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

PCD-08

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

PCD-08

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

PCD-08

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

=∂ ∂ ∂

= + + =∂ ∂ ∂

∂= =

∂

PCD-08

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?

PCD-08

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

PCD-08

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

PCD-08

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

PCD-08

Principle of least action: Hamilton’s principle“actionaction” as an additive property:

L=L(x,v)

PCD-08

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.

PCD-08

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”

PCD-08

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

PCD-08

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.

PCD-08

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.

PCD-08

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

PCD-08

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

PCD-08

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.

PCD-08

( , , )

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

PCD-08

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.

PCD-08

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

PCD-08

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!

PCD-08

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

= −

PCD-08

Kinds of equilibrium

unstable

stable

a bc e d

stableunstable

stableneutral

neutral

PCD-08

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

PCD-08

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

PCD-08

( )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

PCD-08

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.

PCD-08

dUF kxdx

= − = −

mx kx

kx xm

= −

= −

&&

&&

Potential for a Linear harmonic oscillator

x

U(x)

PCD-08

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.

PCD-08

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

⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠&

PCD-08

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!

PCD-08

Uniform circular motion and SHM

www.physics.uoguelph.ca

www.answers.com

2ω πν=Intrinsic natural frequency

‘reference circle’ for the Simple Harmonic Oscillator.

PCD-08

Qmax

Qmin

I=0 I Imax I I=0

PCD-08

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

− + =

+ + =

= −&&

PCD-08

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