Presymplectic Lagrangian systems subject to non-holonomic constraints (Q2778000)

This is a well-written paper in which systems with non-holonomic constraints are studied within the framework of a gauge-invariant formulation of Lagrangian mechanics [see, e.g., \textit{E. Massa, E. Pagani} and \textit{P. Lorenzoni}, Transport Theory Stat. Phys. 29, 69-91 (2000; Zbl 0968.70014)]. This formulation is based on the construction of two principal line bundles over the first jet space of the configuration space-time, which are called the Lagrangian and the co-Lagrangian bundle. In this setting, the concept of Lagrangian function is replaced by that of Lagrangian section, i.e., a section of the Lagrangian bundle. Gauge-equivalent Lagrangians then simply correspond to representations of the same Lagrangian section with respect to different gauges. Moreover, a Lagrangian section induces a connection on the co-Lagrangian bundle, whose curvature \(2\)-form equals (up to a sign) the Poincaré-Cartan \(2\)-form associated to the class of gauge-equivalent Lagrangians represented by the given Lagrangian section. NEWLINENEWLINENEWLINEIn the paper under review this gauge-invariant approach is extended to (time-dependent) Lagrangian systems subjected to non-holonomic constraints. It is shown that in the case of a regular Lagrangian this description is equivalent to the standard one. The analysis is further extended to non-holonomic systems with a singular Lagrangian. For this case, necessary and sufficient conditions are derived for the solvability of the constrained dynamics by applying a suitable implementation of the presymplectic constraint algorithm in the gauge-invariant formulation of mechanics, as developed previously by the author [J. Phys. A 33, 5117-5135 (2000; Zbl 0972.37046)]. Also, the so-called second-order differential equation problem is discussed within the present framework.