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Course, academic year 2018/2019
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Symmetries of Equations of Mathematical Physics and Conservation Laws - NTMF064
Title in English: Symetrie rovnic matematické fyziky a zákony zachování
Guaranteed by: Institute of Theoretical Physics (32-UTF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2014 to 2019
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Karel Houfek, Ph.D.
Annotation -
Last update: T_UTF (14.05.2008)
Symmetries of equations of mathematical physics and their solution using these symmetries. General differential equations of a given symmetry. General conservation laws for systems of differential equations and their relation to symmetries of these equations.
Literature -
Last update: T_UTF (14.05.2008)

Bluman G. W., Anco S. C.: Symmetry and Integration Methods for Differential Equations, Springer, New York 2002

Olver P. J.: Applications of Lie Groups to Differential Equations, 2nd Ed, Springer, New York 1993

Stephani H.: Differential Equations, Their Solutions using Symmetries, Cambridge University Press, Cambridge 1989

Syllabus -
Last update: T_UTF (14.05.2008)

In this course, students will learn

  • how to find point (generalized) symmetries of a given differential equation (of a system of differential equations),
  • how to use these symmetries (which form a Lie group) to simplify or to solve given differential equations,
  • how to find conservation laws (integrals of motion) using point (generalized) symmetries of Euler-Lagrange differential equations which are also symmetries of a corresponding variational functional,
  • how to find a general form of (linear or nonlinear) differential equations of a given order which are invariant under a given Lie group of symmetries.

Basic notions of the theory of Lie groups of transformations which students will learn during lectures:

  • one-parameter and r-parameter Lie group of point transformations,
  • infinitesimal transformations and generators of point transformations,
  • Lie theorems, Lie algebra of a Lie group of transformations, solvable Lie algebra,
  • prolongations of point transformations and their infinitesimal generators,
  • symmetry group of differential equations and infinitesimal criterion of invariance of differential equations,
  • canonical coordinates and their use to reduce, or to solve differential equations,
  • differential invariants and their use to reduce differential equations,
  • generalized symmetries of differential equations,
  • invariant solutions of differential equations, reduction of the number of variables,
  • variational symmetry, infinitesimal criterion of invariance,
  • general conservation laws and their characteristics, Noether theorem for point and generalized symmetries.

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