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
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
