Variational Principles for Lie—Poisson and Hamilton—Poincare Equations

Variational Principles for Some Nonlinear Wave Equations

Generally speaking, there exist two basic ways to describe a physical problem [1]: (1) by differential equations (DE) with boundary or initial conditions; (2) by variational principles (VP). The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration...

Dual Variational Principles for Diffusion Equations

Variational Principles for Lie–poisson and Hamilton–poincaré Equations

As is well-known, there is a variational principle for the Euler–Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie–Poisson equations on g∗, the dual of g, and also to generalize this construction. The more general situation...

Generalized Variational Principles on Oscillation for Nonlinear Nonhomogeneous Differential Equations

and Applied Analysis 3 Lemma 2.1 see 10 . Suppose that X and Y are nonnegative. Then λXYλ−1 −X ≤ λ − 1 Y, λ > 1, 2.1 where the equality holds if and only if X Y . Now, we will give our main results. Theorem 2.2. Assume (S2) holds. Suppose further that for any T ≥ t0, there exist T ≤ s1 < t1 ≤ s2 < t2 such that e t ⎧ ⎨ ⎩ ≤ 0, t ∈ s1, t1 , ≥ 0, t ∈ s2, t2 . 2.2 Let u ∈ C1 si, ti , and nonnegative...

New variational principles for locating periodic orbits of differential equations.

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes...