Syllabus
* Exposition, motivation, and outline
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).
* Motion of point particles subject to constraints
Acting forces versus forces of constraints. Parametric description of curved surfaces, normal vectors. Lagrange's equations of the first kind (intuitive definition, analysis and illustration through simple examples). General form of the equations for N masses and v constraints. Classification of constraints: one-sided - double sided, holonomic - non-holonomic, scleronomic - rheonomic. Virtual displacement and dynamics of a system with constraints: d'Alembert's principle. Consequences: Newton's equations of motion without constraints, looking for equilibrium by means of the principle of virtual work, equivalence with Lagrange's equations of the first kind.
* Lagrange's equations of the second kind
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: cycloidal pendulum.
* The rules, methods, and tricks of the Lagrangian formalism
Recipe for deriving equations of motion (choice of appropriate generalized coordinates, expressing T, V, and thus L, performing corresponding derivatives, their substitution into Lagrange's equation of the second kind). Illustration: motion of a particle in the field of a central force. Methods and tricks of integrating the equations of motion: approximate solutions assuming linearization (mathematical pendulum), integrals of motion, i.e., the 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 principle
Elements of the calculus of variations (motivation and explanation of the main ideas: Fermat's principle, brachistochrone, geodesics in general relativity). Extremum condition: the Euler-Lagrange equations. Definition of action, Hamilton's principle of least action. Its main consequences: Lagrange's equations of the second kind and of the first kind. Symmetries and the conservation laws (the theorem of Emmy Noether for invariant L). Briefly about gauge transformations and fields. Outline of generalization of variation approach to classical and quantum field theories.
* Hamilton's canonical equations and the Poisson bracket
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 both from Hamilton's principle and from Lagrange's 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.
* Canonical transformations and the Hamilton-Jacobi theory
Canonical transformations, generating functions and necessary conditions (summary of basic algorithms, examples). Analogy with thermodynamical potentials. Derivation of the Hamilton-Jacobi equation as a consequence of a suitable canonical transformation, solution methods (separation of variables), example (free fall). Applications in physics: optics (wave-surfaces -> rays), quantum mechanics (semi-classical approximation: the Schroedinger equation -> the Hamilton-Jacobi equation), the Feynman formulation of the quantum theory using path integrals.
* Mechanics of rigid bodies
Recalling vectors and tensors in Euclidean space. Group of finite rotations and algebra of infinitesimal rotations. Their representation in terms of antisymmetric matrices, definition of the vector of angular velocity as dual to them. 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. Decomposition of motion into translation and rotation (the Chasles' theorem). A consequence for the kinetic energy (Koenig's theorem). A simple derivation of equations of motion in non-inertial systems from the Lagrangian.
* Euler's equations and spinning tops
Euler's angles and Euler's kinematic equations. The Lagrange function for a rigid body and derivation of Euler's dynamical equations. Explicit examples: motion of a symmetrical gyroscope and symmetrical top.
* Description of continuous media
Transition from a finite system of point masses to a continuous system. Illustration: density of Lagrangian for transverse oscillations of a string. Derivation of the Euler-Lagrange equations for continuum from Hamilton's principle. 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). Perspectives: classical fields and their quantization. Two possible descriptions of continuous media: Lagrange versus Euler. The displacement and velocity fields.
* Basic equations for continuum
Recalling the tensor of small deformations and the stress tensor. The equation of motion of a general continuous media and the continuity equation, equilibrium conditions. Classification (from a solid state to an ideal fluid). Generalized Hook's law for isotropic bodies with interpretation of the coefficients.
* The most interesting consequences of equations of continuous media
The equation of motion for isotropic continuum. Euler's equation for perfect fluid. Waves in the latter and the speed of sound. Bernoulli's theorem as the first integral. The d'Alembert hydrodynamic paradox for non-rotating, incompressible perfect fluid. The Naviere-Stokes equation for viscous fluids. Illustration: flow along a tube (parabolic velocity profile and the Poiseuill-Hagen law). Laminar versus turbulent flow, the Reynolds number.
Annotation
In this course the students will learn analytic formulations of classical mechanics of point particles and rigid body, and an introduction to fluid mechanics. The core of the course is the Lagrangian and Hamiltonian formalism, including their most important applications and key concepts of quantum and relativistic physics.