Analytical Mechanics -- Fall 2007
| Material covered: (see also Analytical Mechanics -- Fall 2006) |
| Week | Monday | Wednesday |
| Aug 27 | Overview of Mechanics.
Lagrangian formalism
 |
Galilean invariance. Conservation laws;
Energy, momentum, angular momentum. |
| Sep 03 | Labor Day - Please, work at home! | Examples of holonomic and nonholonomic constraints. Lagrangian of a charged particle in an electromagnetic field. Examples of setting up Lagrange equations: Pendula with moving point of support. Resonance and parametric resonance. |
| Sep 10 | Potential forces, criteria of potentiality. Energy conservation from
Newton's law. Problem on Lagrangian formalism: Rotating ring with a bead
|
Rosh Hashanah |
| Sep 17 | One-dimensional motion, turning points, period of motion (example: Pendulum with finite amplitude). Phase portraits, separatrix. Effective mass. | Motion in a central field. Bounded and unbounded motion; open and closed trajectories. Kepler's problem. |
| Sep 24 | 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 1 | Small oscillations in one dimension. | Small oscillations in many dimensions
Home study: Microscopic model of dissipation
|
| Oct 8 | Columbus Day | Parametric resonance
Motion in a rapidly oscillating field |
| Oct 15 | Rotational motion of rigid bodies:
General properties of rotations. Noncommutativity of finite rotations. Commutativity of infinitesimal rotations, angular velocity. Rotational kinetic energy |
Rotational motion of rigid bodies: Lagrangian formalism, examples with one angle. Newtonian formalism: Angular momentum and its equation of motion, torque and rotational potential energy. Examples with one angle. |
| Oct 22 | Rotational motion of rigid bodies: Euler angles. Lagrangian formalism, examples with symmetric tops |
Rotational motion of rigid bodies: Newtonian formalism in general case. Euler equations. Wheel rolling on a plane |
| Oct 29 | Test 1 | |
| Dynamical Chaos |
| Weekly problem sets: |
| Sep 03 | Sep 17 | Sep 24 | Oct 01 | Oct 15 | Oct 29 |
| Set-01 | Set-02 | Set-03 | Set-04 | Set-05 | Midterm Exam |
| Solution | Solution | Solution | Solution | Solution | Solution |
| Selected advanced problems (no solutions yet; to be extended) |
Literature
| 1. Landau and Lifshitz, Mechanics 2. Goldstein, Classical 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. |