3 Lagrange’s Equations of Motion 9 3.1 Lagrange’s Equations Via The Extended Hamilton’s Principle . . . . . . . . . . . . . . . . 9 3.2 Rayleigh’s Dissipation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Kinematic Requirements of Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Lagrange Equation Examples . . . . . . . . . . . . . . . ...

We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

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...

We use the fractional integrals to describe fractal solid. We suggest to consider the fractal solid as special (fractional) continuous medium. We replace the fractal solid with fractal mass dimension by some continuous model that is described by fractional integrals. The fractional integrals are considered as approximation of the integrals on fractals. We derive fractional generalization of the...

The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler-Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler-Lagrange equation ...

In 1940 (and 1964) S. M. Ulam proposed the well-known Ulam stability problem. In 1941 D. H. Hyers solved the Hyers-Ulam problem for linear mappings. In 1992 and 2008, J. M. Rassias introduced the Euler-Lagrange quadratic mappings and the JMRassias “product-sum” stability, respectively. In this paper we introduce an Euler-Lagrange type quadratic functional equation and investigate the JMRassias ...

In this paper we study the existence of a solution in Lloc(Ω) to the Euler-Lagrange equation for the variational problem (0.1) inf ū+W 1,∞ 0 (Ω) Z Ω (1ID(∇u) + g(u))dx, with D convex closed subset of Rn with non empty interior. By means of a disintegration theorem, we next show that the Euler-Lagrange equation can be reduced to an ODE along characteristics, and we deduce that there exists a sol...

A certain functional–difference equation that Runyon encountered when analyzing a queuing system was solved in a combined effort of Morrison, Carlitz, and Riordan. We simplify that analysis by exclusively using generating functions, in particular the kernel method, and the Lagrange inversion formula.

We prove higher integrability properties of solutions to the problem of minimizing ∫ Ω L(x, u(x),∇u(x))dx, where ξ → L(x, u, ξ) is a convex function satisfying some additional conditions. As an application, we prove the validity of the Euler–Lagrange equation for a class of functionals with growth faster than exponential.