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.

In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange–d’Alembert principle, a process that better appro...

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...

Abstract. We study the composite membrane problem in all dimensions. We prove that the minimizing solutions exhibit a weak uniqueness property which under certain conditions can be turned into a full uniqueness result. Next we study the partial regularity of the solutions to the Euler– Lagrange equation associated to the composite problem and also the regularity of the free boundary for solutio...

Abstract In this paper, we present a new simple method for solving two integral equations of Love’s type that have many applications, especially in electrostatic systems. The approach the solution is based on an innovative technique using matrix algebra barycentric Lagrange interpolation. unknown function expressed through product four matrices. kernel interpolated twice, so get it five Additio...