Analytical Mechanics -- Fall 2015
| 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. |