(b) Consider the Lagrangian function on R² (defined by the Cartesian coordinates (x, y)) given by 1 L = - ¡m (x² − ÿ²) + a(y² — x²), where m and a are constants. (i) Show to first order in € (that is, ignore terms of order €² and higher), that L is invariant under the transform (x, y)(x+ey, y + ex).

(b) Consider the Lagrangian function on R² (defined by the Cartesian coordinates (x, y)) given by 1 L = - ¡m (x² − ÿ²) + a(y² — x²), where m and a are constants. (i) Show to first order in € (that is, ignore terms of order €² and higher), that L is invariant under the transform (x, y)(x+ey, y + ex).