dr hab. Javier de Lucas Araujo, prof. UW

Differential Geometric Methods in Physics

 

 

 

Summer Term

 

Duration: 45 hours

Required previous knowledge:  First-year analysis and linear algebra.

Basic knowledge on differential geometry

Lectures are conducted in English.

 

 

       The subject is focused on the application of modern differential geometric techniques in differential equations and classical mechanics. Special stress is made on providing methods to solve practically and effectively systems of ordinary and partial differential equations, e.g. by the characteristic method, Lie symmetries, reduction, etc. Methods will be employed to study problems of physical interest, e.g. homogeneous equations, pendulum with frictions, small oscillations, solitons, constrained systems, etcetera.

 

Program

 

Chapter I -- Introduction: Vector fields, integral curves, flows, distributions, integrable distributions, Frobenius Theorem, Stefan--Sussmann distributions and their integrability, Lie groups, Lie algebras, Lie group actions and infinitesimal group actions, fundamental vector fields, the exponential map, contact and symplectic structures.

 

Chapter II -- Geometry of first-order ordinary differential equations: Autonomous systems and vector fields, particular solutions and integrals curves, Lie groups of transformations, autonomous Lie symmetries, reductions, constants of motion, resolution of singularities, integrating factors, critical points of autonomous systems, the Legendre transformation, implicit differential equations.

 

Chapter III -- Geometry of higher-order differential equations: The jet bundle, contact structure on the jet bundle, the Cartan distribution, geometry of PDEs, prolongations of vector fields, Lie symmetries, reductions, the characteristics method.

 

Chapter IV -- The Painleve integrability: Movable singularities, the Painleve property, Painleve equations, Darboux transformations, Lax pairs, solitons, chaotic behavior.

 

Chapter V -- Geometric methods in classical mechanics: Principle of least action, Lagrangian functions,  regular, hyperregular and singular Lagrangians, Euler--Lagrange equations, constrained systems, Hamiltonian mechanics, Hamilton's equations, canonical transformations, Hamilton-Jacobi theory.

 

 

 

SLIDES

 

 March 3, 2015     Tangent spaces and vector fields.

 

 

 

 

 

 March 10, 2015    Vector fields, integral curves, associated systems.

 

 

 

 

 

 March 17, 2015    Flows and Lie brackets. 

 

 

 

 

 

 March 24, 2015    Lie derivative of vector fields, Lie algebras.

 

 

 

 

 

 April 21, 2015    The characteristics method II, integrating factors,

 

Jacobi multipliers and Lagrangians

 

 

 

 

 

 

 

 

 May 15, 2015       Higher-order differential equations

 

 

 

 

 

 

 

 

 June 2, 2015       Symplectic geometry

 

 

 

 

 

 

 

 

 

 

 

 

 

EXERCISES

 

 

 

 List I  - Integration of vector fields, flows and Lie brackets

 

 

 

 

 

 

 

 List II - First-integral, the characteristics method, integrating factors, 

 

Jacobi multipliers and Lagrangians

 

 

 

 

 

 

 List III - Stability points

 

 

 

 

 

 

 

 List IV - Lie symmetries 

 

 

 

 

 

 

 

 

Bibliography

 

 

V.I. Arnold, Geometrical methods in the theory of ordinary differential equations. Second Edition.

A series of comprehensive studies in mathematics 250, Springer, New York, 1988.

 

P. Olver, Applications of Lie groups to differential equations. Second Edition. Graduate texts in mathematics {\bf 107}, Springer, New York, 1993.

 

M. Nakahara, Geometry, topology and Physics. Second Edition. Graduate student series in physics, IOP Publishing, Bristol, 2003.

 

H. Stephani, Differential equations: their solution using symmetries. Cambridge University Press, Cambridge, 1989.

 

H. Stephani, Symmetry methods for differential equations. A beginner's guide. Cambridge Text in Applied Mathematics}, Cambridge University Press, Cambridge, 1989.

 

V.I. Arnold and A. Weinstein, Mathematical Methods of Classical Mechanics. Second edition. Graduate Texts in Mathematics 60, Springer, New York, 1989

 

M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction,

John Wiley & Sons, Inc., New York, 1989.

 

P.G. Drazin and R.S. Johnson, Solitons: an introduction. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.

 

P. Liebermann and Ch.-M. Marle, Symplectic geometry and analytical mechanics, D. Reidel

Publishing Company, Dordrecht, 1987.

 

W.M. Tulczyjew, Geometric formulation of physical theories. Monographs and Textbooks in Physical Science. Lecture Notes 11, Bibliopolis, Naples, 1989.