Symmetries of Equations of Mathematical Physics and Conservation Laws (NTMF064)

Summer semester 2020/2021

Lectures will be given in English each Thuesday from March 3 to June 1 at 14:50 via ZOOM
Meeting ID: 915 7116 9924, Passcode: SophusLie

Materials to download

Warning: Before running the notebooks presented at lectures in Mathematica, it is necessary first to open NTMF064.Package.m and to press Run All Code!

Notes and Mathematica notebooks

Right now I have complete notes only in Czech, but I will add here the English notes during the semester. Here you can download all notes in English up to date as one PDF file.

Topic     Notes     Notebooks in Mathematica
Introduction and motivation   scan    
Point transformations of independent and dependent variables and transformations of functions and their derivatives   scan   Change.Of.Variables.nb (pdf)
Local Lie group of transformations   scan    
Lie's fundamental theorems   scan    
Extension (prolongation) of infinitesimal transformations into the space of derivatives   scan    
Invariance of differential equations and infinitesimal criterion   scan   Test.of.point.symmetries.nb (pdf)
Point symmetries of a free particle in one dimension   scan   Point.symmetries-Free.particle.in.1D.nb (pdf)
Point symmetries of the heat and Schrödinger equation   scan   Point.symmetries-Heat.equation.in.1D.nb (pdf)
Point symmetries of the classical central problem   scan   Point.symmetries-Classical.central.problem.in.3D.nb (pdf)
Partial differential equations of given symmetry - Klein-Gordon equation   scan   Poincare.group.and.Klein-Gordon.equation.nb (pdf)
Solving ODEs using point symmetries - introduction   scan    
Solving ODEs using point symmetries - canonical variables   scan    
Solving ODEs using point symmetries - differential invariants   scan    
Solving PDEs using point symmetries   scan    
Variational symmetry and conservation laws   scan    
Characteristics of point symmetries and of conservation laws, Noether's theorem for point symmetries   scan    
Generalized symmetries, Noether's theorem for generalized symmetries   scan    
Direct method of construction of conservation laws   scan    

Examples of the exam problems

Here is a set of problems which was used as the exam problems in 2008. Most of these problems are now solved during lectures.

Here are some other problems without solutions.
yk denotes k-th derivative of the function y(x) with respect to x.

    Lie groups of transformations and their generators
  1. Show that the projective transformations x' = (αx + β)/(γx + δ) form the Lie group of point transformations. How many independent parameters are there? Determine the infinitesimal generators of this group and their commutators.
  2. Determine the oneparametric Lie group of transformations of the plane R2, the generator of which is X = x2∂x + 2xy∂y and find a general curve given by F(x,y) = 0 which is invariant under this transformation.


    3. Reducing and solving ODEs using pont symmetries
  3. Use translational symmetry to reduce the order of the equation y2 - (y1)2 + y = 0.
  4. Use the scaling symmetry generated by X = 2 t ∂t - x ∂x to solve the equation t dx/dt = x - tx3.


    5. Determination of point symmetries of ODEs
  5. Find twoparametric group of point symmetries of the Blasius equation y3 + 1/2 yy2 = 0 (see [1], str. 135-136.)
    Use the following theorem: If the ODE of at least third order has the form yn = g(x,y)yn-1 + h(x,y,y1,...,yn-2) and if X = ξ(x,y)∂x + η(x,y)∂y is the generator of its point symmetry, then ∂ξ/∂y = 0 and ∂2η/∂y2 = 0.
  6. Proof the theorem from the previous problem.


    7. Determination of PDEs of given symmetries
  7. Determine a scalar nonlinear PDE of the first order for the function u(x,y) in the form u = F(x,y,ux,uy) that is invariant under the Euclidean group of transformations of the plane (x,y), i.e. under all translations and rotations in (x,y). We do not transform the dependent variable u.


    8. Particular solutions of PDEs using point symmetries
  8. Check that the wave equation uxx = utt is invariant under scaling x' = αx, t' = αt, u' = u and find the particular solution invariant under this transformation.

Recommended literature

The following recommended books are not usually available in the library, but you can download their electronic versions from a hidden directory. If you want to access it, write me an e-mail.

[1]Bluman G W, Anco S C - Symmetry and Integration Methods for Differential Equations, Springer, New York 2002
especially chapters 2, 3.1-3.5, 4.1-4.3
[2]Bluman G W, Cheviakov A F, Anco S C - Applications of Symmetry Methods to Partial Differential Equations, Springer, New York 2009
especially chapter 1
[3]Olver P J - Applications of Lie Groups to Differential Equations, 2nd Ed, Springer, New York 1993
especially chapter 4 about conservation laws
[4]Stephani H - Differential Equations, Their Solutions using Symmetries, Cambridge University Press, Cambridge 1989
supplemental reading for those, who consider the books [1, 2] and especially [3] too mathematical