Chapter 4 AM: Canonical Equations

Consider a conservative mechanical system
Jawad Cheayto
E having s degreesoffreedom Audio 1


In the
Lagrangianformulation
this system isdescribed

by s independent equations
ofmotionof the form

I I Yg
0 i i t s


the Hamiltonianformulationthe
In system is
described


by first orderdifferential equations Tre are 2s
independentfirst orderdifferential equations with 25
independent variables The variablesarethe sgeneralized
momentum
2119g
Pi 29

from a mathematical view point the passagefrom
Lagrangianformulation to Hamiltonicformulation isdone
via a variables in the Lagrangian's
change of
function


LIFEH LIFEH
This procedure is called the Legender
Transformation

,Let 493944 betheLagrangian's function Audio2
the under consideration E
of system

Let us writethe total differential of Li


Es d9
d
f Hit fog t
Idt
As
Tai dat Yai
f ni

d EE fpiidq.tpidgi.lt dt att

writethe second number the right handside
of
of 1H as follows
dlpi9il pid9it didpi
pidaii dlpi.ci i9dpi
So Lt becomes


dL siEpi.dgit.EE dlpi9il oiidpilgtffd

EEipi.dqitdfEgpi9i Esg dni t dt
ft

, digpioi 4 ECoiidni iridqj g.at
Hamilton's
function
the
of system


q pit
HITpitta 4 pig g pig Hai Gilapi't t

considerthetotal differential Audio3
of
HIftp.t
these
DH
Effftp.dqitfltpidpilgtfttdt Comparing we

will get Hamilton's


dtt dt
equations
of motion

EEf pidq.todpi ft
dit dit 0 dt
1,43 Hq traildq.tlftp 9i drift t the
13


Airing independent tipi so
ff't II f
t so
ftp.oii.o

, So uaaeh
sguaiionaseso.ae

aagggs.fi inilgEaenoni
at simplicity and high symmetry as
pi joy
Stated
byJacobi


Ht 2L
It It
Remark Hamilton's
physicalmeaning of function of a conservative
system
observe that

piety Thea




Audioll
Iffesystemissubjectonly to time
independent holonomic constraintslinwhich

casethe transformation relatingthecartesian
coordinates toBe generalized coordinates



Xx 7dam Ms

ya fala Ms 2 2 N
is independent
offing
Ziefalgs gs