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On regularization of variational problems in first-order field theory. (English) Zbl 0979.35005
Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15-22, 2000. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 133-140 (2001).
Summary: Standard Hamiltonian formulation of field theory is founded upon the Poincaré-Cartan form. Accordingly, a first-order Lagrangian \(L\) is called regular if \(\text{det}\left(\frac{\partial^2 L}{\partial y^\sigma_i\partial y^\nu_j}\right)\neq 0\); in this case the Hamilton equations are equivalent with the Euler-Lagrange equations. Keeping the requirement on equivalence of the Hamilton and Euler-Lagrange equations as a (geometric) definition of regularity, and considering more general Lepagean equivalents of a Lagrangian than the Poincaré-Cartan equivalent, we obtain a regularity condition, depending not only on a Lagrangian but also on 2-contact parts of its Lepagean equivalents. In this way one gets a possibility to “regularize” many Lagrangian systems which are singular in the standard sense – this concerns e.g. all Lagrangians linear in the first derivatives of the field variables, among others the Dirac field Lagrangian. Also, with help of the present procedure, one can generate new regularity conditions for Lagrangians. Some examples of such regularity conditions, differing from the standard one, are stated explicitly.
For the entire collection see [Zbl 0961.00020].
35A15 Variational methods applied to PDEs
49N60 Regularity of solutions in optimal control
58Z05 Applications of global analysis to the sciences