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Course, academic year 2018/2019
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Analytical Mechanics - NOFY032
Title in English: Analytická mechanika
Guaranteed by: Laboratory of General Physics Education (32-KVOF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2003
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/1 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Additional information:
Guarantor: doc. RNDr. Jiří Langer, CSc.
Class: Fyzika
Classification: Physics > General Subjects
Annotation -
Last update: T_KVOF (26.05.2003)
Analytical mechanics of particles and rigid bodies. For the 2nd and 3th year students of mathematics.
Aim of the course -
Last update: T_KVOF (28.03.2008)

Analytical mechanics of particles and rigid bodies.

For the 2nd and 3th year students of mathematics.

Literature - Czech
Last update: T_KVOF (26.05.2003)

[1]H.Goldstein, C. P. Poole, C. P., Jr. Poole, J. L. Safko: Classical Mechnics , Prentice Hall, N.Y. 2002.

[2] L.D.Landau, E.M.Lifsic: Mechanika , Fizmatgiz, Moskva, 1958, Mechanics, Pergamon Press, Oxford 2000.

[3] K.R.Symon: Mechanics , Addison-Wesley, Reading, 1971.

Teaching methods - Czech
Last update: T_KVOF (28.03.2008)

přednáška + cvičení

Syllabus -
Last update: T_KVOF (26.05.2003)
Exposition, motivation, and outline
Advantages of alternative formulations of some problem in physics. Recalling of the main ideas and principles of Newtonian mechanics. Limits of classical mechanics (relativistic and quantum mechanics).

Lagranege formalism, Lagrange equations
Acting forces versus forces of constraints. Virtual displacement and dynamics of a system with constraints: d'Alembert's principle.

Generalized coordinates,Configuration space, independence of generalized velocities on generalized coordinates. Derivation of Lagrange's equations of the second kind from d'Alembert's principle. Lagrange's function L : cases without potential, with a potential, with a generalized potential (motion of a particle in given electromagnetic field). Illustration: cycloidal pendulum, motion of a particle in the field of a central force, Binet's equation.

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 n-body problem and celestial mechanics: few words about deterministic chaos.

Hamilton's principle
Elements of the calculus of variations (motivation and explanation of main ideas: Fermat's principle, brachistochrone, geodesics in general relativity). Condition for the extreme: the Euler-Lagrange equations. Definition of action, Hamilton's principle of least action. Its main consequences: Lagrange's equations of the second kind, symmetries and the conservation laws (the theorem of Emmy Noether for invariant L ). Briefly about gauge transformations and fields.

Hamilton's canonical equations and the Poisson bracket
Generalised momentum as a canonically conjugate variable. 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 and statistical physics (partition function). Definition, basic properties, and the algebra of Poisson's brackets.

Mechanics of rigid bodies
Recalling vectors and tensors in Euclidean space. Finite rotations. Infinitesimal rotations and 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 a rotational motion. 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.

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