Geometric Methods in Physics 1

Garant předmětu:

Libor Šnobl

Lecturer:

Libor Šnobl

Tutor:

Ondřej Kubů, Libor Šnobl

Supervisor:

Department of Physics

Synopsis:

Foundations of geometric methods in physics on manifolds. Differential forms.

Requirements:

The course of theoretical physics (02TEF1, 02TEF2)

Syllabus of lectures:

1. Manifolds.

2. Tangent vectors, tangent spaces.

3. Tangent bundle, vector fields as its sections, integral curves, vector fields as derivation on the space of smooth functions, commutator.

4. Covectors, p-forms, corresponding fibre bundles.

5. Differential forms, wedge product, outer derivation, closed and exact forms.

6. Induced maps of tensorial objects.

7. Lie derivative.

8. Geometric formulation of Hamilton ́s mechanics, symplectic form, Hamiltonian vector fields, Poisson brackets, integrals of motion.

9. Orientation on a manifold, decomposition of a unit, integration of forms, Stokes ́ theorem on p-chains.

10. Metrics, affine connection and and curvature.

Syllabus of tutorials:

Solving problems on the following topics:

1. Manifolds.

2. Tangent vectors, tangent spaces.

3. Vector fields.

4. Covectors, forms.

5. Differential forms, wedge product, outer derivation.

6. Induced maps of tensorial objects.

7. Lie derivative.

8. Geometric formulation of Hamilton´s mechanics.

9. Integration of forms, Stokes´ theorem

10. Metrics and curvature.

Study Objective:

Knowledge:

Foundations of analysis on manifolds

Skills:

Application of geometrical methods in theoretical physics

Study materials:

Key references:

[1] L. W. Tu: Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics), Springer 2017

[2] Th. Frankel, The Geometry of Physics: An Introduction, Cambridge University Press 2011

Recommended references:

[3] M. Nakahara: Geometry, Topology and Physics, IOP Publishing, Bristol 1998

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• BS Matematické inženýrství - Matematická fyzika (compulsory course of the specialization)
• Matematické inženýrství - Matematická fyzika (PS)