Analytical Mechanics - NOFY032
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Last update: T_KVOF (26.05.2003)
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Last update: T_KVOF (28.03.2008)
Analytical mechanics of particles and rigid bodies. For the 2nd and 3th year students of mathematics. |
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Last update: Mgr. Hana Kudrnová (14.06.2019)
Ústní zkouška. |
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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.
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Last update: T_KVOF (28.03.2008)
přednáška + cvičení |
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Last update: Mgr. Hana Kudrnová (14.06.2019)
Otázky zkoušky se shodují se sylabem. |
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Last update: T_KVOF (26.05.2003)
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. |