Lagrangian reduction of nonholonomic discrete mechanical systems by stages

Javier Fernández, ,Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A., Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP, Argentina, Cora Tori, Marcela Zuccalli, ,Depto. de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata, Calle 116 entre 47 y 48, 2$ ^{\underline{o}} $ piso, La Plata, Buenos Aires, 1900, Argentina, Centro de Matemática de La Plata (CMaLP), ,Depto. de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calles 50 y 115, La Plata, Buenos Aires, 1900, Argentina, Centro de Matemática de La Plata (CMaLP)
2020 Journal of Geometric Mechanics (JGM)  
In this work we introduce a category LDP d of discrete-time dynamical systems, that we call discrete Lagrange-D'Alembert-Poincaré systems, and study some of its elementary properties. Examples of objects of LDP d are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincaré systems. We also introduce a notion of symmetry group for objects of LDP d and a process of reduction when symmetries are present. This reduction process extends
more » ... reduction process of discrete Lagrange-Poincaré systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in LDP d to the system obtained by a one-stage reduction by the full symmetry group. 2020 Mathematics Subject Classification. Primary: 37J06, 70G45; Secondary: 70G75. Key words and phrases. Geometric mechanics, discrete nonholonomic mechanical systems, symmetry and reduction. This research was partially supported by grants from Universidad Nacional de Cuyo (#06/C567 and #06/C574) and Universidad Nacional de La Plata. 607 608 JAVIER FERNÁNDEZ, CORA TORI AND MARCELA ZUCCALLI add constraints -in the form of a non-integrable subbundle D ⊂ T Q-to the variational principle (see, for instance, [3] and [6]). Numerical integrators and discrete mechanical systems. As in many applications it is essential to predict the evolution of a mechanical system, the equations of motion that can be derived from the corresponding variational principle must be solved. Solving these ordinary differential equations can be quite difficult in practice, so numerical integrators are used to find approximate solutions to those equations. The standard methods for numerically approximating solutions of ODEs do not necessarily preserve the structural characteristics of the solutions of the equations of motion of mechanical systems (see [18] ). Discrete mechanical systems were introduced as a way of modeling discrete-time analogues of mechanical systems; the evolution of a discrete mechanical system is also defined in terms of a variational principle for the discrete Lagrangian L d : Q × Q → R; this formalism is extended to deal with more general systems, including forced discrete systems as well as discrete nonholonomic ones (see [24] and [9]). The equations of motion of discrete mechanical systems are algebraic equations whose solutions are numerical integrators for the corresponding continuous system. In many cases, these integrators have very good structural characteristics (especially when considering long-time evolution), that resemble those of the continuous system ([29] and [18]). Symmetries and symmetry reduction. It is a natural idea to think that when a mechanical or, more generally, a dynamical system has some degree of symmetry, it should be possible to gain some insight into its dynamics by studying some other "simplified system" obtained by eliminating or locking the symmetry. This process is usually known as the reduction of the given system and the resulting system is known as the reduced system. In the case of Classical Mechanics, this idea seems to go back as far as the work of Lagrange. Over time, it has become a technique that has been applied in both the Lagrangian and Hamiltonian formalisms, for unconstrained systems as well as for holonomically and nonholonomically constrained ones (see among many other references, [3] [6], [2], [30, 31] [28], [25], [26], [4] and [23]). The reduction process has also been applied to discrete-time mechanical systems with and without constraints (see, for instance, [21], [27], [20] and [13]). It is well known that, in most instances, the reduction of a mechanical system is not a mechanical system but, rather, a more general dynamical system: that is, while the dynamics of a mechanical system on Q is defined using a variational principle for the Lagrangian, defined on T Q in the continuous case or on Q × Q in the discrete case (and, maybe, other additional data), the dynamics of the reduced system is determined by a function that is usually not defined on a tangent bundle or a Cartesian product (of a manifold with itself). This can be problematic if one expects to analyze the reduced system with the same techniques as the original one. That issue has usually been solved by passing from the family of mechanical systems to a larger class of dynamical systems, where there is a reduction process that is closed within this larger class. Such is the case, for example, of mechanical systems on Lie algebroids and Lie groupoids (see [19] and [20]). In this paper we follow this guiding principle, but choose the larger class following ideas adapted from [7] and [14] . Reduction by stages. Sometimes, it may be convenient to eliminate part of the symmetric behavior of a mechanical system, while keeping some residual symmetry
doi:10.3934/jgm.2020029 fatcat:nuqtrzcpdrhxpffybj7st6k4ca