Euler-Lagrange equations and geometric mechanics on Lie groups with potential

Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a rigid body with a fixed point in the gravitational field. This Mechanical systems is usually know as symmetric heavy top. Then we show that the extracted equations by this theory coincide with the known equations of heavy top. Finally, as an infinite dimensional example, we study the Camassa-Holm equations on Bott-Virasoro group at the presence of potential. Bott-Virasoro group is the product of the group of diffeomorphisms of the circle of Sobolev class by the real line and by a potential on a Lie group G we mean a differentiable function from G to the real line R.