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Chapter 2 AM: Lagrangian Mechanics $8.03
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Chapter 2 AM: Lagrangian Mechanics

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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
  • 71
  • 2020/2021
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General Jawad Cheayto
Coordinates


Holonomic

Constraints

Non Holonomic



System of N particles




p



fu Xsly its Xnyn tn t O

3N Cartesian coordinates
h constraints 3N has
X2 Zz Xn
YnZn
e
Y
si numbers ofdegrees

So we need s generalized offreedom
the
of
system
coordinates
to describethe

the
configuration
of system

at anygivenmoment
of time

, Lagrangian Audio 2
Mechanics




Let E be a mechanical


system composed of Nparticles
subject toKeholononicconstraints

All constraints are holonomic
All applied forces actingon
the system are assumed
1 Subdivision
conservative of forces into
Constraintforces and appliedforces



Constraintforce Due to the presence

a constraints Constraintforces theirexistence depends
of
on otherforces

Appliedforce forces notdue to gfonstraint
a constraints
it canchangethe rt
the
by charge of
massorte length the
Generalized coordinates spring
of veg
applied
911 gs where s 3N force

xi filgu.o.o.gs't T

gs.tt gTEEIrI
yi.g.iq
rTu. Ziehi
9ne.ee 9s t 2 1 is

,Derivation
ofthe Audio3
Lagrangian equations


firstDerivation Variational

principle


Action
If E has s degrees
of
freedom it hasgeneralized
coordinates
q As

Assumethat E starts with a

configuration
qui ai at tests

and ends at a

configuration
gin qY at Tst
Audio y
q
9
qui fff
i r
Along
Definseq

p
path
aff
ca asia iasitHt

f
te ta along a certain
Path19
9s tt 4944 gg b
where dga
dt

, Liske Lagrangianof
the where H
system
z1y Miu
K U
kinetic potential i mass
my
energy energy




Let E'bete total applied
force acting on the particle i
ten No potential energyfor
E gTadiU constraint forces


Ei
iii YI't'S.IT Hi
Audios
rEIX.IIy.jo ziIig
positionvector

offeparticlei

1
Audio 7
Leastaction 2 s
Principle
choosingmakes
that
A system E evolves between two path
settled or minimal

lattimets and
configurations
S Edt is
extremal
HEYattimeta sothat theaction

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