Syllabus
* The principle of virtual work.
Configuration space, constrains, virtual displacement, applications, d'Alembert's principle. ~ Lagrange's equations. Generalized coordinates, generalized forces, Lagrangian, Lagrange's equations of the 2nd kind. Applications.
* Lagrange's equations of the 1st kind.
* Small oscilations of systems of point masses.
Linearization of equations. Lagrangian for small oscillations. Normal oscillations.
* Motion in a central field. 2-body problem, separation of Lagrange's equations. Cyclic coordinates. Binet's formula. Scattering: Rutherford formula, scattering at rigid sphere, differential cross-section.
* Hamilton's equations.
Generalized momentum, phase space. Hamiltonian (and energy). Hamilton's canonical equations.
* Elements of deterministic chaos.
Determinism of classical mechanics. Stability of solution of differential equations. Attractors. Examples: a planet near a binary star, double pendulum, Lorentz attractor. Population dynamics model, doubling of periods; universality in chaos.
* Variational principles.
Hamilton's principle, action. Euler-Lagrange's equations. Brachistochrone. Variational principles in other parts of physics.
* Kinematics and dynamics of rigid body.
Tensor of inertia, motion of free symmetrical gyroscope.
* Waves.
Equation of motion of a string and its solution.
* Elements of mechanics of continuum.
Stress tensor, strain tensor, generalized Hook's law. Equation of hydrostatic equilibrium; application to spherically symmetric star. Continuity equation. Euler's hydrodynamic equations.
Annotation
Introduction to concepts and methods of analytical mechanics and their use for solving of selected problems: The principle of virtual work, Lagrange and Hamilton equations, variational principles, kinematics and dynamics of rigid bodies, basic ideas how to describe continuous systems.
For students of the 2nd year of combinations Math and Physics and Physics/Informatics.