Poisson ceometery and some physical applications

Abstract

In this research we use the Lie symmetries in Hamilton's principle to derive symmetry – reduced equations of motion and analyze their solutions. We investigate that the Legendre transformation provides the Hamiltonian formulation of these equations in terms of Lie – Poisson brackets with some examples. We present the Euler – Poincare' equations , and then the standard Euler – Poincare' examples are treated. Also we discuss the semidirect – product Euler – Poincare' reduction theorem for the ideal fluid dynamics , with applications to geophysical fluid.