(in Polish) Mechanika Geometryczna II - Teoria Pola 1100-MGTP
This course presents the geometric foundations of classical and modern field theories, with a particular focus on multisymplectic structures, k-symplectic and k-contact formalisms, and the role of Lie groupoids and algebroids. Each chapter is illustrated with selected physical models.
Chapter I (6h) — Vector, Principal, and Associated Bundles
Vector bundles as field configuration spaces. Principal bundles and associated bundles. Connections, curvature, and holonomy. Interpretation of fields as sections of bundles. Electromagnetism as a connection theory on the principal U(1) bundle. Yang–Mills models. Spinor fields on associated bundles.
Chapter II (5h) — Lie Groupoids and Algebroids
Definition of a Lie groupoid. Lie algebroid. Algebroid morphisms. Reduction by symmetry. Relative structures. Gauge symmetries, reduction of field theory, models with constraints.
Chapter III (7h) — Variational Formalism of Field Theory
Jet bundles of the first and higher orders. Hamilton's principle. Euler–Lagrange equations. Poincaré–Cartan form. Higher-order theories. Wave equation, scalar field, elasticity, Euler–Bernoulli beam.
Chapter IV (5h) — Multisymplectic Geometry
Multisymplectic manifolds. Hamilton–De Donder–Weyl equations. Covariant description of field dynamics. Scalar field, electromagnetism, sigma models, Noether currents.
Chapter V (4h) — k-Symplectic Formalism
k-Symplectic structures. HDW equations. Symmetries and moment map. (3+1) models, multi-time formalisms, continuum mechanics, reductions.
Chapter VI (3h) — k-Contact Formalism
Generalization of contact geometry. Dissipative systems. Linear damping, gradient flows, reaction-diffusion equations.
Main fields of studies for MISMaP
physics
Mode
Prerequisites (description)
Course coordinators
Learning outcomes
The student knows the geometric foundations of field theory, the formalism of jet bundles, and multisymplectic, k-symplectic, and k-contact structures, and is able to interpret partial differential equations as Hamiltonian equations, taking into account symmetries and reductions.
Assessment criteria
Exercises (50 points: 40 for the colloquium + 10 for activity), half the points allows you to take the written exam.
- Written exam (50 points), passing allows you to take the oral exam.
- Oral exam (50 points).
The final grade is the average of the grades from the colloquium and exams. Grades can be improved through a make-up exam and a final paper.
Final grade
[45.50, 50] - 5!
[41, 45[ - 5
[37, 41[ - 4.5
[33, 37[ - 4
[29, 33[ - 3.5
[25, 29[ - 3
< 25 - 2
Colloquium
Written exam
Oral exam
Written make-up exam
Oral make-up exam
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: