Syllabus
* Introduction and motivation
Advantages of alternative formulations of some problems in physics. Illustrated by theories of gravity: Newton's gravitational force -> Poisson's equation (potential field) -> Einstein's equation (metric field, general relativity). Theoretical mechanics as formulation of Newton's laws of motion in various formalisms: for point masses, rigid bodies, and continuum. Motivation and outline of the course. Recalling the main ideas and principles of Newtonian mechanics. Limits of classical mechanics (relativistic and quantum mechanics).
* Lagrangian formalism and Lagrange's equations
Generalized coordinates: don't use only (x,y,z). Occam's razor: don't use more coordinates than necessary. Configuration space: Zeno's paradox and independence of generalized velocities on generalized coordinates. Derivation of Lagrange's equations of the second kind. Lagrange's function L: cases without potential, with potential, with generalized potential (motion of a particle in a given electromagnetic field). Illustration: motion of a particle in the field of a central force. First integrals (cyclic coordinate -> conservation of the corresponding generalized momentum, explicit independence of L on time -> conservation of a generalized energy). Illustration: Binet's equation for motion in a central field.
* Motion of planets and further applications
Kepler's problem: revolution of planets around the Sun. Derivation of Kepler's laws of planetary motion. Effective potential method. Comparison of classical and relativistic mechanics: motion around the Sun versus motion around a black hole, perihelion shift. Simplification of the problem of two bodies to motion of a single particle with reduced mass. The 3-body problem and celestial mechanics: a few words about deterministic chaos. Scattering of particles, the Rutherford formula for cross-section.
* Hamilton's canonical equations and the Poisson brackets
Generalized momentum as a canonically conjugate variable. The concept of phase space with some illustrations (oscillator, damping, chaos). Hamiltonian function. Derivation of Hamilton's canonical equations. Illustrations of canonical equations (harmonic oscillator, particle in electromagnetic field). Importance of Hamiltonian formalism for quantum theory (the Schroedinger equation, the Feynman diagrams as an expansion of interaction Hamiltonian) and statistical physics (partition function). Definition, basic properties, and the algebra of Poisson's brackets. Their analogy with commutators in quantum mechanics.
* Mechanics of rigid bodies
Recalling vectors and tensors in Euclidean space. Group of finite rotations and algebra of infinitesimal rotations. Definition of the vector of angular velocity. The rotation of a rigid body around fixed axis, the inertia tensor. Eigenvalues and eigenvectors, including an interpretation of the inertia ellipsoid. Kinetic energy of rotational motion. Euler's angles and Euler's kinematic equations. Euler's dynamical equations. Explicit examples: motion of a symmetrical gyroscope and symmetrical top.
* Wave equation and its solutions
Transition from a finite system of point masses to a continuous system. Illustration: longitudinal and transverse oscillations of a string. Wave equation and basic methods of its solution: a) the d'Alembert method, b) separation of variables (normal modes, boundary and initial conditions, the Fourier analysis).
* Foundations of relativistic mechanics
Annotation
Classical mechanics of particles: Lagrangian and Hamiltonian description. Kinematics and dynamics of rigid bodies (inertia tensor, Euler angles and equations). Vibrations of a string; solutions of the wave equation. Foundations of relativistic mechanics. Outline of syllabus:
1. Introduction and motivation.
2. Lagrangian formalism and Lagrange's equations.
3. Motion of planets and other applications.
4. Hamilton's canonical equations and Poisson brackets.
5. Rigid body mechanics.
6. Equation of a vibrating string and its solution.
7. Foundations of relativistic mechanics.