Canonical theories of Lagrangian dynamical systems in physics

Kastrup H.A.

By reformulating the variational problem for a given classical Lagrangian field theory in the framework of differential forms, one can show (Lepage) that for m a2 independent and for n>2 dependent (field) variables z” =f(x) a much wider variety of Legendre transformations v% – ditf(x)-*Pa, L^>H, exists than has been employed in physics. The different canonical theories for a given Lagrangian can be classified according to the rank of the corresponding basic canonical m-form. Each such canonical theory leads to a Hamilton-Jacobi theory, the “wave fronts” of which are transversal to solutions of the field equations. Two canonical theories are discussed in more detail: The one by DeDonder and Weyl which employs the conventional canonical momenta pi = dL/dVp and the more sophisticated one by Caratheodory, the HJ theory of which is more intimately related to that of mechanics than the conventional one. Generalizing results from mechanics one can show that each solution of a HJ equation which depends on a parameter generates a conserved current for those extremals which are transversal to that wave front. The geometrically very rich, but algebraically rather complicated canonical formalism of Caratheodory provides interesting new approaches for the “qualitative” analysis of classical field theories. For instance: solutions of the field equations which give a vanishing Lagrangian density L are associated with singularities in the transversality relations between wave fronts and extremals. A number of examples (strings, gauge theories etc.) illustrates the wealth of possible physical applications of these more general canonical formalisms for field theories, which, up to now, have been ignored almost completely by physicists.