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
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
