Phys 701 Classical Mechanics 2009
Lecture 1. (Aug 21) Constraints (Sec. 1.4)
Lecture 2. (Aug 24) Lagrange Formalism (Sec. 1.4)
Lecture 3.PDF (Aug 26) Lagrangian of a particle in an EM field. Some vector algebra, eijk tensor (Sec. 1.5)
Lecture 4.PDF (Aug 28) Viscous friction and dissipation function (Sec. 1.5). Sample problem on Lagrange formalism. Mathematics: functionals and functional derivatives (Sec. 2.2).
Lecture 5.PDF (Aug 31) Example of a functional minimization: shape of soap film (Sec. 2.2). Least action principle (Sec. 2.3)
Lecture 6.PDF (Sept 2) Finding reaction forces in Lagrange formalism.
(Sept 4) continue same topic. Discussion of the method of Lagrange multipliers.
Lecture 7.PDF (Sept 5) Least action principle for system with explicit constraints
Lecture 8.PDF (Sept. 9) Cyclic coordinates and conserved generalized momenta. Conservation of angular momentum. (Sec. 2.6). Discussion of an undergraduate problem: finding the height of the projectile explosions.
Lecture 9.PDF (Sept. 11) Conservation of
h. Hamiltonian form of the equations of
Lecture 10.PDF (Sept. 14) Discussion of the energy conservation problem. Minimum action principle for Hamiltonian equations. Independence of the q(t) and p(t) variables (Sec. 8.5). About the further direction of the course.
Lecture 11.PDF (Sept. 16) Hamiltonians in different systems of generalized coordinates (Sec. 8.2). Conservation of momenta of cyclic coordinates (Sec. 8.2). Symplectic notation (Sec. 8.1). Motivation for the Jacobi form of the principle of least action (Sec. 8.6).
Lecture 13.PDF (Sept. 21) Jacobi form of the principle of least action (Sec. 8.6). Example of a projectile motion.
Lecture 14.PDF (Sept. 23) Small oscillations preliminaries (Ch.6).
Lecture 15.PDF (Sept. 25) Small oscillations. Eigenvecotors and eigenvalues. Normal coordinates. (Ch.6).
Lecture 16.PDF (Sept. 28) Small oscillations. Example of three masses connected by two springss. (Ch.6).
Lecture 17.PDF (Sept. 30) Forced small oscillations. (Ch.6).
Lecture 18.PDF (Oct. 2) Small oscillations with friction. Discussion of the example of degenerate eigenfrequencies with insufficient number of eigenvectors.
Recording: http://breeze.sc.edu/p85577201 (Voice only. Sorry, I pressed the wrong buttons...)
Lecture 19.PDF (Oct. 5) Forced oscillations with friction. Form of the resonance curve for a single oscillator and a system with many fundamental modes. Example.
Lecture 20.PDF (Oct. 7) Examples of oscillations.
Recording: http://breeze.sc.edu/p42943404 (Voice only. Sorry, I pressed the wrong buttons...)
Lecture 21.PDF (Oct. 12) Oscillations of a mass on a spring in 2 dimensions. Discussion of Taylor expansion of the energy.
Lecture 22.PDF (Oct. 14) Oscillation of a mas on a spring in 2 dimensions. "Mexican hat" potential.
Lecture 23.PDF (Oct. 19) Discussion of the midterm problems. Canonical transformations (Ch. 9).
Recording: http://breeze.sc.edu/p16572165 (please fast-forward the video to the beginning of the lecture)
Lecture 24.PDF (Oct. 21) Canonical transformations. Solving the harmonic oscillator with C.T. (Ch. 9).
Lecture 25.PDF (Oct. 23) Symplectic criteria for C.T. (Ch. 9).
Lecture 26.PDF (Oct. 26) Geometric meaning of the symplectic criteria. Poisson brackets. Invariance of the Poisson brackets under canonical transformations.
Lecture 27.PDF (Oct. 28) Symplectic criteria provides the existence of the generating function (please see the PDF note for the correct treatment). Poisson brackets of variables themselves, [q,p]=1, etc. Evolution operator and the example of the motion in a gravity field.
Lecture 28.PDF (Oct. 30) Generating function of an inifnitesimal canonical transformation. Change of system characteristics (e.g. Hamiltonian) under the transformation. du = [u,G]
Lecture 29.PDF (Nov. 2) Correspondence between the conserved quantitites and the generators of transformations that conserve the Hamiltonian. Example: conserved linear momentum.
Lecture 30.PDF (Nov. 4) Geometric view on conservation laws and I.C.T. generators. Next example: angular momentum.
Recording: problems with breeze.sc.edu. Will post the link when they fix them.
Lecture 31-32.PDF (Nov. 6, Nov. 9) Lioville's theorem (Ch.9 Sec.9). Hamilton-Jacobi equation. Example of an oscillator. (Ch. 10 Sec. 1)
Recording: http://breeze.sc.edu/p91162954/ Nov. 6 Lioville's theorem, Hamilton-Jacobi equation
Recording: http://breeze.sc.edu/p50718375/ Nov. 9 Example of a harmonic oscillator
Lecture 33.PDF (Nov. 11) Solution of the Hamilton -- Jacobi equation and the action S(q,t). Connection with the quasiclassical approximation in quantum mechanics. Periodic motions and multivalued W(q). Expression for the period T = dJ/dh.
Lecture 34.PDF (Nov. 13) Motion in a central field. Conservation of L, planar character of the motion, polar coordinates. Effective potential and periodic motion along the radius. Bernard's theorem about closed orbits. Integral and explicit expression for the trajectory.
Lecture 35.PDF (Nov. 16) Proof of the trajectory being elliptic. Parameters of the ellipse expressed through E and Lz. Eccentric anomaly and Kepler's equation for the motion. Calculation of the period and Kepler's laws.
Lecture 36.PDF (Nov. 18) Rotational motion. Orthogonal transformation. Euler theorem. Istantaneous axis of rotation. Infinitesimal rotations. Instantaneous angular velocity.
Lecture 37.PDF (Nov. 20) Addition of angular velocities. Ficticious forces in rotating frames. Coriolis force.
Lecture 38.PDF (Nov. 23) Examples of the Coriolis force actions. Expressions for the angular momentum of a rotating body and moment of inertia.
Lecture 39.PDF (Nov. 30) Moments of inertia around different points. Motion of bodies without a fixed point. Calculation of kinetic energy: examples.
Lecture 40.PDF (Dec. 2) Kinetic energy and angular momentum sof bodies without fixed points.
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