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Course info KMA / 8GVAN : Course description

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Department/Unit / Abbreviation	KMA / 8GVAN			Academic Year	2023/2024
Academic Year	2023/2024
Title	Global Variational Analysis			Form of course completion	Exam
Form of course completion	Exam
Accredited / Credits	Yes, 15 Cred.			Type of completion	Oral
Type of completion	Oral
Time requirements	Lecture 2 [Hours/Week] Tutorial 2 [Hours/Week]			Course credit prior to examination	No
Course credit prior to examination	No
Automatic acceptance of credit before examination	No
Included in study average	NO
Language of instruction	Czech, English
Occ/max				Automatic acceptance of credit before examination	No
Summer semester	0 / 0	0 / 0	0 / 0	Included in study average	NO
Winter semester	0 / 0	0 / 0	0 / 0	Repeated registration	NO
Repeated registration	NO
Timetable	Yes			Semester taught	Winter + Summer
Semester taught	Winter + Summer
Minimum (B + C) students	not determined			Optional course	Yes
Optional course	Yes
Language of instruction	Czech, English			Internship duration	0
No. of hours of on-premise lessons				Evaluation scale	S|N
Periodicity	every year
Specification periodicity				Fundamental theoretical course	No
Fundamental course	No
Fundamental theoretical course	No
Evaluation scale	S|N
Substituted course	None
Preclusive courses	N/A
Prerequisite courses	N/A
Informally recommended courses	N/A
Courses depending on this Course	N/A
Histogram of students' grades over the years: Graphic PNG ,  XLS

Course objectives:
Study of variational functionals on fibred spaces, local and global properties, applications in geometry and in physics. Annotation: Variational funcionals on fibred manifolds, Lepage forms, Lagrange, Hamilton and Hamilton-Jacobi theory, symmetries and conservation laws, variational sequences and bicomplexes, the inverse problem of the calculus of variations, applications in first and higher order mechanics and field theory.
Requirements on student
Self-study, consultations. The course is completed with an oral exam.
Content
- Fibred manifolds and jet bundles, the contact structure, contact symmetries - Lepage forms, the Euler form, the first variation formula - Global Euler-Lagrange equations, Hamilton differential systems - Regular variational problems, regularization, dual jets, Legendre map - Jet fields, the geometric Hamilton-Jacobi theory, fields of extremals - Second variation, Jacobi fields - Harmonic maps, minimal immersions - Symmetries, reduction, conservation laws, Noether theorem - Applications (variational functionals in classical mechanics, higher order mechanics, continuum mechanics, hydrodynamics, elasticity, relativity, gauge theories, string theory, etc.) - Euler-Lagrange mapping, trivial Lagrangians, the inverse problem of the calculus of variations - Presheaves, sheaves on paracompact spaces, resolution, sheaf cohomology, Abstract De Rham theorem - The variational sequence, variational morphisms, Euler-Lagrange and Helmholtz map, the interior Euler operator, representations of the variational sequence - Symmetries in the variational sequence, symmetries of the Helmholtz form, applications to dynamical forms (nonvariational equations) - The variational bicomplex - Homogeneous variational problems in mechanics and field theory - Variational geometric structures: Riemannian and Finsler manifolds, sub-Riemannian geometry, nonholonomic geometry, geometric optimization.
Activities
Fields of study
Guarantors and lecturers
Guarantors: doc. Diana Schneiderová, PhD. (100%),  Lecturer: doc. Diana Schneiderová, PhD. (100%),  Tutorial lecturer: doc. Diana Schneiderová, PhD. (100%),
Literature
Basic: R. Bryant, P. Griffiths, D. Grossman. Exterior Differential Systems and Euler-Lagrange PDE's. The Univ. of Chicago Press, 2003. ISBN 0226077934. Basic: D. Krupka. Introduction to Global Variational Geometry. Atlantis Press, 2015. ISBN 978-94-6239-072-0. Basic: Saunders, D. J. The Geometry of Jet Bundles. Cambridge University Press, 2nd, 2005. ISBN 0-521-36948-7. Basic: O. Krupková. The Geometry of Ordinary Variational Equations. Springer, Berlin, 1997. ISBN 3540638326. Recommended: Y. Choquet-Bruhat, C. DeWitt-Morette. Analysis, Manifolds and Physics, II. Applictions.. North-Holland, 1989. ISBN 0444870717. Recommended: Olver, P. Applications of Lie Groups to Differential Equations. Springer, 2000. ISBN 0-387-95000-1. Recommended: M. Giaquinta, S. Hildebrandt. Calculus of Variations, I, II,. Springer, Berlin, 1997. ISBN 354050625X. Recommended: Krupka, D., Saunders D. J. Handbook of Global Analysis. Elsevier, 2008. ISBN 978-0-444-52833-9. Recommended: I. Kolář, P.W. Michor, J. Slovák. Natural Operators in Differential Geometry. Springer, 1993. ISBN 9783540562351. Recommended: G.E. Bredon. Sheaf Theory. Springer, 2003. ISBN 9781461268543. On-line library catalogues
Prerequisites
Learning outcomes
Assessment methods
Knowledge - knowledge achieved by taking this course are verified by the following means: IC6 - Oral examiantion
Teaching methods
Knowledge - the following training methods are used to achieve the required knowledge: Individual tutoring