Lagrangian–Hamiltonian unified formalism for field theory

Abstract

The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems ͑including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.͒. In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations.

Unified formalism for nonautonomous mechanical systems

“…Cortés et al [3] used the Skinner and Rusk formalism to consider vakonomic mechanics and the comparison between the solutions of vakonomic and nonholonomic mechanics. Finally, in [4,9,15] the Skinner-Rusk model was developed for classical field theories.…”

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

“…Euler-Lagrange and Hamilton-De Donder-Weyl multivector fields can be recovered from these Lagrange-Hamiltonian multivector fields (see [24]). …”

“…As an example of application of these formalisms we consider a classical system which has been taken from [24]: minimal surfaces (in R 3 ). Other examples of application of the multisymplectic formalism are explained in detail in [39,43,91] as well as in many other references (see, for instance, [16,25,26,27,28,30,69,71] and quoted references).…”

“…This formalism has been generalized for non-autonomous mechanics [7,20,45] and recently for classical field theories [24,71]. The main features of this formulation are explained in Section 4.…”

Bv Quantization of Covariant (Polysymplectic) Hamiltonian Field Theory

“…The Hamiltonian counterpart of classical first-order Lagrangian field theory is covariant Hamiltonian formalism which is developed in the polysymplectic, multisymplectic and Hamilton -De Donder variants (see [1,2,3,4,5] and references therein). In order to quantize covariant Hamiltonian field theory, one usually attempts to construct the multisymplectic generalization of a Poisson bracket [6,7,8].…”