Introduction to Classical Mechanics · Problem 6.7

The Lagrangian method — Morin

Physics — Mechanics · noether-theorem-thrown-ball

Problem

Consider a thrown ball with Lagrangian

L = \frac{m}{2}(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - mgz

. This is invariant under translations in

x

(i.e.,

x \to x + \epsilon

) and also under translations in

y

(i.e.,

y \to y + \epsilon

). Use Noether's theorem to identify the conserved quantities associated with these symmetries.