Analytical Mechanics

1. Lagrangian formalism: Lagrange's equations of the first and the second kind, classification of constraints, generalized coordinates, configuration space

2. Differential principles of mechanics: d'Alembert's principle, principle of virtual work and static equilibrium of constrained mechanical systems, virtual displacements

3. The two-body problem: reduction of the problem of an isolated set of 2 point particles using 10 basic conservation laws, the Kepler problem

4. Elementary problems of mechanics: small oscillations of mechanical systems, modes, normal coordinates; Rigid body, angular velocity vector, Euler's equations, free top

5. Variational principles of mechanics: basics of calculus of variations, Hamilton's principle in classical mechanics, Jacobi's variational principle in classical mechanics as an analogue of Fermat's principle in geometric optics

6. Hamiltonian formalism: Hamilton's equations of motion, Legendre transformation, Poisson bracket and integrals of motion

7. Motion of charged particle in electromagnetic field in Lagrangian and Hamiltonian formalism, gauge invariance of corresponding Lagrange equations

8. Noether's Theorem in Lagrangian and Hamiltonian formalism, integrals of motions as generators of symmetries of the Hamiltonian function

9. Canonical transformations and their invariants, Liouville's theorem of statistical mechanics, Hamilton-Jacobi equation

10. Special theory of relativity: Lorentz transformations, Minkowski spacetime and Lorentz group, light cone, four-velocity, four-momentum, four-force, relativistic expression for Lorentz force, equation of motion of a point particle in Minkowski space-time, motion of charged relativistic particle in electromagnetic field