A class of Liouville-integrable Hamiltonian systems with two degrees of freedom (Q2738225)

The Liouville integrability of two-dimensional Hamiltonian systems with the rather special Hamilton function \(H= \frac{1}{2} p_1^2+ \frac{1}{2} p_2^2+ V(q_1,q_2)\) is studied. The following conditions are shown to be equivalent: (i) The Hamiltonian vector field corresponding to \(H\) admits the bi-Hamiltonian representation defined by constant Poisson bi-vectors \(\sum \partial_i \wedge \partial^i\) and \(\sum \partial_j \wedge \partial^k\) \((i,j,k=1,2\); \(j\neq k)\), (ii) \(H\) is Liouville-integrable, (iii) \(V\) satisfies the wave equation, (iv) the dynamics of \(H\) can be described by the Lax pair \(L_t= LM-ML\) for certain \(4\times 4\) matrices \(L,M\) (they are involved and cannot be stated here). Three examples are discussed: \(V= (q_1-q_2)^{-2}\) (the Carogero-Moser model), \(V= Cq_1^2+ Dq_2^2+ Aq_1q_2^2- \frac{1}{3}Bq_1^3\) (the Hénon-Heiles potential), \(V= \exp(q_1-q_2)\) (the Toda lattice).