Integrable Euler top and nonholonomic Chaplygin ball

We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball.

💡 Research Summary

The paper provides a comprehensive comparative study of two cornerstone models in classical mechanics: the integrable Euler top, a holonomic system describing the free rotation of a rigid body, and the non‑holonomic Chaplygin ball, a sphere rolling without slipping on a plane. The authors place both systems within the modern framework of integrable dynamics, focusing on Poisson geometry, Lax representations, r‑matrices, bi‑Hamiltonian structures, separation of variables, and related algebraic structures.

The first part revisits the Euler top. Starting from the inertia tensor (I) and the angular momentum vector (M), the Hamiltonian (H=\frac12 M\cdot I^{-1}M) is introduced together with the canonical Lie‑Poisson bracket ({M_i,M_j}=\varepsilon_{ijk}M_k). This defines the standard Poisson tensor (\omega_1) which is isomorphic to the Lie algebra (\mathfrak{so}(3)). The authors construct a Lax matrix (L(\lambda)=M\cdot\sigma+\lambda A\cdot\sigma) (where (\sigma) are Pauli matrices and (A) encodes the inertia parameters) and an (r)-matrix (r(\lambda,\mu)=\frac{C}{\lambda-\mu}P). They verify the classical Yang‑Baxter relation (