The time-evolution operator for singular Lagrangians
In the Lagrangian approach to classical dynamics, one is given a function \(L\) on the tangent bundle \(TQ\) of the configuration manifold \(Q\). In contrast to the Hamiltonian approach on the cotangent bundle \(T^*Q\) which has a canonical symplectic structure, there is no guarantee that the dynamics can be defined by a vector field on \(TQ\), unless \(L\) satisfies global regularity conditions. If some weaker conditions are satisfied, we get a constrained vector field, i.e., defined on some submanifold of \(TQ\). This vector field has the property that it can be translated into a constrained Hamiltonian vector field via the Legendre transformation \(FL: TQ\to T^*Q\) associated to \(L\). The authors give here an intrinsic characterization of an operator \(K\) defined by \textit{C. Batlle} et al. [J. Math. Phys. 27, No. 12, 2953--2962 (1986; Zbl 0613.70012)] for singular Lagrangian systems, in order to relate the Hamiltonian with the Lagrangian constraints. It gives roughly the time evolution of functions in \(T^*Q\) in terms of that of functions in \(TQ\). The idea is to work instead with the Whitney sum \(W=T^*Q\otimes TQ\) over \(Q\) and the dynamical formalism on \(W\) developed by \textit{R. Skinner} and \textit{R. Rusk} [J. Math. Phys. 24, 2589--2594 (1983; Zbl 0556.70012)], so that velocities and momenta are treated on an equal footing. The classical approach had the shortcoming that the constraint on \(TQ\) requiring the vector field to be a second order differential equation (equivalent to Euler-Lagrange equations) was not obtained without resorting to \(W\), because of not being projectable on \(TQ\).
scientific article; zbMATH DE number 4040874
scientific article