Someone argues: "the action variable has units of joule-seconds

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables

Problem

Someone argues: "the action variable

I

has units of joule-seconds, exactly the units of the action

S=\int L\,\mathrm{d}t

of Chapter 1. So

2\pi I

must just be that action, evaluated over one period:

2\pi I = \displaystyle\int_0^\tau L\,\mathrm{d}t

." Test this for a ball of mass

m = 0.10

kg bouncing elastically on a floor from a height

h = 2.0

m, with

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

, potential energy

V = mgx

and

x

its height above the floor. ((a)) Compute

2\pi I

from the area of the loop the motion traces in the

(x,p)

plane. ((b)) Compute

\displaystyle\int_0^\tau L\,\mathrm{d}t

directly from the motion, over one full bounce period

\tau

. ((c)) Say where the argument goes wrong, and check your two numbers against

\oint p\,\mathrm{d}x = \displaystyle\int_0^\tau(L+H)\,\mathrm{d}t

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