Analytical Mechanics -- Fall 2016
| Material covered: (see also Analytical Mechanics -- Fall 2015) |
| Week | Monday | Wednesday |
| Aug 29 | Overview of Mechanics. Holonomic and nonholonomic constraints. Newtonian
mechanics. 
Single-particle problems. |
Newtonian mechanics. Single-particle problems. Viscous drag. Harmonic oscillator. |
| Sep 05 | Labor Day - Please, work at home! | Resonance. Charged particle in a magnetic field. Momentum and angular momentum. |
| Sep 12 | Work and energy. Potential forces, criteria of potentiality. Energy conservation from Newton's law. One- and two-particle interactions. | Center of mass, reduced mass. One-dimensional motion, turning points, period of motion (example: Washboard potential). Phase portraits, separatrix. |
| Sep 19 | One-dimensional motion: pendulum Constraints and equations of motion in special coordinate systems: polar |
Constraints and equations of motion in special coordinate systems: spherical |
| Sep 26 | 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 3 | Yom Kippur Make-up on Friday: Problem solving on Newtonian mechanics |
Lagrangian mechanics.
The least-action principle. Lagrange equations. Lagrange function. Examples.
Non-holonomic constraints. Lagrangian of a particle in electromagnetic field. |
| Oct 10 | Columbus Day | No classes |
| Oct 17 | Gauge transformation. Galilean transformation. Invariance of Lagrangians and conservation laws. | Small oscillations in many dimensions
|
| Oct 24 | Problem solving on Lagrangian mechanics & small oscillations. Rotating ring
with a bead
|
Midterm 1 |
| Oct 31 | 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 07 | Rotational motion of rigid bodies: Rotatonal kinetic energy. Tensor of inertia. Angular momentum and its equation of motion, torque and rotational potential energy. |
Rotational motion of rigid bodies: Precession and spin of a symmetric top. Equation of motion for Euler angles of a free asymmetric top. Stability of rotations. Larmor equation. |
| Nov 14 | Rotational motion of rigid bodies: Lagrangian formalism. Symmetric top with gravity. | Rotational motion of rigid bodies: Equation of motion for Euler angles of a free asymmetric top. Stability of rotations. Problem solving: cone. |
| Nov 21 | Hamiltonian formalism:
Hamiltonian function and equations, Variational principle; Poisson brackets |
Hamiltonian formalism:
Canonical transformations |
| Nov 28 | Hamiltonian formalism:
Action as function of coordinates; Hamilton-Jacobi equation |
Hamiltonian formalism:
Hamilton-Jacobi equation; Separation of variables |
| Dec 05 |
Hamiltonian formalism:
Integrable and nonintegrable systems; Angle-action variables and adiabatic invariant |
Hamiltonian formalism:
Parametric resonance Parametric resonance via natural and angle-action variables |
| Dec 13 | Dynamical Chaos
Motion in a rapidly oscillating field |
Midterm 2 |
| Problems with solutions:
|
| 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. |