Kovalevskaya top – an elementary approach

The goal of this note is to give an elementary and very short solution to equations of motion for the Kovalevskaya top [1]. For this, we use some results from the original papers by Kovalevskaya [1], Kötter [2] and Weber [3] and also the Lax representation from the note [4]. 1. The Kovalevskaya top [1] is one of the most beautiful examples of integrable systems. This is the top for which the principal momenta of inertia J1, J2, J3 satisfy the relation J1 = J2 = 2J3 = J, (1) and the center of mass lies in the equatorial plane of the body (for the simplicity, we put further J = 1). The dynamical variables are components m1, m2, m3 of angular momentum and components n1, n2, n3 of the center mass vector in the system related to the principal axes of the body. This system is Hamiltonian relative to the Poisson structure for the Lie algebra e(3) of motion of three-dimensional Euclidean space {mi, mj} = εijk mk, {mi, nj} = εijk nk, {ni, nj} = 0, (2) where εijk is a standard totally skew-symmetric tensor. The Hamiltonian has the form H = 1 2 ( m21 +m 2 2 + 2m 2 3 − n1 ) (3) ∗On leave of absence from the Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia. Current E-mail address: [email protected]