The motion of a particle of mass m in a constant magnetic field directed along the 2 axis can be describedby the LagrangianL = 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 magneticfield).(a) Write down the Euler-Lagrange equations, and use them to show that the three quantitiesI₁ := 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 byI = m(xy-yi) +k(x² + y²).

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