dr hab. Javier de Lucas Araujo, prof. UW

This course introduces some of the main geometric structures appearing in the description of classical mechanics and classical field theories. Basic knowledge on differential geometry is assumed:  e.g. vector and principal bundles, Poincare lemma, calculus on manifolds,. A little more than half of the course covers some standard topics in symplectic, Poisson, and Dirac geometry, while the final parts are aimed at structures appearing in recent research problems that have been attracting much interest lately, such as contact and multisymplectic geometry, along with their Marsden-Weinstein reductions. In fact, contact geometry seems to have had a revival in recent years, and there is much interest in its use to study Hamiltonian systems. Moreover, its analogue for field theories, the k-contact geometry, has been drawing some interest too. I will also devote some time to modern attempts to devise a multisymplectic reduction. The multisymplectic reduction problem has been open for around 30 years now, and it is worth noting that multisymplectic geometry, which draws quite much attention, was devised for the first time at the Department of Mathematical Methods in Physics of the University of Warsaw by Tulczyjew and Kijowski.

 

Chapter 1 - Symplectic Geometry (6 hours): Basic definitions, Poisson algebras, Types of submanifolds: Lagrangian/Coisotropic/Isotropic/Symplectic, Momentum maps, Marsden-Weinstein reductions 

Chapter 2 - Poisson geometry (7 hours): Generalising Symplectic Geometry, Poisson bivectors  and Schouten brackets,  Symplectic stratification, Normal form for Poisson bivectors, Poisson Hamiltonian systems, Poisson reduction

Chapter 3 - Dirac Geometry (5 hours): Dirac geometry as a generalisation of symplectic and Poisson geometry, Dirac manifolds as manifolds stratified by presymplectic manifolds, Dirac brackets, Dirac reduction, Jacobi manifolds

Chapter 4 - Modern trends and challenges (12 hours): Contact manifolds, Multisymplectic manifolds, k-symplectic and k-contact manifolds: old and new approaches, Modern open problems and related questions: multisymplectic reduction.

 

Literature:

 

R. Abraham and J.E. Marsden, Fundamentals of Mechanics, Benjamin, Reading, Mass. 1978.

C. Albert, Le theoreme de reduction de Marsden-Weinstein en geometrie cosymplectique et de contact, J. Geom. Phys. 6, 4 (1989).

H. Bursztyn, A brief introduction to Dirac manifolds, in: Geometric and Topological Methods for Quantum Field Theory. Proceedings of the 2009 Villa de Leyva Summer School (Villa de Leyva, 2009),Cambridge University Press, Cambridge,

2013, pp. 4–38.

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, Springer, Heidelberg, 2004.

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319, 631–661 (1990).

P.A.M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Math. 2, 129–148 (1950).

A. Echeverrıa-Enrıquez, M.C. Munoz Lecanda, and N. Roman-Roy, Remarks on multisymplectic reduction, Rep. Math. Phys. 81, 415–424 (2018).

E García-Toraño Andrés and T. Mestdag, Conditions for symmetry reduction of polysymplectic and polycosymplectic structures, J. Phys. A 56, 335202 (2023).

P. Libermann and C.M. Marle, Symplectic geometry and analytical mechanics, in: Mathematics and its Applications Vol. 35, D. Reidel Publishing Co., Dordrecht, 1987.

J. de Lucas, X. Gràcia, X. Rivas, N. Román-Roy, S. Vilariño, Reduction and reconstruction of multisymplectic Lie systems, J. Phys. A  55, 295204 (2022).

C.-M. Marle, The Schouten-Nijenhuis bracket and interior products, J. Geom. Phys. 23, 350–359 (1997).

J.E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121--130 (1974). 

J. E. Marsden and A. Weinstein, Comments on the history, theory, and applications of symplectic reduction. In Quantization of singular symplectic quotients, volume 198 of Progr. Math., pages 1-19. Birkhauser, Basel, 2001. 

J. Ortega and T. Ratiu, Momentum maps and Hamiltonian reduction, in: Progress in Mathematics 222. Birkhauser Boston Inc., Boston, 2004.

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118, Birkhauser Verlag, Basel, 1994.