Much effort in recent years has gone into generalizing the classical Hamiltonian and Euler-Lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints. These generalizations, in which nonsmoothness abounds and gradients are systematically replaced by subgradients, have succeeded in furnishing necessary condi...

The goal of this paper is to demonstrate how the internal symmetry of the N-body celestial-mechanics problem can be exploited in orbit calculation. We start with summarising research reported in (Efroimsky 2002, 2003; Newman & Efroimsky 2003; Efroimsky & Goldreich 2003) and develop its application to planetary equations in non-inertial frames. This class of problems is treated by the variationo...

-Using the He’s variational iteration method, it is possible to find the exact solutions or better approximate solutions of the partial differential equations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. In this paper, this method is used for solving a nonlinear partial differential equation, a three-di...

We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of this variational principle, we get the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical ...

We investigate the differentiability of minimal average energy associated to the functionals S ε(u) = ∫ Rd 1 2 |∇u|2 + εV(x, u) dx, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to det...

We obtain the equations of motion for a Lagrangian dynamical system under nonholonomic constraints making use of the D’Alembert principle. We show that the Lagrange multipliers can be expressed in terms of the Poisson bracket of the Hamiltonian and the constraint. This appealing result greatly simplifies the derivation of the equations of motion. The existence of at least two conserved quantiti...

A numerical method for solving nonlinear Fredholm-Volterra integral equations is presented. The method is based upon Lagrange functions approximations. These functions together with the Gaussian quadrature rule are then utilized to reduce the Fredholm-Volterra integral equations to the solution of algebraic equations. Some examples are included to demonstrate the validity and applicability of t...

The objective of this paper is path planning of a 3 DOF planer robot with hydraulic actuator using genetic algorithm. First the geometric and kinematic parameters of robot were established. The equations of motion are derived by Lagrange method. We proposed the model for proportional valve and hydraulic actuators. Then using the genetic algorithm we minimized the hydraulic energy consumption as...

Let X be a closed smooth 4-manifold. In the Theory of the SeibergWitten Equations, the configuration space is Cα = Aα × Γ (S + α ), where Aα is a space of u1-connections defined on a complex line bundle over X and Γ (S α ) is the space of sections of the positive complex spinor bundle over X. The original SWα-equations are 1 -order PDE fitting into a variational principle SWα : Aα × Sα → R, whi...

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...