Stefan Rauch-Wojciechowski

When knowledge of one integral of motion is sufficient for integrability?

A system of differential equations is integrable by quadratures when solutions can be expressed using integrations, algebraic operations and taking inverse functions. A general autonomous system of $n$ equations requires knowledge of $n-1$ integrals of motion and one extra integration determines time dependence of solutions.

Usually the notion of integrability is associated with Hamiltonian integrability in $2n$ dimensional phase space when knowledge of only $n$ independent and involutive integrals of motion is sufficient for Liouville integrability. This is due to the special nature of the vector-field which is determined by one function, the Hamiltonian.

But there are known systems of equations when only $2$ or $1$ integral of motion suffice for integrability due to special algebraic form of the equation. There is a trade off between number of integrals and algebraic features of equations.

The purpose of this talk is to make you aware of elegant, little known classes of $n$ $2^{nd}$ order Newton equations for which only $1$ quadratic in velocities integral of motion implies existence of further $n-1$ integrals. This renders equations integrable and solvable by quadratures through separation of variables. These equations have the triangular form: $d^2q_r/dt^2= M_r(q_1, \ldots , q_r)$, $r=1,\ldots ,n$ where the $r^{th}$ equation depends only on the preceding variables $q_j$, $j=1, \ldots, r$.

XXXIX Workshop on Geometric Methods in Physics	19.06–25.06.2022
XI School on Geometry and Physics	27.06–1.07.2022

Event sponsored by:
University of Bialystok