Regularization of Hamilton's principle for higher-order Lagrangian systems

SEBASTIÁN J. FERRARO

Rio de Janeiro

Encuentro; IV Iberoamerican Meeting on Geometry, Mechanics and Control; 2014

IMPA (Instituto Nacional de Matemática Pura e Aplicada)

Let L: T^{(k)}Q -> R be a Lagrangian function defined on the k-th order tangent bundle of the configuration manifold Q. A trajectory of the corresponding unconstrained higher-order Lagrangian system is a C^k curve q: [0,h] -> Q such that the action $int_0^h L( q(t), dot q(t), ..., q^{(k)}(t)) dt$ has a critical point at q with respect to variations arising from deformations of q with fixed endpoint values for q and its first k-1 derivatives. For h=0, the constrained problem becomes nonregular since the constraint $q mapsto (q(0), dot q(0), ..., q^{(k-1)}(0); q(h), dot q(h), ..., q^{(k-1)}(h))$ is not a submersion. For a regular Lagrangian, meaning that ({partial^2 L}/{partial q^{(k)2}}) is a regular matrix, we use a procedure similar to that given by G. Patrick for first-order systems (Lagrangian mechanics without ordinary differential equations, Rep. Math. Phys. 57, no. 3, 437--443, 2006) to regularize the variational principle at h=0. From this we obtain the existence and uniqueness of solutions for small enough, positive h. This is important, for instance, when studying geometric integrators for higher-order systems, since it follows that the exact discrete Lagrangian is well-defined.