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Chapter 5 AM: Canonical Transformation $8.14   Add to cart

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Chapter 5 AM: Canonical Transformation

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The main objective of this course is to introduce students of physics to the 'modern' formalism of classical (or Newtonian) mechanics, especially Lagrangian and Hamiltonian mechanics. An important part of the course will focus on introducing the variational methods and principles in mechanics, or m...

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  • January 8, 2023
  • 48
  • 2020/2021
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Jawad Cheayto
E System with
Mechanical s DOF
Holonomic conservative Audio 1

TH
Hamilton's equations
ofa mechanicalsystem
fipi
IPi where H H g Ft


Consider a coordinate transformation
It
Qi Qi gift What are the conditionsthat this

I haveto satisfy inorderthat
Pi Pi g pi t Hamilton's equations
inthe new variable
maintiontheir canonical form




Qi Jlt i Die 211
JP Jai
where H'sICE Et
isthe new Hamiltonian


A transformation satisfying
therequired
conditions is called canonical

obtained earlierthe Hamiltonian's ta 2

We
equations through a principle ofleastaction 0 85 8 flat 8 Epida Hdt
t I

,Hamilton's equations
inthe new variables Audio 2
must also follow from the principle
of
leastaction

85 Edt
SITEPiQi H'Idt

since the new Hamilton's equations

describethe same the
system corresponding

Lagrangefunction I must be related
tobe old Lagrangian
by
DE
dt

etagangest


f function
thesame
coordinates andtimedescribe
ofmechanical
system



Remark The
Lagrangian of a system is
upto additive
of a total time
determined

derivative
of an arbitraryfunction oftime
and coordinates

,proof Audio3


Consider two Lagrange's functions LHiq.tl
and
uqiqtt uq.aeHt adit


The corresponding
actions Sands are

S
L'dtfftldtffdfdt IFdttfifsstfatf.us

flakflattalita
85 85 8 fearful
futffittslith 8tF
Sfa ftp.sqiiialtff SfiD o


Ss S
It follows Hat
85 0 85 0
iff

, Canonical transformations Audio



I
It 49307,4


8 Idt.gs 1dt gfIg8FlH
Efg x89iltalt.Egfq
8Qicta 8S
0
tffsta
85tSfH gs.gg
85 0 iff 88 0


L df
dt
Ldtstidttdf

Epidgi Hdt PidQi H'dttdf
Ficharacterizes the transformation


Qi Q.iq
pit Pi
PilgTp7 andfiscalled1fe
the transformation
generatingfunction of
dfsqpidgi qpidQ.at CH Allot

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