CONSTRAINT ALGORITHM FOR SINGULAR FIELD THEORIES

It is well known that for systems of ODE’s describing singular dynamical systems, the existence and uniqueness of solutions are not assured. In many of these cases, there are geometrical constraint algorithms that, in the most favourable cases, give a maximal submanifold of the phase space of the system, where consistent solutions exist. The same problems arise when considering system of PDE’s associated with field theories described by singular Lagrangians (both in the Lagrangian and Hamiltonian formalisms), as well as in some other applications related with optimal control theories. Working in the framework of the multisymplectic description for this kind of theories, we present a geometric algorithm for finding the maximal submanifold where there are consistent solutions of the system, reducing the problem to another one in the realm of multilinear algebra. 1 MULTILINEAR THEORY 1.1 Statement of the problem W and E R-vector spaces. dim E = m, dim W = m+ n. σ: W → E surjective morphism, V(σ) = ker σ, : V(σ) ↪→W natural injection. η ∈ ΛmE ∗ volume element; ω = σt(η). Ω ∈ Λm+1W∗