The Monge problem: From quadrature-free integration of underdetermined nonlinear ODEs to efficient car parking

Abstract: This talk is about an old topic of finding closed-form solutions of UNDERDETERMINED systems of nonlinear ordinary differential equations, started by G.Monge in 1784 and later followed by Goursat (1905), Hilbert (1913) and Cartan (1914).

In the last decades of the XX century these problems draw attention of specialists in nonlinear control. In particular, the technique of this problem was used in developing motion algorithms for nonholonomic mechanical systems, a typical example being a car with N trailers. Parking such a "car train" moving back is a popular difficult task! Modern results based on the old investigations of Goursat make automatic control of such vehicles possible.

For those interested in the problem of integration of ODEs and PDEs: using the results described one can often remove (unnecessary) quadratures in the final expressions for the complete solution of a C-integrable nonlinear PDEs.

We expose the classical results by Cartan and Hilbert showing the intruiging details of the Monge problem.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods