Physics601Homework4­­­DueFridayOctober1Goldstein:9.7,9.25Inaddition:

1. Thisproblemusestheresultinproblem2ofhomework3inthecontextofa6‐dimensionphasespaceassociatedwithasingleparticlein3dimensions.Thecanonicalvariablesare

€

x,y,z, px, py, pz .Inthisproblemvarioustransformationswithclearphysicalmeaningsareproposed.Statethephysicalmeaning,showthattheyarecanonicalandfindthegeneratorofthetransformation(asdefinedinin

problem2ofhomework3)andshowitsatisfiestherelation

€

∂ ξ ( η ;ε)∂ε

= [ ξ ,g]PB

given

inproblem2ofhomework3.Inalloftheseεisaconstant.

€

X = x + ε

Y = yZ = zPx = pxPy = pyPz = pz

a.

€

X = x cos(ε) + y sin(ε)Y = −x sin(ε) + y cos(ε)

Z = zPx = px cos(ε) + py sin(ε)Py = −px sin(ε) + py cos(ε)

Pz = pz

b.

€

X = xY = yZ = z

Px = px + ε

Py = pyPz = pz

c.

€

X = (1+ ε)xY = (1+ ε)yZ = (1+ ε)z

Px = (1+ ε)−1 pxPy = (1+ ε)−1 pyPz = (1+ ε)−1 pz

d

2. Showthatifaone‐parameterfamilyoftime‐independentcanonical

transformationsleavestheHamiltonianinvariantanditsgeneratorhasnoexplicittimedependencethenitsgeneratorisconserved.Thatisgivenacanonicaltransformation

€

ξ ( η ;ε)with

€

H( ξ ( η ;ε)) = H( η )andwithgasagenerator‐‐‐

€

∂ ξ ( η ;ε)∂ε

= [ ξ ,g]PB ‐‐‐‐thengisconserved.

3. Inproblem1ofHW2weshowedanexplicitexampleofaconservedquantity

whichwasnotassociatedwithapointtransformation.Usetheresultshownaboveinproblem2toshowthattheconservedquantityΔisassociatedwithafamilyofcanonicaltransformationsbyexplicitlyconstructingthefamilyoftransformations.

4. Considerthecanonicaltransformationforasystemwithonedegreeoffreedom

generatedby: .

a. Findtheexplicitformforthecanonicaltransform.b. VerifythatitsatisfiesthecanonicalPoissonbracketrelations.

Fortheremainderofthisproblemconsiderparticlemovinginonedimensioninaconstantgravitationalfield

€

L(q, ˙ q ) = 12 m ˙ q 2 −mgq .

c. UsethefactthatforanyA,

€

dAdt

= [A,H]PB +∂A∂ttoshowthat and

.

d. Partc.impliesthatKtheHamiltonianassociatedwithQ,PmustbezerouptoapossiblytimedependentconstantindependentofQ,P.Showfromthe

explicitformof thatthisisinfactthecase.

e. WhatisthephysicalinterpretationofQ,P.f. Showthat

€

F 2(q,P,t) satisfiesaHamilton‐Jacobiequation:

€

H q,∂F2

∂q

+

∂F 2

∂t= 0 .

g. Introducethefunction

€

f (Q,P,t) = F 2(q(Q,P, t),P,t) .Showthat

€

∂f (Q,P,t)∂t

= L q(Q,P, t), ˙ q (Q,P,t)( ) .

5. InclasswederivedtheHamiltonJacobiequation

€

H q , ∇ S( q ,

P )( ) +

∂S( q , P )

∂t= 0

where

€

P isaconstantofthemotion.Deriveananalogusexpression

€

Q

€

H − ∇ p ˜ S (

Q , p ), p )( ) +

∂ ˜ S ( Q , p )∂t

= 0where

€

Q isaconstantofthemotionandthetildeis

on

€

˜ S todistinguishitfromS.

!

F2(q,P,t) = (q + 1

2gt

2)(P "mgt) "

P2t

2m

!

dQ

dt= 0

!

dP

dt= 0

!

"F 2(q,P,t)

"t