Hamilton application
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Transcript of Hamilton application
Central problem in ‘Mechanics’:
How is the ‘mechanical state’ of a system described,and how does this ‘state’ evolve with time?
Formulations due to Galileo/Newton, Lagrange and HamiltonPCD-08
coordinate , velocity
is related to momentum
dqq qdt
q p
⎛ ⎞=⎜ ⎟⎝ ⎠&
&
Equation of motion: relation between , and q q q& &&
Causality and determinism, Newton’s second law
state of the system in Quantum mechanics: ‘Position/Momentum uncertainty’
p
q
. (q,p)Point in ‘phase space’specifies the ‘state’ of the system.We need dq/dtand dp/dt‘Mechanics’ by L&L, III Edition
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Homogeneity with respect to time/space translations and isotropy of space, inertial frame :
The laws of mechanics are the same in an infinity of inertial reference frames moving, relative to one another, uniformly in a straight line.
If the position of particle is given by the vector ( ) in one frame of reference, and by (t) in another frame of
reference moving at a constant velocity v with respect the previous one,
then
r tr
r
′
rur
r
'( ) ( ) ; is 'absolute' in the two frames:
t r t tTIME t t
= +′=
ur rr v
“GALILEAN PRINCIPLE OF RELATIVITY”
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In an inertial frame,
Time is homogeneous
Space is homogenous and isotropic
Every mechanical system is characterized by a function ( , , ), the Lagrangian of the systemL q q t&
2
1
( , , ) .t
t
S L q q t dtaction = ∫ &
Mechanical state of a system 'evolves' (along a 'world line') in such a way that
' ', is an extremum
HamiltonHamilton’’s principles principle
‘‘principle of least principle of least (rather, (rather, extremumextremum)) actionaction’’PCD-08
0
SS
Sδ =
would be an extremum when the variation in is zero;
i.e. 2
1
( , , )t
t
S L q q t dtaction = ∫ &
motion takes place in such a way that
' ', is an extremum
2 2
1 1
( , , ) ( , , ) 0t t
t t
S L q q q q t dt L q q t dtδ δ δ= + + − =∫ ∫& & &
( )2 2
1 1
. . 0t t
t t
L L L L di e S q q dt q q dtq q q q dt
δ δ δ δ δ⎧ ⎫ ⎧ ⎫∂ ∂ ∂ ∂
= = + = +⎨ ⎬ ⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭ ⎩ ⎭∫ ∫&
& &
( )2 2
1 1
22 2
1 11
. . 0
0
t t
t t
tt t
t tt
d qL Li e S q dt dtq q dt
L L d Lq dt q q dtq q dt q
δδ δ
δ δ δ
⎧ ⎫⎧ ⎫∂ ∂= = +⎨ ⎬ ⎨ ⎬∂ ∂⎩ ⎭ ⎩ ⎭
⎧ ⎫⎧ ⎫ ⎡ ⎤ ⎛ ⎞∂ ∂ ∂= + −⎨ ⎬ ⎨ ⎬⎜ ⎟⎢ ⎥∂ ∂ ∂⎩ ⎭ ⎣ ⎦ ⎝ ⎠⎩ ⎭
∫ ∫
∫ ∫
&
& &
Integration by parts
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2 2
11
. . 0 t t
tt
L L d Li e q qdtq q dt qδ δ
⎧ ⎫⎡ ⎤ ⎛ ⎞∂ ∂ ∂= + −⎨ ⎬⎜ ⎟⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎝ ⎠⎩ ⎭
∫& &
( ) ( )1 2Now, = 0, and is an arbitrary variation.
Hence, 0 '
q t q t qL d L Lagrange s Equationq dt q
δ δ δ=⎛ ⎞∂ ∂
− =⎜ ⎟∂ ∂⎝ ⎠&
Lagrange’s equation of motion
21 2
2
( , , ) ( ) ( )
( )2
- ,
so, . Also, , the momentum
L d Lq dt q
L q q t f q f qm q V q
T VL V LF mq pq q q
⎛ ⎞∂ ∂= ⎜ ⎟∂ ∂⎝ ⎠= +
= −
=∂ ∂ ∂
= − = = =∂ ∂ ∂
&
& &
&
&&
i.e., : in 3D: ( ) ' dp dpF F V q Newton s II Lawdt dt
= = = −∇ ⇔r r r
Homogeneity & Isotropy of space
⇒L can depend only quadratically on the
velocity.
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Law of conservation of energy arises from the homogeneity of time.
dL d (1) 0 + q + dt dt
d(2) 0 qdt
(1) (2) d - 0dt
-
ii
ii
L L L Lq q qq q q q
Lq
Equations andLq Lq
Lq Lq
∂ ∂ ∂ ∂= = =
∂ ∂ ∂ ∂⎡ ⎤∂
= ⎢ ⎥∂⎣ ⎦⇒
⎡ ⎤∂=⎢ ⎥∂⎣ ⎦
⎡ ⎤∂⎢ ⎥∂⎣ ⎦
∑
∑
&& && &&& & &
&&
&&
&&
is a CONSTANT:
- - i i ii
ENERGY
HamiltonianLH q L q p Lq
⎡ ⎤∂ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦∂⎣ ⎦∑ ∑& &
&
Summation over i: degrees of freedom
USING LAGRANGE’s EQUATION
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Time is homogeneous: Lagrangian of a closed system does not depend explicitly on time.
Time is homogeneous: Lagrangian of a closed system does not depend explicitly on time.
Law of conservation of energy arises from the homogeneity of time.
dL d (1) 0 + q + dt dt
d(2) 0 qdt
(1) (2) d - 0dt
-
ii
ii
L L L Lq q qq q q q
Lq
Equations andLq Lq
Lq Lq
∂ ∂ ∂ ∂= = =
∂ ∂ ∂ ∂⎡ ⎤∂
= ⎢ ⎥∂⎣ ⎦⇒
⎡ ⎤∂=⎢ ⎥∂⎣ ⎦
⎡ ⎤∂⎢ ⎥∂⎣ ⎦
∑
∑
&& && &&& & &
&&
&&
&&
is a CONSTANT:
- - i i ii
ENERGY
HamiltonianLH q L q p Lq
⎡ ⎤∂ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦∂⎣ ⎦∑ ∑& &
&
USINGLAGRANGE’sEQUATION
Summation over i: degrees of freedom
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Hamiltonian (Hamilton’s Principal Function) of a system
k kk
k k k k k kk k k kk k
k k kk k k
k k k kk k
H q p L
L LdH p dq q dp dq dqq qLq dp dqq
q dp p dq
= −
∂ ∂= + − −
∂ ∂∂
= −∂
= −
∑
∑ ∑ ∑ ∑
∑ ∑∑ ∑
&
& & &&
&
& &
k kk kk k
k kk k
H Hdp dqp q
H Hq pp q
∂ ∂+
∂ ∂
∂ ∂∀ = = −
∂ ∂
∑ ∑
& &
k kBut, H=H(p ,q )
so dH =
Hence k: and
Hamilton’s equations of motion
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since 0, this means . . is conserved.
. ., is independent of time, is a constant of motion
L d L Li e pq dt q q
i e
⎛ ⎞∂ ∂ ∂− = =⎜ ⎟∂ ∂ ∂⎝ ⎠& &
In an inertial frame, Time is homogeneous; Space is homogenous and isotropic
Law of conservation of momentum,arises from the homogeneity of space.
the condition for homogeneity of space : ( , , ) 0
. ., 0
which implies 0 where , ,
L x y zL L Li e L x y zx y z
L q x y zq
δ
δ δ δ δ
=∂ ∂ ∂
= + + =∂ ∂ ∂
∂= =
∂
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NOETHER’s THEOREM:
Emmy Noether1882 to 1935
SYMMETRY CONSERVATION PRINCIPLE
Homogeneity of time
Energy
Homogeneity of space
Linear Momentum
Isotropy of Space
Angular momentum
CPT Theorem: Standard ModelPCD-08
DYNAMICAL SYMMETRY, (‘accidental’ symmetry)rather than GEOMETRICAL SYMMETRY
Laplace Runge Lenz Vector – constant for a strict 1/r potential
Force: -1/r2
Why is the ellipse in the Keplerproblem fixed?
What ‘else’ is conserved?
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Laplace Runge Lenz Vector is constant for a strict 1/r potential.
Reference: Goldstein’s ‘Classical Mechanics’, Section 9, Chapter 3.
2 2 21 ( ) ( )2
( )
L T V m V
kV
ρ ρ ϕ ρ
ρρ
= − = + −
= −
& &
ˆA p L mkeρ= × −ur ur ur
2
0,
one requires
d p ˆdtDYNAM ICAL SYM M ETRY
dAFordt
k eρρ
=
= −
uur
r
For (angular momentum vector) to be conserved, any central force would do. [Geometrical Symmetry]
Lur
LRL figure from http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector
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pur L×
ureρ
Aur
p L×ur ur
ˆmkeρ−
Unit 5 (Sept. 1-5): Kepler Problem.Laplace-Runge-Lenz vector, ‘Dynamical’ symmetry.Conservation principle ↔ Symmetry relation.
U5L1: Kepler Problem.Laplace-Runge-Lenz vector, ‘Dynamical’ symmetry.
U5L2: Conservation principle ↔ Symmetry relation.T5: on 8th September, Monday.
T4: 1st September, Monday
Pierre-Simon Laplace1749 - 1827
Carl David TolméRunge
1856 - 1927
Wilhelm Lenz
1888 -1957
Symmetry of the H atom: ‘old’quantum theory. En ~ n-2
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A simple illustration: one-dimensional motionalong Cartesian x-axis
– this example highlights ‘additivity’ of the action integral as limit of a sum.
Hanc, Taylor and Tuleja, Am. J. Phys., 72(4), 2004
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Principle of least action: Hamilton’s principle“actionaction” as an additive property:
L=L(x,v)
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Principle of least action: Hamilton’s principle
Leads to LAGRANGE’s Eq.
To first order, the first term is the average value ∂L/∂x on the two segments A and B.
In the limit ∆t→0, this term approaches the value of the partial derivative of L at x.
In the same limit, the second term isthe time derivative of the partial derivative of the Lagrangianwith respect to velocity d(∂L/∂v)/dt.
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special case:L is NOT a function of x : “ignorable” or “cyclic” coordinate
( )
0
( )
L Ld Ldt
L m p
=∂⎧ ⎫ =⎨ ⎬∂⎩ ⎭
∂=
∂
Then the Lagrangian
and the Lagrange's equation reduces to
which means is a contant of motion.
v
v
v =v
( )L V xx x
L m p
force
linear momentum
∂ ∂= − =
∂ ∂∂
= = =∂
meaning and physical significane of the two terms?
note that the
and the vv
Newton’s Second Law!
Thus translational symmetry ( i.e. L being independent of x )
leads to the conservation of linear momentum!PCD-08
Hanc, Taylor and Tuleja, Am. J. Phys., 72(4), 2004
HOMOGENEITY WITH RESPECT TO “TIME”
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i.e. is a constant of motionL L∂⎧ ⎫−⎨ ⎬∂⎩ ⎭vv
2 2 21 1( ) ( )2 2
L L m m V x m V x∂⎧ ⎫ ⎧ ⎫− = − − = +⎨ ⎬ ⎨ ⎬∂⎩ ⎭ ⎩ ⎭
but v v v vv
The total energy of the system is a constant of motion ( is conserved)
Hanc, Taylor and Tuleja, Am. J. Phys., 72(4), 2004
Symmetry Conservation Principle
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q1
q2
X
v2
v1
Two positive charges q1 and q2 are moving along orthogonal directions as shown. They exert the Lorentz force q(E + vxB) on each other.
The coulomb repulsion between them is directed away from each other, in opposite directions.
The magnetic vxB force that the magnetic fields generated by the moving charges is however not in opposite directions.
ACTION IS NOT OPPOSITE TO REACTION !F12 ≠ - F21PCD-08
( )
1 2
12 21
1 2 0
'
d p d pdt dt
p p
Newton s III Lawas statement ofconservation oflinear momentum
= −
= −
+ =
uur uur
r r
ur ur
F Fddt
HOWEVER, WE HAVE JUST SEEN THATACTION IS NOT ALWAYS OPPOSITE TO REACTION !F12 ≠ - F21
We shall see now that it is firmly placed on HOMOGENIETY of SPACE, thus expressing the relation between ‘SYMMETRY’ and ‘CONSERVATION LAWS’ (Noether’s theorm).
Conservation of Momentum must be placed on a more robust principle.
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1
N
k k jj
F f=
=∑ur uuur
In an N-particle closed system, force on the kth particle is the sum of forces due to all other particles.
WE SHALL NOT ASSUME WHETHER OR NOT F12 = (OR ≠) - F21
Consider ‘virtual’ displacement of the entire N-particle system in homogenous space.
In such a displacement of the entire system in homogeneous space, the internal forces can do no work.
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1 1 10 . . . .
0
N N N Nk
k k jk k j k
dp dPs f s s sdt
dP
δ δ δ δ= = =
= = =
=
∑ ∑∑ ∑uur ur
uuur uuur uuur uuur uuur uuur
ur
F = dt
dt
Consider ‘virtual’ displacement of the entire N-particle system in homogenous space.
Conservation of LINEAR MOMENTUM arises from HOMEGENEITY of SPACE.
SYMMETRY CONSERVATION LAW
Noether’s TheoremPCD-08
U1L3: Applications of Lagrange’s/Hamilton’s Equations
Entire domain of Classical Mechanics
Enables emergence of ‘Conservation of Energy’and ‘Conservation of Momentum’on the basis of a single principle.
Symmetry Conservation Laws
Governing principle: Variational principle – Principle of Least ActionThese methods have a charm of their own and very many applications….
Constraints / Degrees of Freedom- offers great convenience!
‘Action’ : dimensions ‘angular momentum’ :
: :h Max Planckfundamental quantityin Quantum Mechanics
We shall now illustrate the use of Lagrange’s / Hamilton’s equations to solve simple problems in Mechanics
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Manifestation of simple phenomenain different unrelated situationsDynamics of
spring–mass systems, pendulum, oscillatory electromagnetic circuits, bio rhythms, share market fluctuations …
radiation oscillators, molecular vibrations, atomic, molecular, solid state and nuclear physics, electrical engineering, mechanical engineering …Musical instruments
ECG
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SMALL OSCILLATIONS
1581:Observations on the swaying chandeliers at the Pisa cathedral.
Galileo (then only 17) recognized the constancy of the periodic time for small oscillations. PCD-08
:
:
:
q
q
p
&
( , , )L L q q t= &
( , , )H H q p t=
Generalized Coordinate
Generalized Velocity
Generalized Momentum
Lpq
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠&
Use of Lagrange’s / Hamilton’s equations to solve the problem of Simple Harmonic Oscillator.
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( , , )
0 '
L L q q t Lagrangian
L d L Lagrange s Equationq dt q
=
⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠
&
&
( , , )
'
k kk
k kk k
H q p L
H H q p t Hamiltonian
H Hq pp q
Hamilton s Equations
= −
=
∂ ∂∀ = = −
∂ ∂
∑ &
& &
k: and
2nd order
differentialequation
TWO
1st order
differentialequations
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2 2
( , , )
2 2
0 '
2 2 02 2
0
L L q q t Lagrangian
m kL T V q q
L d L Lagrange s Equationq dt q
k d mq qdt
kq mq
mq kq
Equation of Motion for a simple harmonic
=
= − = −
⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠
⎛ ⎞− − =⎜ ⎟⎝ ⎠
− − =
= −
&
&
&
&
&&
&&
oscillator
2nd order
differentialequation
i.e.
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2 2
2 2 2
2 2
22
( , , ) ( , , )
:2 2
2 2
2 2
2 2
'
L L q q t LagrangianH H q p t Hamiltonian
m kLagrangian L T V q q
Lp mqq
m kH pq L mq q q
m kH q q
p kH qm
Hamilton s Equat
==
= − = −
⎛ ⎞∂= =⎜ ⎟∂⎝ ⎠
= − = − +
= +
= +
&
&
&&
& & &
&
ions of Motionfor a simple harmonic oscillator
22
22
( . . )'
k
H p pqp m m
H kp qq
i e f kqHamilton s Equations
TWO first order equations
∂= = =∂
∂= − = −
∂
= −
&
&
and
( , , )
( , , )
!
L L q q t
H H q p t
VERYIMPORTANT
=
=
&
p kq= −&1
2
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2 2
22
:2 2
2 2
m kLagrangian L T V q q
Lp mqq
p kH qm
= − = −
⎛ ⎞∂= =⎜ ⎟∂⎝ ⎠
= +
&
&&
( , , )
( , , )
!
L L q q t
H H q p t
VERYIMPORTANT
=
=
&
Generalized Momentum is interpreted only as and not a product of mass with velocity
Lpq
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠&
Be careful about how you write the Lagrangianand the Hamiltonian for the Harmonic oscillator!
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In an INERTIAL frame of reference recognized first as one in which motion is self-sustaining, determined entirely by initial conditions alone,
equilibrium denotes the state of rest or of uniform motion of an object along a straight line;
motion at a constant angular momentum.
Stable unstable neutral
Absolute maximum
Local vs. Absolute (Global) Extrema
Local maximum
Localminimum
Absoluteminimum Local
minimum
a bc e d
in one dimensiondUFdx
= −
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Kinds of equilibrium
unstable
stable
a bc e d
stableunstable
stableneutral
neutral
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Is a point mid-way between two equal positive point charges a point of equilibrium for a unit point positive test charge?
Can it be unambiguously classified as a point or ‘stable’ / ‘unstable’equilibrium?
what if the test charge is negative?
+ +
Saddle point
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equilibria
the point of inflexionthe tangent cuts the curve
neutral equilibrium is not possible for regular potential curves in one dimension
direction of restoring force
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( )2222 RyxKf −+=points of equilibria?
equilibria
Mexican hat
www.CartoonStock.com
Smoking is injurious to health and wealthconsider points on the circle f = 0
push tangentially, push radially
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0 0 0
2 32 3
0 0 0 02 3
1 1( ) ( ) ( ) ( ) + ( ) ...2! 3!x x x
U U UU x U x x x x x x xx x x
∂ ∂ ∂= + − + − − +
∂ ∂ ∂
meaning of ‘small oscillations’
approximations
0
22 2
0 02
0
1 1( ) ( )+ ( ) = 2! 2
choosin ( ) 0x
UU x U x x x kxx
by g U x
∂≈ −
∂=
dUF kxdx
= − = −
kx xm
= −&&
Potential for a Linear harmonic oscillator
x
U(x)
The constant term is of no physical significance.
It only adds a constant value to the potential and does not contribute to the physical force.
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dUF kxdx
= − = −
mx kx
kx xm
= −
= −
&&
&&
Potential for a Linear harmonic oscillator
x
U(x)
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l : length
E: equilibrium
S: support
θ
mg
cosθl
cos (1 cos )
h θθ
= −= −l ll
2 2 2
2 2
2 2
( , , , )
1 ( ) (1 cos )21 (1 cos )21 cos2
L L r rL T V
L m r r mg
L m mg
L m mg mg
θ θ
θ θ
θ θ
θ θ
== −
= + − −
= − −
= − +
&&
&& l
&l l
&l l l
(1 cos )V mgh mg θ= = −l
Remember this!ALWAYS, the first thing to do is to set-up the Lagrangian in terms of the generalized coordinates and the generalized velocities.
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2 21 cos2
L m mg mgθ θ= − +&l l l
2
=0rL prL p mlθ θθ
∂=
∂∂
= =∂
&
&&
Subsequently, we can find the generalized momentum for each degree of freedom.
: fixed lengthr = l
0 L d Lq dt q
⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠&
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l : length
E: equilibrium
S: support
θ
mg
cosθl
cos (1 cos )h θ
θ= −= −l ll
2 2
( , , , )1 cos2
L L r r T V
L m mg mg
θ θ
θ θ
= = −
= − +
&&
&l l l
0.
sin
LrL mgl mglθ θθ
∂=
∂∂
= − ≈ −∂
2
=0rL prL p mlθ θθ
∂=
∂∂
= =∂
&
&&
0 L d Lq dt q
⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠&
2
2
( ) 0dmgl mldt
ml mglgl
θ θ
θ θ
θ θ
− − =
= −
= −
&
&&
&&0 0
0
(1) (2) Solution: Substitute (2) in (1)
i t i tq q
q Ae Beω ωα
ω α
−= −
= +⇒ =
&&
0gl
ω =PCD-08
- - i i ii
Hamiltonian ApproachLH q L q p Lq
⎡ ⎤∂ ⎡ ⎤= =⎢ ⎥ ⎣ ⎦∂⎣ ⎦∑ ∑& &
&
2 21 cos2
(cos )(cos )(cos )
( sin )
H p m mg mg
H mgl
mgl mglH mgl
θθ θ θ
θθθ θ θ
θ θ
θθ
= − + −
⎡ ⎤∂ ∂ ∂= − ⎢ ⎥∂ ∂ ∂⎣ ⎦= − − ≈ +
∂≈
∂
& &l l l
2p mlθ θ= &
2p mlθ θ= &&&
2 Hp ml mgl
gl
θ θ θθ
θ θ
∂= = − = −
∂
= −
&&&
&&
Hpθ θ∂
= −∂
&
0gl
ω =PCD-08
gl
θ θ= −&&
0gl
ω =
For the simple pendulum oscillating in the gravitational field where the acceleration due to gravity is g, we must, and do, get the same answer regardless of which approach we employ:
(1) Newtonian(2) Lagrangian
(3) Hamiltonian
Note! We haven’t used ‘force’, ‘tension in the string’ etc. in the Lagrangian and Hamiltonian approach!
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Uniform circular motion and SHM
www.physics.uoguelph.ca
www.answers.com
2ω πν=Intrinsic natural frequency
‘reference circle’ for the Simple Harmonic Oscillator.
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Qmax
Qmin
I=0 I Imax I I=0
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max
2
2
( )
is proportional to ,
not to V, as in the case of a resistor.
mac
L
QVCdI d QV L L LQdt dtd d dVI Q Q CV Cdt dt dt
dVIdt
=
= − = − = −
= = = =
&&
&
Unlike what happens in a
resistor
the current and voltage in
an inductance L
and in a capacitor C
do not peak together.
CV
LVI Voltage lags the current in a capacitor by 900,
but leads the current in an inductor by the same amount.
2
2
0
0
1( )
L CV Vd Q QLdt C
Q QLC
− + =
+ + =
= −&&
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0 0
0
1( )
(1) (2) Most general solution: Substitute (2) in (1)
i t i t
kx xm
Q QLC
q qq Ae Beω ω
α
ω α
−
= −
= −
= −= +
⇒ =
&&
&&
&&
0
0 1
km
LC
ω
ω
=
=
Electro-mechanical analogues:
Inductance mass, inertiaCapacitance 1/k, compliance
Question:
Could we have associated L with 1/k and C with m?
PCD-08
Linear relation between restoring force and displacementfor spring-mass system:
Hooke’s law, after Robert Hooke (1635-1703), (a contemporary of Newton), who empirically discovered this relation for several elastic materials in 1678.
kx xm
= −&&
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Hooke.htmlPCD-08