Problem Set 3

Part II – Symmetry and conservation laws

1. Use symmetry to argue that for a central force F r=F r r , (i) torque (τ) = 0 and (ii) the motion takes place in a plane.

2. A particle of mass m moves under the gravitational attraction of a fixed mass M situated at the origin. Take polar coordinates r , θ as generalised coordinates and obtain Lagrange’s equations. Show that θ is a cyclic coordinate and find (and identify) the conserved momentum pθ.

3. A particle P of mass m slides on the smooth inner surface of a circular cone of semi-angle α . The axis of symmetry of the cone is vertical with the vertex O pointing downwards. Take as generalised coordinates r, the distance OP, and φ , the azimuthal angle about the vertical through O. Obtain Lagrange’s equations. Show that φ is a cyclic coordinate and find (and identify) the conserved momentum pφ .

4. A particle of mass m and charge e moves in the magnetic field produced by a current I flowing in an infinite straight wire that lies along the z-axis. The vector potential A of the induced magnetic field is given by

Ar=Aθ=0 , A z=−(μ0 I2π )ln r

where r,θ, z are cylindrical polar coordinates. Find the Lagrangian of the particle. Show that θ and z are cyclic coordinates and find the corresponding conserved momenta.

5. A particle moves freely in the gravitational field of a fixed mass distribution. Find the conservation principles that correspond to the symmetries of the following fixed mass distributions: (i) a uniform sphere, (ii) a uniform half plane, (iii) two particles, (iv) a uniform right circular cone, (v) an infinite uniform circular cylinder.

6. In spherical coordinates, consider a potential that depends only on r. Is the angular momentum for a particle moving under such a force a constant? Will it be a constant if the potential were to depend on r and θ ? Will angular momentum be conserved if potential V were to vanish?

7. Write the Lagrangian for the one-dimensional free particle and consider the variation x' = x + a, where a may be some function of time. Write the new Lagrangian and show that for it to be invariant, the given transformation would yield the well-known Galilean transformations.

8. Write the Lagrangian for the motion of a particle of mass m which mover vertically in the uniform gravitational field g near the surface of the earth. Show that the action principle is invariant under the transformations (i) z' = z + α, and (ii) z' = z + βt , where α and β are constants. Find the associated constants of motion.

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