In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where aD α t x(t)) and 0 < α < 1, such that the following is the corresponding Euler-Lagrange tD α b ( c aD α t )x(t) + b(t, x(t))( c aD α t x(t)) + f(t, x(t)) = 0. (1) At last, exact solutions for some Euler-Lagrange equations ar...

In this paper, we exploit symmetry properties of multi-agent robot systems to design control laws that preserve connectivity while swarming. We start by showing that the connectivity controller is invariant under the action of the special Euclidean group SE(3) and therefore is amenable to reduction of the dynamics by this action. We then utilize the reduced Euler-Lagrange equations that split t...

The definition and properties of the Euler-Lagrange cohomology groups H EL , 1 6 k 6 n, on a symplectic manifold (M2n, ω) are given and studied. For k = 1 and k = n, they are isomorphic to the corresponding de Rham cohomology groups H1 dR(M 2n) and H dR (M 2n), respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomo...

We prove a version of the variational Euler–Lagrange equations valid for functionals defined on Fréchet manifolds, such as spaces sections differentiable vector bundles appearing in various physical theories.

This paper presents the formulation of time-fractional Klein-Gordon equation using the Euler-Lagrange variational technique in the Riesz derivative sense and derives an approximate solitary wave solution. Our results witness that He’s variational iteration method was very efficient and powerful technique in finding the solution of the proposed equation. The basic idea described in this paper is...

The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version. Submitted to: J. Phys. A: Math. Gen. PACS numbers: 45.10.Na 02.40.Yy The introduction of numbers as coordinates. . . is an act of violence Hermann Weyl An increasi...

A new variational principle for an anisotropic elastic formulation in stress space is constructed, the Euler–Lagrange equations of which are the equations of compatibility (in terms of stress), the equilibrium equations and the traction boundary condition. Such a principle can be used to extend recently obtained configurational balance laws in stress space to the case of anisotropy.

The Euler-Lagrange equations of recently introduced chiral action principles are discussed using Lie algebra-valued differential forms. Symmetries of the equations and the chiral description of Einstein’s vacuum equations are presented. A class of Lagrangians, which contains the chiral formulations, is exhibited. † Mathematics Department, King’s College London, Strand, London WC2R 2LS, UK 1