Theory of Differential Equations: Pfaff equations and applications
The problem that became known as Pfaff’s problem had its origins in the theory of first order partial differential equations, which as a general theory began with Euler, Monge and Lagrange. Nowadays the problem of Pfaff is characterized more specifically as the problem of determining for a given Pfaffian equation ω = 0 in n variables the maximal dimension d of its integral manifolds. This problem is a keystone in all the modern integrability theories in classical mechanics: Euler-Jacoby, Liouville, Lie, Frobenius, Darboux, Cartan and Yang-Baxter theories. We plan to discuss the classical Pfaff theory and its modern applications in Hamiltonian and nonholonomic mechanics.
Lecturer
Andrey Tsiganov
Date
18th September, 2026 ~ 8th January, 2027
Reference
1. Lie S., Theorie der Transformationsgruppen, unter Mitwirkung von Dr. Friedrich Engel, vol.1-3, Leipzig, 1888.
2. Lie S., Scheffers G., Vorlesungen uber continuierliche Gruppen mit geometrischen und anderen Anwendungen, vol.1-3, Leipzig, B G Teubner, 1893.
3. Forsyth A.R., Theory of Differential Equations, vol 1-6, Cambridge University Press (1890-1906), reprinted by Dover Public., New York (1959).
4. Cartan E., Lecons sur les invariants integraux, Hermann, Paris, 1922.
5. Olver P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., vol. 107, New York: Springer, 1993.
6. Gorbatsevich V., Onishchik A., Vinberg E., Structure of Lie groups and Lie algebras, English transl. in Encycl. Math Sc. 41, Springer-Verlag, Berlin, Heidelberg, 1994.
7. A.Agrachev, D. Barilari, U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics Publisher: Cambridge University, 2019.
8. E. Le Donne, Carnot-Carathéodory spaces from the Lie group viewpoint, 2024.
9. E. Le Donne, Lecture notes on Lie groups - from the differential view point, 2019.
2. Lie S., Scheffers G., Vorlesungen uber continuierliche Gruppen mit geometrischen und anderen Anwendungen, vol.1-3, Leipzig, B G Teubner, 1893.
3. Forsyth A.R., Theory of Differential Equations, vol 1-6, Cambridge University Press (1890-1906), reprinted by Dover Public., New York (1959).
4. Cartan E., Lecons sur les invariants integraux, Hermann, Paris, 1922.
5. Olver P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., vol. 107, New York: Springer, 1993.
6. Gorbatsevich V., Onishchik A., Vinberg E., Structure of Lie groups and Lie algebras, English transl. in Encycl. Math Sc. 41, Springer-Verlag, Berlin, Heidelberg, 1994.
7. A.Agrachev, D. Barilari, U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics Publisher: Cambridge University, 2019.
8. E. Le Donne, Carnot-Carathéodory spaces from the Lie group viewpoint, 2024.
9. E. Le Donne, Lecture notes on Lie groups - from the differential view point, 2019.
Video Public
Yes
Notes Public
Yes
Lecturer Intro
Andrey Tsiganov currently works at the Department of Computational Physics, Saint Petersburg State University, Russia. His main research interests are integrable and superintegrable systems in classical and quantum mechanics, nonholonomic and vakonomic mechanics, geometry and topology of dynamical systems, see profile at https://www.researchgate.net/profile/Andrey-Tsiganov. He is one of the organizers of the BIMSA Integrable System Seminar, see https://researchseminars.org/seminar/BIMSA-ISS and https://sites.google.com/view/bimsa-iss.