The Lagrange-d’Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange-d’Alembert-Poincaré equations. In a previous contribution we have shown that both sets of equations fall in the category of so-called ‘Lagrangian systems on a subbundle of a Lie algebroid’. In this paper, we investigate the special case when the reduced system is again invariant under a new ...

A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on G×G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. In this context the reduction of the discrete Euler–Lagrange equations is shown to lead to th...

Hamel’s equations are an analogue of the Euler–Lagrange equations of Lagrangian mechanics when the velocity is measured relative to a frame which is not related to system’s local configuration coordinates. The use of this formalism often leads to a simpler representation of dynamics but introduces additional terms in the equations of motion. The paper elucidates the variational nature of Hamel’...

The classical relativistic wave equations are presented as partial difference equations in the arena of covariant discrete phase space. These equations are also expressed as difference-differential equations in discrete phase space and continuous time. The relativistic invariance and covariance of the equations in both versions are established. The partial difference and difference-differential...

This report provides additional results on local region statistics and accompanies the work in [2]. In particular, we derive the Euler-Lagrange equations of the local Gaussian region model including variance as well as the local nonparametric model. Moreover, we show some experimental results on contour tracking with local region statistics. 1 Euler-Lagrange Equations for Local Region Statistic...

Implicit Euler approximations of the equations governing the porous flow of two imiscible incompressible fluids are shown to be the Euler–Lagrange equations of a convex function. Tools from convex analysis are then used to develop robust fully discrete algorithms for their numerical approximation. Existence and uniqueness of solutions control volume approximations are established.

In this article we investigate the application of high order approximation techniques to one-dimensional boundary layer problems. In particular, we use second order differential equations and coupled second order differential equations as case studies. The accuracy and convergence rate of numerical solution obtained with Lagrange, Hermite, B-spline finite elements and C∞ generalized finite elem...

Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constrain...

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canoni-cal Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange form...

This paper reviews the moving frame approach to the construction of the invariant variational bicomplex. Applications include explicit formulae for the Euler-Lagrange equations of an invariant variational problem, and for the equations governing the evolution of differential invariants under invariant submanifold flows.