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...
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
ya fala Ms 2 2 N
is independent
offing
Ziefalgs gs
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