We consider Lagrangian systems in the limit of infinitely many particles. It is shown that the corresponding discrete action functionals Γ-converge to a continuum action functional acting on probability measures of particle trajectories. Also, the convergence of stationary points of the action is established. Minimizers of the limiting functional and, more generally, limiting distributions of s...

We consider fractional isoperimetric problems of calculus of variations with double integrals via the recent modified Riemann–Liouville approach. A necessary optimality condition of Euler–Lagrange type, in the form of a multitime fractional PDE, is proved, as well as a sufficient condition and fractional natural boundary conditions. M.S.C. 2010: 49K21, 35R11.

We use analytic continuation to derive the Euler-Lagrange equations associated to the Pfaffian in indefinite signature (p, q) directly from the corresponding result in the Riemannian setting. We also use analytic continuation to derive the Chern-Gauss-Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. Complex metrics on...

In this work, we communicate the topic of complex Lie algebroids based on the extended fractional calculus of variations in the complex plane. The complexified Euler–Lagrange geodesics and Wong’s fractional equations are derived. Many interesting consequences are explored.

The aim of the paper is to announce some recent results concerning Hamiltonian theory. The case of second order Euler–Lagrange form non-affine in the second derivatives is studied. Its related second order Hamiltonian systems and geometrical correspondence between solutions of Hamilton and Euler–Lagrange equations are found.

This paper considers the problem of designing explicit measurement feedback H∞ control laws for a class of Euler-Lagrange systems. For these systems the joint positions are assumed as outputs of the system, while velocity measures are to be estimated from an observer+controller structure. The main contribution of this work lies in the explicit formulation of the dynamic structure of a joined ob...

In the Lagrangian framework for symmetries and conservation laws of field theories, we investigate globality properties of conserved currents associated with non–global Lagrangians admitting global Euler–Lagrange morphisms. Our approach is based on the recent geometric formulation of the calculus of variations on finite order jets of fibered manifolds in terms of variational sequences.

We begin by reporting on some recent results of the authors (Frederico and Torres, 2006), concerning the use of the fractional Euler-Lagrange notion to prove a Noether-like theorem for the problems of the calculus of variations with fractional derivatives. We then obtain, following the Lagrange multiplier technique used in (Agrawal, 2004), a new version of Noether’s theorem to fractional optima...

We give twenty eight diverse proofs of the fundamental Euler sum identity ζ(2, 1) = ζ(3) = 8 ζ(2, 1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating the wide variety of techniques fruitfully used to study such sums and the attraction of their study.

We establish a group-invariant version of the variational bicomplex that is based on a general moving frame construction. The main application is an explicit group-invariant formula for the Euler-Lagrange equations of an invariant variational problem.