral invariants of Élie Cartan.
[6] The De Donder–Weyl theory has been a part of the calculus of variations since the 1930s and initially it found very few applications in physics.
[9] In 1999 Igor Kanatchikov has shown that the De Donder–Weyl covariant Hamiltonian field equations can be formulated in terms of Duffin–Kemmer–Petiau matrices.