We will show that after reduction the ODE describing the Lagrange top (possibly with an added harmonic potential term) with appropriately time-dependent moments of inertia and/or potential can be transformed into the Painlevé V equation. It is well known that the Painlevé equations are Hamiltonian with time-dependent Hamiltonians, but the link to the Lagrange top appears to be new. The connection appeared through the quantisation of the Lagrange top. The Schrodinger equation for the Lagrange top is a confluent limit of a Fuchsian equation, specifically the confluent Heun equation. Painlevé V can be thought of as a "de-quantisation" of the confluent Heun equation, and for the Lagrange top these statements can be made precise.
Alain Albouy, Alain Chenciner, Jacques Laskar