Lax forms and the Euler top equations: A new approach

We consider the problem of torque-free spinning of a rigid body in the context of Lax representation from which the linearization of the nonlinear Euler top equations naturally arises. The Lax equation with Hermitian matrices leads to the two equivalent pictures of quantum mechanics, namely, the Schrödinger and Heisenberg pictures. We derive a 3×3 Hamiltonian matrix based on principal moments of inertia and the Jacobi elliptic functions for the case of a 3-dimensional free rotation. We show generalization of our work for the nn-dimensional case.

► We derive new Lax forms of the Euler top equations.
► We obtain matrix mechanics formulation of rigid body dynamics.
► We derive rotational Hamiltonian in a matrix form in three dimensions.
► Generalization of the matrix formulation of the rigid body dynamics is given.