Principle of Least Action

Manoj K. Harbola

Department of physics

Acknowledgement: Varun

LEAST ACTION HERODOES A RAY OF LIGHT KNOW WHERE IT'S GOING?

(Jim Holt, Lingua Franca vol. 9, No. 7,October 99)

Suppose you are standing on the beach, at some distance from the water.You hear cries of distress. Looking to your left, you see someonedrowning. You decide to rescue this person. Taking advantage of yourability to move faster on land than in water, you run to a point at the edgeof the surf close to the drowning person, and from there you swim directlytoward him. Your path is the quickest one to the swimmer − but it is not astraight line. Instead, it consists of two straight-line segments, with anangle between them at the point where you enter the water.

A

B

h1

h2

d

P.E.=0

P. E.= V>0

Which path does a particle of total energy E traveling from A to B take?

3. path of least time

1. path of least distance 2. path of least action

What does the path of least distance give us?

θ1=θ2

Least distance means motion in a straight line which implies

θ1

A

B

h1

h2

d

P.E.=0

P. E.= V>0

θ2

What does the path of least action give us?

Action = ( )22

221

2 )()(2)(2v hxdVEmhxmEdsB

A

+−−++=∫

minimization of action with respect to x gives

EVE

hxdxd

hxx

)(

)()(

22

2

21

2 −=

+−−

+

1vv)(

sinsin

1

2

2

1 <=−

=E

VEθθ

which is equivalent to

θ1

A

B

h1

h2

d

P.E.=0

P. E.= V>0

θ2x

What does the path of least time give us?

Total time =)(2

)(2v

22

221

2

VEmhxd

mEhxds

−

+−+

+=∫

minimization of time with respect to x gives

)()(

)(22

2

21

2

VEE

hxdxd

hxx

−=

+−−

+

which is equivalent to

1vv

)(sinsin

2

1

2

1 >=−

=VE

Eθθ

θ1

A

B

h1

h2

d

P.E.=0

P. E.= V>0

θ2x

When particle strikes the surface, the component of velocity along the surface remains unchanged

θ1

A

B

θ2

v1

v2

v1sinθ1

v2sinθ2

2211 sinvsinv θθ =

1vv

sinsin

1

2

2

1 <=

⇓

θθ

θ1

A

B

θ2

Trajectory of the particle is thepath of least action

Principle of least action

When a particle of fixed energy travels from point A to point B, itstrajectory is such that the corresponding action has the minimumpossible value.

v

xA B

For motion in a straight line

Action=area

Test case 1: Can a particle traveling in a straight line from A to B suddenlyreverse its direction of motion, go back for some distance, reverse its motionagain and reach point B?

Principle of least action prevents that from happening

v

xA B x

v

A B

Test case 2: A cricket ball hit so that it reaches a fielder

x

y ∆y

Let actual path be y(x)

Let a nearby path be y(x)

∫∫

+−==

B

A

B

A

dxdxdymgyE

mds

2

1)(2vAAction for the actual path y(x)

Change in action for a nearby path y(x)

∫∫

′+−==

B

A

B

A

dxymgyEm

ds 21)(2vA δδδ

must be zero if action for the actual path is minimum

Change in action arises from:

(i) Change in the speed )()()( xymgyEdydmgyE δδ

−=−

δy(x+∆x)

δy(x)

x x+∆x

(ii) Change in the length of trajectory

dxyyydddxy ′

′+′

=′+ δδ 22 11

)()()( xydxd

xxyxxyy δδδδ =

∆−∆+

=′

where

Change in action therefore is

∫

′+′

−+−′+=B

A

dxxydxdy

yddmgyExymgyE

dydy )(1)()()(1A 22 δδδ

∫

′+′

−−−′+=B

A

dxxyyyddmgyE

dxdmgyE

dydy )(1)()(1A 22 δδ

Integration by parts leads to

Since δy(x) is arbitrary, δA=0 implies

01)()(1 22 =

′+′

−−−′+ yyddmgyE

dxdmgyE

dydy

This simplifies to

0)1()(2 2 =′++−′′ ymgmgyEy

Equation of the trajectory

0)1()(2 2 =′++−′′ ymgmgyEy

Integration of the equation leads to

1)(12 −−=′ mgyECy

This gives2

21

1

1 )(4

)1( CxmgCmgC

ECy +−−

=

Conditions y(0)=0 and y(a)=0 leads to

21

1

1

24)1(

−−

−=

axmgCmgC

ECy

1)(12 −−=′ mgyECyPut y’=0 in to get

mgCECy1

1max

1−=

2

maxmax 2)(4

−

−−=

axmgyE

mgyy

and

Again the condition y(0)=0 givesmg

agmEEy

442 2222

max−±

=

Thus there are two parabolic trajectories that the ball can take

a x

y

ymax1

ymax2

Comparison with Newtonian approach:

Given initial position and velocity of a particle, Newtonian methodbuilds up its trajectory in an incremental manner by updating thevelocity and position. Energy of the particle may or may not be fixed.

Principle of least action says if a particle of fixed energy has to go frompoint A to point B, the path it takes is that which minimizes the action.

But this can't be right, can it? Our explanation for the route taken by thelight beam (particle in our case) − first formulated by Pierre de Fermatin the seventeenth century as the principle of least time (principle ofleast action in the present case) − assumes that the light (particle)somehow knows where it is going in advance and that it actspurposefully in getting there. This is what's called a teleologicalexplanation. (Jim Holt)

The idea that things in nature behave in goal-directed ways goes backto Aristotle. A final cause, in Aristotle's physics, is the end or telostoward which a thing undergoing change is aiming. To explain a changeby its final cause is to explain it in terms of the result it achieves. Anefficient cause, by contrast, is that which initiates the process ofchange. To explain a change by its efficient cause is to explain it interms of prior conditions.

One view of scientific progress is that it consists in replacingteleological (final cause) explanations with mechanistic (efficientcause) explanations. The Darwinian revolution, for instance, can beseen in this way: Traits that seemed to have been purposefullydesigned, like the giraffe's long neck, were re-explained as theoutcome of a blind process of chance variation and natural selection.

(Least Action Hero, Jim Holt, Lingua Franca vol. 9, No. 7, October 99)

Plan of the talk:Aristotle and the motion of planets

Reflection of light and Hero of Alexandria

Fermat’s principle of least time for light propagation; Descartes versus Fermat

Wave theory and Fermat’s principle

Maupertuis’ principle of least action

Euler-Lagrange formulation

Hamilton’s investigations

Quantum connections

ARISTOTLE (384-322 BC)

Aristotle on the motion of planets:

If the motion of the heaven is the measure of all movementswhatever in the virtue of being alone continuous and regularand eternal, and if, in each kind, the measure is theminimum, and the minimum movement is the swiftest,then clearly, the movement of the heaven must be theswiftest of all movements. Now of lines which return uponthemselves the line which bounds the circle is the shortest;and that movement is the swiftest which follows theshortest line. Therefore, if the heaven moves in a circleand moves more swiftly than anything else, it mustnecessarily be spherical.

REFLECTION OF LIGHT

&

HERO OF ALEXANDRIA(125 BC)

Whatever moves with unchanging velocity moves in astraight line…. For because of the impelling force the objectin motion strives to move over the shortest possibledistance, since it has not the time for slower motion, that is,for motion over a longer trajectory. The impelling force doesnot permit such retardation. And so, by reason of its speed,the object tends to move over the shortest path. But theshortest of all lines having the same end points is a straightline……Now by the same reasoning, that is, by aconsideration of the speed of the incidence and thereflection, we shall prove that these rays are reflected atequal angles in the case of plane and spherical mirrors. Forour proof must again make use of minimum lines.

Let a light ray start from point Aand reach point B after reflection.The true path is AOB such thatrays AO and BO make equalangles from the mirror.

A

B

OO1

B1

C

Draw an alternate path AO1B

Drop a perpendicular BC on the mirror and extend it to B1 so that BC=B1C. Join O and B1 and O1 and B1.

From congruency of ∆BOC and ∆B1OC and the fact that AO and BOmake equal angles from the mirror, it follows that AOB1is a straight line.

In ∆AO1B1: AO1+O1B1 > AB1( = AO+OB1=AO+OB)

Path AOB is the shortest

Proof:

REFRACTION OF LIGHT

&

FERMAT’S PRINCIPLE OF LEAST TIME

Refraction of light and Snell’s law (1621):

constantsinsin

2

1 =θθ

θ1

θ2

medium 1

medium 2For medium 2 denser than medium 1

constant > 1

Historical note: There is evidence that Thomas Hariot in England had also discovered the same law around 1600.

Newton (1642-1727)

Descartes’ explanation (1637) of Snell’s law:

Argument given by Descartes is a mechanical one, based on the fact that the component of velocity along the surface remains unchanged

Descartes explained the constancy of the ration of sine of angles in terms ofthe ratio of the speed of light in the two media

By Descartes’ explanation, light had to be moving faster in the densermedium

2211 sinvsinv θθ =

1

2

2

1

vvconstant

sinsin

==θθ

θ1v1

v1sinθ1

v2sinθ2

v2

θ2

Descartes (1596-1650) Newton (1642-1727)

Fermat’s principle of least time (1658):

Generalization of Hero’s explanation of reflection to include refraction of light.

During refraction a light ray does not take the path of least distance; thatwould be a straight line.

Between two points, a light ray travels in such a manner that it take theleast time.

For reflection this leads to equal angles of incidence and reflection

θ1

A

θ2

B

2

1

2

1

vvconstant

sinsin

==θθ

For refraction this implies

By Fermat’s principle of least time, light moves slower in the densermedium.

Fermat (1601-1665) Newton (1642-1727)

Descartes versus Fermat: Descartes (1596-1650) Fermat (1601-1665) Newton (1642-1727)

Descartes believed that light traveled infinitely fast.

Fermat on Descartes:

1. “of all the infinite ways to analyze the motion of light the author hastaken only that one which serves him for his conclusion; he hastherefore accommodated his means to his end, and we know as littleabout the subject as we did before.”

2. rejects Descartes’ assertion of infinite speed of light and therefore hisillogical conclusion that light travels faster in water than in air .

According to Fermat light traveled at finite speed in air and slowed down in water.

Experimental verification of finiteness of speed of light – 1675 by Roemer

Measurement of speed of light in water – 1850 by Fizeau and Foucault

These observations CONFIRM Fermat’s principle of least time

Objections of Cartesians to Fermat’s theory

From Clerselier’s letter to Fermat:

“That path, which you reckon the shortest because it is the quickest, isonly a path of error and bewilderment, which Nature in no way follows andcannot intend to follow. For, as Nature is determinate in everything shedoes, she will only and always tend to conduct her works in a straight line”

Clerselier on the velocity of light:

“M. Descartes − in 23rd page of his Dioptrique − proves and does notsimply suppose, that light moves more easily through dense bodies thanthrough rare one”

Fermat’s reply

“I have often said to M. de la Chambre and yourself that I do not claimand that I have never claimed, to be in the private confidence ofNature.

She has obscure and hidden ways that I have never had the initiativeto penetrate; I have merely offered her a small geometrical assistancein the matter of refraction, supposing that she has need of it.

But since you, Sir, assure me that she can conduct her affairs withoutthis, and that she is satisfied with the order that M. Descartes hasprescribed for her, I willingly relinquish my pretended conquest ofphysics and shall be content if you will leave me with a geometrialproblem, quite pure and in abstracto, by means of which there can befound the path of a particle which travels through two different mediaand seeks to accomplish its motion as quickly as it can.”

HUYGENS’ WAVE THEORY

&

FERMAT’S PRINCIPLE

Huygens’ wave theory (1629-1695) and the principle of least time

According to Huygens’ theory, light travels as a wave with path of light ray being in the direction perpendicular to the wavefront

A

BConsider the true ray path from A to B and also an alternate path. Because true path is perpendicular to the wavefronts,

AB (true path) < AB (alternate path)

Newton (1642-1727)

Actual principle is the principle of stationary (minimum, maximum or saddle point) time

Consider a light beam that starts in adiverging manner from point A and thenconverges to point B.

A

B

All nearby paths of light should not havedifferent time of arrivals implyingstationarity of time of travel.

Example 1: In an elliptical mirror, light starting from one focus of theellipse and reaches the other focus after reflection from the mirror. Thereare many possible paths for this and all of them are equal.

F1

F2

Example 2: An elliptical mirror

minimum time maximum time

Example 3: A circular mirror A B

minimum time stationary time

45°

stationary time

60°

How does light know which path to take?

Wave Theory: Light does not know which path to take. It takes allpossible paths with certain probability amplitude and phase and theseprobability amplitudes interfere. The phase depends on the time ofpassage.

Path of least time is where the interference is constructive to thelargest extent possible. This is because phase for other nearby pathsdoes not vary much.

Light source Image

Does light really go through all possible paths?

Experiment : Take a pointdifferent from the image.No light from the sourcereaches the that point.

Light source No light

Now blacken parallel stripson the mirror to remove lightof opposite phase. It formsa grating and light reachesmany different points.

Light source Image

PRINCIPLE OF LEAST ACTIONMAUPERTUIS, LAGRANGE & EULER

Principle of Least Action

Nature acts in a way so that it renders a quantity called action aminimum

Maupertuis in “The agreement between different laws of Nature that had,until now, seemed incompatible” read on April 15, 1744 to Académie dessciences.

Newton (1642-1727)

Action is defined as the product of the mass, the velocity and thedistance.

sm ××= vAction

Comment: Maupertuis’ attempts was to explain the propagation of lightand movement of a particle by a single principle.

Example 1: To find the final velocity of two masses involved in a perfectinelastic collision.

u1 u2m1 m2

v

Treat distance as that covered in one second, that is as velocity.

222

211 )v()v(actioninchange −+−= umum

Minimization of change in action with respect to v leads to

21

2211vmm

umum++

=

Example 2: To find the relationship between final velocities of twomasses involved in a perfect elastic collision.

u1 u2m1 m2v1

v2m1 m2

2222

2111 )v()v(actioninchange −+−= umum

In a perfect elastic collision: 1221 vv uu −=−

Minimization of change in action with respect to v1 leads to

22112211 vv mmumum +=+

Example 3: Refraction of light

θ1

A

θ2

B

O

OBAO ×+×= 21 vvAction

Minimization of action leads to

constantvv

sinsin

1

2

2

1 ==θθ

Snell’s law is verified through the principle of least action, and agreeswith Descartes’ conclusions.

Lagrange (1736-1813) on Maupertuis’ principle in Mécanique Analytique, 1788

“This principle, looked at analytically, consists in that, in themotion of bodies which act upon each other, the sum of the product ofthe masses with the velocities and with the distances travelled is aminimum. The author deduced from it the laws of reflection andrefraction of light, as well as those of the impact of bodies.

But these applications are too particular to be used forestablishing the truth of a general principle. Besides, they havesomewhat vague and arbitrary character, which can only render theconclusions that might have been deduced from the true correctness ofthe principle unsure………………

But there is another way in which it may be regarded, moregeneral, more rigorous, and which itself merits the attention of thegeometers. Euler gave the first hint of this at the end of his Traité desisopérimètres, printed at Lausanne in 1744.”

Principle of Least Action in Mechanics

Proper mathematical foundation is provided by Euler (1744)

Before paying attention to this problem, Euler had already developedCalculus of Variations and given the Euler condition for making thevariation of an integral of the form

∫ ′),(

),(

22

11

),,(yx

yx

dxxyyf

between two fixed points (x1,y1) and (x2,y2) vanish with respect to arbitraryvariations in y(x)

Euler condition

0=∂∂

−

′∂

∂yf

yf

dxd

Consider a particle moving in xy plane under the influence of a force with x-component Fx and y-component Fy

Centripetal force

2

yx2

1v

y

FyFr

m′+

−′=

Aim is to see if the principle of least action gives the same answer

Fx

Fy

X

Y

∫∫ ′+== dxymdsm 21vvAction

Euler condition gives

( ) ( ) 01v1v 22 =′+∂∂

−

′+

′∂∂ y

yy

ydxd

Use the relations

y2

x2 v

21,v

21 Fm

yFm

x=

∂∂

=

∂∂

and

yy

xdxd

∂∂′+

∂∂

=

( ) 2

yx232

2

11v

y

FyF

yym

′+

+′−=

′+

′′

( )yyr′′′+

=21curvatureofradius

Fx

Fy

X

Y

2

yx2

1v

y

FyFr

m′+

−′=

MAKING ACTION STATIONARY LEADS TO THE CORRECT FORCE BALANCE EQUATION

What does the minimum action principle imply for one-dimensional motion?

∫= dxxA )(v

)()(v)(v ∫∫ += dxxdxxA δδδ

t

xδx

If the total energy is constant

( )

( )

∂∂

−=

∂∂

−−=

−∂∂

=

xxxU

m

xxxU

xUEm

xxUEmx

x

δ

δ

δδ

)(v

1

)()(2

1

)(2)(v

{ } { }

( ))(

)()()()(

xdxxxx

xxxxxxxxxdx

δ

δδδ

=−∆=∆−∆=

−−∆+−∆+=−∆+=

δx

δ(x+∆x)

∆x

∆x

δ(∆x) ∆(δx)

∆t

x

∫∫

∫∫

+∂

∂

−=

+∂

∂

−=

xdxdxxxU

m

xdxdxxxU

mA

δδ

δδδ

v)(v

1

)(v)(v

1

If the end points of the trajectory are kept fixed

∫∫ +∂

∂

−= xdxdx

xxU

mA δδδ v)(

v1

Making Action stationary is absolutely equivalent to a particle’s equation of motion

For many interacting particles also, minimization of leads to the equation of motion (Lagrange)

∑∫i

iii dsvm

{ }( )dt

dmr

rU ii

i

i α

α

v=

∂∂

−

Making Action stationary is equivalent to Newton’s second law

Since δx(t) is arbitrary, δA=0 implies that

v)(v

1 ddxxxU

m=

∂∂

− or

dtdm

xxU v)(

=∂

∂− using dtdx

=v

Does the principle of least action teach us anything new?

In a characteristic way, the principle of least action did not at firstexercise an appreciable effect on the advance of science, even afterLagrange had completely established it as a part of mechanics. It wasconsidered more as an interesting mathematical curiosity and anunnecessary corollary to Newton’s laws of motion. Even in 1837Poisson could only call it “a useless rule”. (From an essay byPlanck)

This, however, changed when Hamilton (1805-1865) entered the scenein 1830’s

HAMILTON’S PRINCIPLE OF VARYING ACTION

PARALLEL BETWEEN GEOMETRIC OPTICS AND MECHANICS

Consider the action integral

dsxA ∫= )x,x,v(x)( 321∆s

as a function of the end points of the true path. The integral is obviously taken along the true path.

As the path is increased by ∆s to the next point, we have

)(vOR)(v xdsdAsxA =∆=∆

Can this equation be used instead to find the path taken by light?

Function v(x1,x2,x3;α1,α2,α3) is considered to be a function of the directionalcosines {αi} of the ray of light. Making action stationary with respect tovariations in {xi; αi} gives the equation for {αi} .

Conventional approach (Hamilton):

Recall that in an earlier minimization, the integrand was taken to be function ofy(x) and y′(x). Thus

∫ ′+= dxyyA 21)(vy(x) is then found by making the variation of the action vanish with respect tovariations δy(x).

Now the independent variables are taken to be {xi; αi} instead. Thus

∫= dsxA ii });({v α

{αi(x)} are found by making the variation of the action vanish with respect tovariations {δxi} and {δαi}. Using make v({xi;αi}) homogeneous

Of degree 1 in {αi}

123

22

21 =++ ααα

Take the true path and a varied path around itobtained by shifting the line element by {δxi);by changing its length by δ(ds) and bychanging its directional cosines by {δαi}

∫ ∫∫ +== )(vvv dsdsdsA δδδδδx

δy

dx

dx+δ(dx)

∫∑∫ +

∂∂

+∂∂

= )(vvv dsdsdsxx

Ai

ii

ii

δδαα

δδ

)()()()( iiiii xddxdsdsds δδαδδαδα ===+

)()( dsxdds iii δαδδα −=⇒

∫ ∑∑∫

∂∂

−+

∂∂

+∂∂

= )(vv)(vv dsxddsxx

Ai

iii

ii

ii

δαα

δα

δδ

Now consider the variation of action integral between the true path and the varied path

Integration by parts leads to

∫ ∑

∫∑∑

∂∂

−+

∂∂

−∂∂

+

∂∂

−

∂∂

=

)(vv

vvvv 01

ds

xddsx

xxA

ii

i

iiii

iii

ii

δαα

δα

δα

δα

δ

δx1(0)= variation of coordinate at the final (initial) point of the path

∑ ∂∂

=i i

i αα vv since v is a homogeneous function of {αi} of degree 1

Demand that δA vanish for arbitrary variations with the end points of the pathfixed i.e. δx1/0=0. This gives

∫ ∑

∫∑∑

∂∂

−+

∂∂

−∂∂

+

∂∂

−

∂∂

=

)(vv

vvvv 01

ds

xddsx

xxA

ii

i

iiii

iii

ii

δαα

δα

δα

δα

δ

Differential equation for the path of light rayii

ddsx α∂

∂=

∂∂ vv

23

22

21321 ),,(v);(v αααα ++= xxxxThus Note: 12

322

21 =++ ααα

∑∑∫

∂∂

−

∂∂

=i

iii

ii

xxds 01 vvv δα

δα

δ

Now consider the integral

with the initial point fixed and δx1 non zero and in the direction of the path.

)0(1 == iii sx δαδαδ

ssxdsi

iii

ii

δδαα

δα

δ vvvv 1 =

∂∂

=

∂∂

= ∑∑∫Then

The action is a function of the end points of the true path∫ dsvCONCLUSION: Stationary action implies existence of a characteristic function A(x) such that

iixxA

α∂∂

=∂∂ v)(

1

How to find the path if A(x) and v(x,α) are given?

From the equation solve for as a function of

(x1,x2,x3)iix

xAα∂∂

=∂∂ v)(

1),,( 321 ααα

Differential equation for the characteristic function A(x)

vv)(1 i

iixxA α

α=

∂∂

=∂∂

123

22

21 =++ ααα

)(v)()()( 22

3

2

2

2

1

xx

xAx

xAxxA

=

∂∂

+

∂∂

+

∂∂

v(x) is the refractive index of the medium

Direction of light ray and surfaces of constant Action:

Direction of light ray ( )

)(

ˆAˆAˆA

ˆvˆvˆvˆˆˆvˆˆˆ

33

22

11

33

22

11

332211332211

xA

xx

xx

xx

xxx

xxxxxx

∇=

∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

=

++=++

ααα

αααααα

Thus light ray moves in the direction of the gradient of the characteristicfunction

EQUIVALENTLY

If the points of equal action for each ray are joined together, light raysmove in the direction perpendicular to surface so formed i.e.perpendicular to the surfaces of constant action

constant),,( 321 =xxxA

Light rays and surfaces of constant action

light rays

surfaces of constant action

light rays

wavefronts

Huygens theory of light waves

ACTION

is

EQUIVALENT

to

SPACE-PART

of

PHASE

Mechanical systems:

In going from point A to point B, a particle also satisfies the principle ofleast action

0v =∫ dsδ

23

22

21321

23

22

21 ),,(vvvvv ααα ++=++= xxx is the speed of the particle

Thus there exists a characteristic function A(x) for a mechanical systemalso such that

})({v})({v ii

ii xAOR

xxA

∇=∂

∂=

And the path of a particle can be determined if we know the characteristic function

Equation for the characteristic function

)(v)()()( 22

3

2

2

2

1

xx

xAx

xAxxA

=

∂∂

+

∂∂

+

∂∂

From the energy conservation equation

})({v21 2

ixUmE +=

Thus the equation for the characteristic function is

mExmUx

xAx

xAxxA

i 2})({2)()()(2

3

2

2

2

1

=+

∂∂

+

∂∂

+

∂∂

Example: A projectile thrown with initial speed v0 at an angle φ0 in agravitational field

mgyyU =)(

The equation for the characteristic function

)2v( 20

22

gymyA

xA

−=

∂∂

+

∂∂

Solve the equation by separation of variables to get

23220

23220 )2v(

31)(v

31),( gyk

gxkk

gyxA −−−+−=

Values of A(x,y) are obtained by substituting for x and y, the coordinates ofa trajectory

23220

23220 )2v(

31)(v

31),( gyk

gxkk

gyxA −−−+−=

Integration of these equations leads to

20000 2

1sinvcosv gttytx −== φφ

These give the trajectory and the action for projectile motion

gygykyA

kxA

y

x

2sinv2vv

cosvv

022

022

0

00

−=−−=∂∂

=

==∂∂

=

φ

φ

Velocity of the projectile

Action

Trajectories are lines perpendicular to the surfaces of constant action

Surfaces of constant action

Question: Can we associate the action of a particle with a phase?

Trajectories of the projectile

Mechanical motion of a particle is like the motion of ray of light andtherefore equivalent to Geometric optics.

QUANTUM CONNECTIONS

Fast forward to 1920s: When it was discovered that particles have awave associated with them, Hamilton’s theory became the natural choice toaccount for it and develop the quantum-mechanical wave equation.

How Schrödinger obtained the wave equation (Ist paper by Schrödinger)

mExmUx

xAx

xAxxA

i 2})({2)()()(2

3

2

2

2

1

=+

∂∂

+

∂∂

+

∂∂

Start with the equation for the action

Treating action like phase, take the wavefunction Ψ as

Ψ==Ψ log)(OR)/)(exp( KxAKxA

Substitute this wavefunction in the equation for action to obtain a quadratic form in Ψ, which is equal to zero

( ) 0})({2)()()( 22

3

2

2

2

1

=Ψ−+

∂Ψ∂

+

∂Ψ∂

+

∂Ψ∂ ExUm

xx

xx

xx

i

Rather than looking for solutions of this equation, seek a function Ψ such thatthe integral of the quadratic form above over all space is stationary for anyarbitrary variations of Ψ.

( ) 0})({2)()()( 22

3

2

2

2

1

=

Ψ−+

∂Ψ∂

+

∂Ψ∂

+

∂Ψ∂

∫ rdExUmx

xx

xx

xi

δ

For well-behaved Ψ vanishing at infinity, this leads to the Schrödinger equation

Ψ=Ψ+Ψ∇− EUm

K 22

2

Space part of the phase of matter waves = A(x)

Frequency of the waves = hE ; h = Planck’s constant

Direct connection (IInd paper by Schrödinger):

As the wavefront moves with phase velocity uphase covering distance ∆x intime ∆t, we have

0)( =∆−∆=∆ tExAmφ

Calculate the phase velocity of the wave treating surfaces of constantaction as wavefronts

−

=h

EtxmA )(2πφ ; m = mass of the particle

0)( =∆−∆=∆ tExAmφ xxxAxA particle∆=∆∂∂

=∆ v)(

This gives

)(2mv UEmEE

txu

particlephase −

==∆∆

=

particle

phaseparticlephase m

hUEm

hhE

uu

v)(2ANDv =

−==≠ λ

Group velocity of the waves

particlegroup UEmEh

E

ku v

)(21=

−∂∂

=

∂

∂

=∂∂

=

λ

ω

And finally the wave equation

01 22

2

2 =Ψ∇−∂Ψ∂tu phase

leads to the Schrödinger equation

Ψ=Ψ+Ψ∇− EUm

h 22

2

8π

−Ψ=Ψ

−= t

hEixtx

UEEuphase π2exp)();(AND

)(2

Substituting

Classical Mechanics: Path of a particle is that of least action and thereforenormal to the surfaces of constant action

Quantum Mechanics: Because of the waves associated with a particle, itdoes not know which path to take. It takes all possible paths with certainprobability amplitude and phase and these probability amplitudes interfere.The phase depends on the action.

Path of least action is where the interference is constructive to the largestextent possible.

Classically we see only those result when amplitudes interfereconstructively giving a large final amplitude

A comparison between Classical and Quantum Mechanics (Feynman):

Planck on the Principle of Least Action :

As long as physical science exists, the highest goal to which it aspires isthe solution of the problems of embracing all natural phenomena,observed and still to be observed, in one simple principle which will allowall past and, especially, future occurrences to be calculated.

Among the more or less general laws, the discovery of which characterizethe development of physical science during the last century, the principleof Least Action is at present certainly one which, by its form andcomprehensiveness, may be said to have approached most closely to theideal aim of theoretical inquiry.

Thank you

According to Planck:"on this occasion everyone has to decide for himself which point of view hethinks is the basic one."

You can be a teleologist if you wish. You can be a mechanist if that bettersuits your fancy. Or you may be left wondering whether this is yet anothermetaphysical distinction that does not make a difference.