A particle of mass falls under uniform gravity along a vertical line

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch Symmetry, Noether's theorem and what is conserved

Problem

A particle of mass

m=2.0\ \mathrm{kg}

falls under uniform gravity

g=9.8\ \mathrm{m\,s^{-2}}

along a vertical line, with

L=\tfrac12 m\dot z^2-mgz

z

measured upward from a fixed origin. Consider the transformation

z\to z+\epsilon

, a shift by a small constant

\epsilon

that does not depend on

t

. ((a)) Show that this shift changes

L

\epsilon\,\mathrm{d}f/\mathrm{d}t

with

f=-mgt

, and use the general result

Q=p_z\zeta-f

(with

\zeta=1

here) to write down the conserved quantity

Q

in terms of

m

\dot z

g

and

t

. ((b)) The particle is thrown upward at

t=0

with

\dot z(0)=5.0\ \mathrm{m\,s^{-1}}

. Evaluate

Q

and identify, in words, what constant of the motion it represents.