Variational development of the semi-symplectic geometry of nonholonomic mechanics †

The geometry of constrained Lagrangian systems is developed using the Lagrange-d'Alembert principle, extending the variational approach of Marsden, Patrick and Shkoller [26] from holonomic to nonholonomic systems. It emerges that the instrinsic geometry of nonholonomic systems corresponds to the geometry of the distributional Hamiltonian systems of Śniatycki [35], here called semi-Hamiltonian. The principle physical reason that nonholonomic systems exhibit non-symplectic dynamics is exposed, leading to curvature conditions for the presence of holonomic subsystems. The underlying geometry of semi -Hamiltonian systems is semi-symplectic. An abstract exposition of the semi-symplectic category is developed. This is closed under a reduction scheme able to incorporate conserved quantities which are not momenta but are often structurally implied by symmetry. The variational development is continued to include nonlinear constraints irrespective of whether or not they are obtained from Chetaev's rule. Even though these systems are not semi-Hamiltonian, their geometry is still semi-symplectic.