Ph-507. Homework 8 (due: Monday, April 11).
PROBLEM 8-1 (4 pts) Two identical oscillators are placed in viscous medium. They are coupled due to dissipative hydrodynamic interactions (-terms in the equations of motion below). One of the oscillator is
Ph-507. Homework 9 (due: Wednesday, April 20).
PROBLEM 8-1 (15 pts.) Perform numerical study of the pendulum subjected to time-dependent torque: H= p2 + 2
2 0
cos + cos t
a) Investigate the transition from periodic to chaotic behavior by construc
Physics 507: MIDTERM EXAM
PROBLEM 1. (5 pts.) A classical particle of mass m moves in the presence of the following force eld in one dimension: F (x) = kx 2x2 a2 .
At moment t = 0 the particle is found at point x0 = a, with zero velocity. Find i
Ph-507. Homework 6 (due: Wednesday, March 23).
PROBLEM 6-1 (5 pts) Right after a symmetric spinning top (in gravity) has been kicked by a nutty professor, the nutation and precession rates had values = 0 and = , respectively. The orientation of
Ph-507. Homework 4 (due: Friday, February 25).
PROBLEM 4-1 (4 pts) Episode II: A long time ago, in a galaxy far, far away. In 2D space, the gravitational potential energy should have a logarithmic form. In particular, for a system of N particles, X U
Physics 507: Theoretical Mechanics
Contents
I. Lagrangian Mechanics A. Equations of Motion B. Symmetries and Conservation Laws C. Back to Newton II. Classical Integrable Problems A. Motion in 1D B. Motion in Central Field 1. Two-body problem 2. Mapp
Ph-507. Homework 1 (due Mon, January 24).
PROBLEM 1-1 (1 pts). Starting with the Newtons laws, prove additivity of mass. PROBLEM 1-2 (3 pts). By minimizing the functional S, derive the dierential equation for z (t): ZT
0
S=
dt
"
d2 z dt2
2
+
Ph-507. Homework 3 (due: Monday, February 14).
PROBLEM 3-1 (3 pts) Four point particles of the same masses m, are placed at the corners of a square of size d d. They are released with no initial speed at time t = 0. After what time will the particle
Ph-507. Homework 7 (due: Friday, April 1).
PROBLEM 7-1 (4 pts.) A particle of mass m is moving in two dimensions under the inuence of asymmetric harmonic potential: U (x, y) = k1 x2 + k2 y 2 2
Here (x, y) are the coordinates of the particle in a rot
Ph-507. Homework 5 (due: Friday, March 11).
PROBLEM 5-1 (3 pts) The distribution of dust in Solar system is roughly uniform. This results in the following small correction to regular gravitational potential energy of a planet of mass m: U (r) = 2 Gmr
Ph-507. Homework 2 (due: Friday, February 4).
PROBLEM 2-1 (3 pts) Considetr a relativistic particle in an external potential, U = k |x|: r x2 2 L = mc 1 2 k |x| c Find the period of its oscillations as a function of amplitude x0 . PROBLEM 2-2 (2 p