PHYS 301 Final, Fall 2007

Problem 1: A system with two degrees of freedom x and y has the La- grangian L = x _y + y _x2 + _y _x Derive Lagrange’s equations. Obtain the Hamiltonian H (x; y; px; py) : Derive Hamilton’s equations and show that they are equivalent to Lagrange’s equations. Problem 2: Write down the properties of Poisson brackets. Let f(p; q; t) be some function of co-ordinates, momenta and time, express its time evolution using Poisson brackets. Prove that if f (p; q; t)and g(p; q; t) are two integrals of motion, then their Poisson bracket is likewise an integral of motion. Determine the Poisson brackets formed by the components of angular momenta. Problem 3: Is the following transformation canonical (verify your answer)? P = 1 2 � p2 + q2� ; Q = tan�1 �q p � Problem 4: The angular velocity is expressed using the Euler angles by ! = 0 @ _� sin sin � + _� cos _� cos sin � � _� sin _� cos � + _ 1 A Express the kinetic energy T of a rotating rigid body, using the principal mo- ments of inertia I1; I2; I3; in terms of the Euler angles. Assuming there is no torque, write down Lagrange’s equation with respect to to obtain the third component of the Euler’s equation of motion. Problem 5: Write down the action for an electromagnetic …eld A� with sources: Using the principle of least action together with the Bianchi identity, derive Maxwells equations of electromagnetism in terms of the electric and mag- netic …elds. Write down the Lagrangian and derive the relativistic equation of motion for a charge particle in an electromagnetic …eld. Show that this rela- tivistic equation reproduces the Lorentz force law. Working in cylindrical polar coordinates, consider the vector potential A = (0; rg(z); 0) where g(z) > 0: Ob- tain two constants of motion. Show that if the electron is projected from a point (r0; �0; z0) with velocity _r = _z = 0 and _� = 2eg(z0)=m; then, provided some condition is satis…ed, it will describe a circular orbit. 1