PHY5210 W15

Chapter 7: Lagrange's Equations

Reading:

Recall

S = ∫₁² L(q_i, q_idot, t) dt.

∂L/∂q_i - (d/dt)(∂L/∂q_idot) = 0

L = T - U

Generalized Momenta and Ignorable Coordinates

Recall that we introduced the terminology that ∂L/∂q_i = F_i is a generalized force and ∂L/∂q_idot = p_i is a generalized momentum. And with this notation Lagrange's equations read

F_i = dp_i/dt.

It is sometimes the case that the Lagrangian for a system won't depend a coordinate q_k. In this situation, the corresponding generalized force will be zero, F_k = 0 and the generalized momentum will be conserved. Such a coordinate is said to by cyclic or ignorable since the generalized momenta are just constants of the motion.

This situation is a simple of example of a broader theorem about conserved quantities in a system called Noether's Theorem (named for Emmy Noether, a late 19^th / early 20^th century female mathematician/physicist).