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Analytical Mechanics -- Fall 2015

CUNY Graduate Center


Material covered:  (see also Analytical Mechanics -- Fall 2006)
Week Monday Wednesday
Aug 31 Overview of Mechanics. Holonomic and nonholonomic constraints. Newtonian mechanics. Single-particle problems. Newtonian mechanics. Single-particle problems. Viscous drag. Harmonic oscillator. 
Sep 07 Labor Day - Please, work at home!
Makeup on Th: Momentum and angular momentum. Potential forces, criteria of potentiality. Energy conservation from Newton's law. Center of mass, reduced mass.
Resonance. Charged particle in a magnetic field.
Sep 14 Rosh Hashanah One-dimensional motion, turning points, period of motion (example: Washboard potential). Phase portraits, separatrix.
Sep 21 Constraints and equations of motion in special coordinate systems: polar, spherical. Yom Kippur
Sep 28 Motion in a central field. Bounded and unbounded motion; open and closed trajectories. Kepler's law. Precession of orbits as a result of perturbations. Scattering problem, differential scattering cross-section. Rutherford formula for scattering on a Coulomb center. Scattering on a rigid sphere. Small-angle scattering.
Oct 5 Lagrangian mechanics. The least-action principle. Lagrange equations. Lagrange function. Examples. Non-holonomic constraints. Lagrangian of a particle in electromagnetic field. Galilean transformation. Invariance of Lagrangians and conservation laws. Problem on Lagrangian formalism: Rotating ring with a bead
Oct 12 Columbus Day Small oscillations in many dimensions
 
Oct 19 Problem solving Midterm 1
Oct 26 Rotational motion of rigid bodies:
General properties of rotations. Single rotation. Noncommutativity of finite rotations. Commutativity of infinitesimal rotations, angular velocity. Rolling constraint. Euler angles.
Rotational motion of rigid bodies:
Rotation matrices. Active and passive transformations.
 
Nov 02 Rotational motion of rigid bodies:
Rotatonal kinetic energy. Tensor of inertia. Angular momentum and its equation of motion, torque and rotational potential energy. Precession and spin of a symmetric top.
Rotational motion of rigid bodies:
Equation of motion for Euler angles of a free asymmetric top. Stability of rotations. Larmor equation.
 
Nov 09 Rotational motion of rigid bodies:
Lagrangian formalism. Heavy symmetric top. 
Newtonian formalism. Euler equations.
Home Study: Wheel rolling on a plane
Hamiltonian formalism:
Hamiltonian function and equations, 
Variational principle; Poisson brackets
Nov 16 Hamiltonian formalism:
Canonical transformations
Hamiltonian formalism:
Action as function of coordinates; Hamilton-Jacobi equation
Nov 23 Hamiltonian formalism:
Hamilton-Jacobi equation; Separation of variables
Hamiltonian formalism:
Integrable and nonintegrable systems;
Angle-action variables and adiabatic invariant
Nov 30  Hamiltonian formalism:
Parametric resonance
Hamiltonian formalism:
Parametric resonance via natural and angle-action variables
Dec 07 Dynamical Chaos Home study:   Motion in a rapidly oscillating field Midterm 2
Dec 14 Microscopic model of dissipation  

 
Problems:

Selected advanced problems (no solutions yet; to be extended)

Literature

1. Dmitry Garanin, Classical Mechanics
2. Landau and Lifshitz, Mechanics
3. David Tong, Lectures on Classical Dynamics

Heavy wheel rolling on a plane without slipping and with no horizontal force. This is only one of many different types of motion - the so-called "drunk wheel" that does not have enough spin to support steady motion. The trajectory of the center is shown in black and that of the contact point in red. Note that here the direction of the wheel's precession changes with time.