Quantum n-body problem: generalized Euler coordinates
(from J-L Lagrange to Figure-Eight by C Moore and K A Ter-Martirosyan, then and today)

Alexander Turbiner (ICN-UNAM)

RESUMEN: The potential of the $n$-body problem, both classical and quantum, depends only on the relative (mutual) distances between bodies. By generalized Euler coordinates we mean relative distances and possible angles. Their advantage over Jacobi coordinates is emphasized. The NEW IDEA is to study both trajectories in classical system, and eigenstates in quantum system which depends on relative distances between bodies ALONE. We show how this study is equivalent to the study of (i) the motion of a particle (quantum or classical) in curved space of dimension $n(n-1)/2$ or the study of (ii) the Euler-Arnold (quantum or classical) - $sl(n(n-1)/2)$ algebra top. The curved space of (i) has a number of remarkable properties. In the 3-body case the {\it de-Quantization} procedure of quantum Hamiltonian leads to a classical Hamiltonian which solves a ~250-years old problem posed by Lagrange on 3-body planar motion.