Variational order for forced Lagrangian systems

Abstract

We are able to derive the equations of motion for forced mechanical systems in a purely variational setting, both in the context of Lagrangian or Hamiltonian mechanics, by duplicating the variables of the system as introduced by Galley (2013 Phys. Rev. Lett. 110 174301) and Galley et al (2014( arXiv:1412). We show that this construction is useful to design high-order integrators for forced Lagrangian systems and, more importantly, we give a characterization of the order of a method applied to a forced system using the corresponding variational order of the duplicated one.

Contact variational integrators

“…• a second order variational but non-contact (VNC) method for forced Lagrangian systems [16] obtained by a Verlet discretization of a Lagrangian in duplicated phase space [22],…”

Lagrangian reduction of nonholonomic discrete mechanical systems by stages

“…, N − 2. As, in this case, ν η d ((W k , e), (W k+1 , e))(D η (W k+1 ,e) ) =ν η d ((W k , e), (W k+1 , e))(−dR W k+1 (e)(d)) 10) where…”

“…Indeed, a first step would be to tackle this same problem with no constraints, that is, for discrete Lagrange-Poincaré systems ( [14]). It should be noted that this error analysis is only known for unconstrained systems (see [29]) and forced mechanical systems (see [10] and [12]). Another avenue for exploration would be the study of possible Poisson structures in DLDPSs: even though DLDPSs do not have a canonical Poisson structure, some of them do (those coming from discrete mechanical systems, for instance) and it would be interesting to see how those structures behave under the reduction process.…”

Error analysis of forced discrete mechanical systems

“…As we are assuming that L j CP,h are smooth, by Proposition 1 in [6], L j CP are smooth 11 . Similarly, as f 2…”

“…We are also able to transfer the error analysis developed for systems defined on T Q to the systems on Q × Q (Theorem 5.18). It is worth mentioning that another approach to this problem has been considered by D. Martín de Diego and R. Sato Martín de Almagro in [11], where they associate a non-forced mechanical system to the forced one and, then, apply the results of Cuell & Patrick to the former.…”