Geometry and Differential Equations (18/19)
Faculteit  Science and Engineering 
Jaar  2018/19 
Vakcode  WMMA14002 
Vaknaam  Geometry and Differential Equations (18/19) 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester II a 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Geometry and Differential Equations (tweejaarlijks 2018/2019)  
Leerdoelen  At the end of the course, the student is able to: 1. Derive the canonical equations of motion from a given Hamiltonian; 2. Apply the Liouville theorem to give qualitative estimates for solutions of initial problems; 3. Inspect whether given dynamical functions are integrals of motion, and generate the Lie algebra of conserved quantities using the Poisson bracket; 4. Solve the HamiltonJacobi equation in standard geometric settings; 5. Perform the canonican transformations and introduce actionangle variables; 6. Find the PoincareCartan integral invariants and use the adiabatic invariants to predict properties of longterm behaviour of a given system. 

Omschrijving  This course in geometry of Hamiltonian differential equations is oriented equally towards mathematicians and physicists. This year, it can be taken independently of or in parallel with the MasterMath course "Symmetries and conservation laws of nonlinear PDE" (which will be more about the EulerLagrange systems). We shall study geometric and analytic aspects of Hamiltonian mechanics: * canonical variables and equations of motion, * the Liouville theorem about phase volume preservation, * Poisson brackets and the Darboux lemma. Particular attention will be devoted to * the HamiltonJacobi equation (with emphasis on practical skills of solving the Cauchy problems for first order quasilinear inhomogeneous PDEs). * We shall also address the problem of finding the action from known solution of the canonical equations of motion and, reciprocally, solving the canonical equations of motion for a known action functional. This will be studied using * the separation of variables: e.g., introduction of cyclic coordinates. The idea of canonical transformations, notion of the PoincareCartan integral invariants, actionangle variables, and adiabatic invariants will conclude the course. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Werkcollege (T)  
Toetsvorm 
Schriftelijk tentamen (WE)
(Students will report on one or several topics typically, a small research question that will have been assigned during lectures; a student can also report on his/her individual supervised research if it pertains to this course and only if approved by the lecturer) 

Vaksoort  master  
Coördinator  A.V. Kiselev  
Docent(en)  A.V. Kiselev  
Verplichte literatuur 


Entreevoorwaarden  Mechanics and Relativity 1,2 and/or Mechanics for mathematics. Knowledge of differential geometry and group theory is advised but not compulsory.  
Opmerkingen  
Opgenomen in 
