A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our...

where λ is the Lagrange multiplier and ψ = 0 the holonomic constraint. The first practical algorithm for constrained dynamics was called SHAKE [3]. In this algorithm the Lagrange multiplier was calculated from the viewpoint, that no exact discrete solution could be obtained. Thus the constraints were satisfied by inserting the discrete propagator into the holonomic constraint, ψ(t+ h) = 0. This...

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affin...

To accelerate molecular dynamics simulations, it is common to impose holonomic constraints on the hardest degrees of freedom. In this way, the time step used to integrate the equations of motion can be increased, thereby allowing longer total simulation times. The imposition of such constraints results in an aditional set of N(c) equations (the equations of constraint) and unknowns (their assoc...

In this paper, a numerical solution based on Haar wavelet quasilinearization (HWQ) is used for finding the solution of nonlinear Euler-Lagrange equations which arise from the problems in calculus of variations. Some examples of variational problems are given and outcomes compared with exact solutions to demonstrate the accuracy and efficiency of the method.

In this paper, we use Petrov-Galerkin elements such as continuous and discontinuous Lagrange-type k-0 elements and Hermite-type 3-1 elements to find an approximate solution for linear Fredholm integro-differential equations on $[0,1]$. Also we show the efficiency of this method by some numerical examples  

We extend Routh’s reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use ...

In 1798 J.-L. Lagrange published an extensive book on the solution of numerical equations. Lagrange had developed a general systematic algorithm for detecting, isolating, and approximating all real and complex roots of a polynomial equation with real coeecients, with arbitrary precision. In contrast to Newton's Method, Lagrange's algorithm is guaranteed to converge. We discuss some lesser known...

As is often the case, some results get rediscovered over time. In particular, some rather striking results of Lagrange are often recreated. For instance, in [6], a result pertaining to the Pell equation for a prime discriminant was recast in the light of nonabelian cohomology groups. Yet, in [1], the authors acknowledged the fact that the result “has been discovered before,” and provided an ele...

This paper applies the recently developed theory of discrete nonholonomic mechanics to the study of discrete nonholonomic left-invariant dynamics on Lie groups. The theory is illustrated with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation of the reduced energy by the discrete flow is observed and the discrete momentum conse...