The motion of a particle of mass m in a constant magnetic field directed along the 2 axis can be described by the Lagrangian L = ½{m(i² + y² + 2²) + k(xÿ — yi) . (the constant k is proportional to the product eB of the electric charge of the particle and the magnetic field). (a) Write down the Euler-Lagrange equations, and use them to show that the three quantities I₁ := mȧ - 2ky, I : my + 2kx, I₂ := m², are conserved. (b) Show that L is invariant under a rotation by an infinitesimal angle € about the 2 axis, 8x = €y, dy = -x . Show that the corresponding Noether invariant is given by I = m(xy-yi) +k(x² + y²).

The motion of a particle of mass m in a constant magnetic field directed along the 2 axis can be described by the Lagrangian L = ½{m(i² + y² + 2²) + k(xÿ — yi) . (the constant k is proportional to the product eB of the electric charge of the particle and the magnetic field). (a) Write down the Euler-Lagrange equations, and use them to show that the three quantities I₁ := mȧ - 2ky, I : my + 2kx, I₂ := m², are conserved. (b) Show that L is invariant under a rotation by an infinitesimal angle € about the 2 axis, 8x = €y, dy = -x . Show that the corresponding Noether invariant is given by I = m(xy-yi) +k(x² + y²).