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Home , Action (physics)

Principle of Least Action

Manoj K. Harbola Department of

Acknowledgement: Varun LEAST ACTION HERO DOES 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 from the water. You hear cries of distress. Looking to your left, you see someone drowning. You decide to rescue this person. Taking advantage of your ability to move faster on land than in water, you run to a point at the edge of the surf close to the drowning person, and from there you swim directly toward him. Your path is the quickest one to the swimmer − but it is not a straight line. Instead, it consists of two straight-line segments, with an between them at the point where you enter the water. Which path does a particle of total E traveling from A to B take?

A

P.E.=0 h1

h2 P. E.= V>0 B d

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

3. path of least What does the path of least distance give us?

A

Least distance means motion in a

straight line which implies P.E.=0 h1 θ1

θ1=θ2

θ2

h2 P. E.= V>0

B d What does the path of least action give us?

B = 2 + 2 + − − 2 + 2 Action = ∫ vds 2mE(x h1 ) 2m(E V )((d x) h2 ) A

A minimization of action with respect to x gives

x 2 2 P.E.=0 h1 x + h (E −V ) θ1 1 = (d − x) E 2 2 (d − x) + h2 θ x 2 h2 which is equivalent to P. E.= V>0 B d sinθ (E −V ) v 1 = = 2 <1 sinθ2 E v1 What does the path of least time give us?

ds x2 + h2 (d − x)2 + h2 Total time = = 1 + 2 ∫ v 2mE 2m(E −V )

A minimization of time with respect to x gives

x 2 2 P.E.=0 + h1 x h1 E θ = 1 (d − x) (E −V ) 2 2 (d − x) + h2 θ2 x h2 which is equivalent to P. E.= V>0 B d sinθ E v 1 = = 1 >1 sinθ2 (E −V ) v2 When particle strikes the surface, the component of along the surface remains unchanged

A v1sinθ1

v1 sinθ1 = v2 sinθ2 v1 θ 1 ⇓

θ sinθ1 v2 v2sin 2 = <1 θ 2 sinθ2 v1

v2 B A

θ1 of the particle is the path of least action

θ2 B Principle of least action

When a particle of fixed energy travels from point A to point B, its trajectory is such that the corresponding action has the minimum possible value.

For motion in a straight line Action=

v

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

Principle of least action prevents that from happening

v v

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

y ∆y Let actual path be y(x)

Let a nearby path be y(x) x 2 B B 2  dy  Action for the actual path y(x) A = ∫ vds = ∫ (E − mgy) 1+   dx A A m  dx 

Change in action for a nearby path y(x)

B B  2 2  δA = δ ∫ vds = δ ∫  (E − mgy) 1+ y′  dx A A m  must be zero if action for the actual path is minimum Change in action arises from:  d  (i) Change in the δ (E − mgy) =  (E − mgy) δ y(x)  dy 

(ii) Change in the of trajectory  d  ′2  ′2  ′ δ ∆ δ 1+ y dx =  1+ y δy dx y(x+ x)  dy′  δy(x) where δ y(x + ∆x) −δ y(x) d δ y′ = = δ y(x) x x+∆x ∆x dx

Change in action therefore is

B  d d d  δA = 1+ y′2 (E − mgy)δ y(x) + (E − mgy) 1+ y′2 δ y(x) dx ∫  ′  A  dy dy dx  Integration by parts leads to

B  d d  d  δA =  1+ y′2 (E − mgy) −  (E − mgy) 1+ y′2  δ y(x)dx ∫  ′  A  dy dx  dy 

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

2 d d  d 2  1+ y′ (E − mgy) −  (E − mgy) 1+ y′  = 0 dy dx  dy′ 

This simplifies to

2y′′(E − mgy) + mg(1+ y′2 ) = 0 Equation of the trajectory

2y′′(E − mgy) + mg(1+ y′2 ) = 0

Integration of the equation leads to

2 y′ = C1(E − mgy) −1

This gives (C1E −1) C1mg 2 y = − (x + C2 ) C1mg 4

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

2 (C E −1) C mg  a  y = 1 − 1  x −  C1mg 4  2  2 C E −1 ′ 1 Put y’=0 in y = C1(E − mgy) −1 to get ymax = and C1mg 2 mg  a  y = ymax −  x −  4(E − mgymax )  2 

2E ± 4E 2 − m2 g 2a2 Again the condition y(0)=0 gives y = max 4mg

Thus there are two parabolic that the ball can take

y

ymax2

ymax1

a x Comparison with Newtonian approach:

Given initial and velocity of a particle, Newtonian method builds up its trajectory in an incremental manner by updating the velocity 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 from point 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 the light beam (particle in our case) − first formulated by Pierre de Fermat in the seventeenth century as the principle of least time (principle of least action in the present case) − assumes that the light (particle) somehow knows where it is going in advance and that it acts purposefully in getting there. This is what's called a teleological explanation. (Jim Holt) The idea that things in nature behave in goal-directed ways goes back to Aristotle. A final cause, in Aristotle's physics, is the end or telos toward which a thing undergoing change is aiming. To explain a change by its final cause is to explain it in terms of the result it achieves. An efficient cause, by contrast, is that which initiates the process of change. To explain a change by its efficient cause is to explain it in terms of prior conditions.

One view of scientific progress is that it consists in replacing teleological (final cause) explanations with mechanistic (efficient cause) explanations. The Darwinian revolution, for instance, can be seen in this way: Traits that seemed to have been purposefully designed, like the giraffe's long neck, were re-explained as the outcome 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 movements whatever in the virtue of being alone continuous and regular and eternal, and if, in each kind, the measure is the minimum, and the minimum movement is the swiftest, then clearly, the movement of the heaven must be the swiftest of all movements. Now of lines which return upon themselves the line which bounds the circle is the shortest; and that movement is the swiftest which follows the shortest line. Therefore, if the heaven moves in a circle and moves more swiftly than anything else, it must necessarily be spherical. REFLECTION OF LIGHT & HERO OF ALEXANDRIA (125 BC) Whatever moves with unchanging velocity moves in a straight line…. For because of the impelling the object in motion strives to move over the shortest possible distance, since it has not the time for slower motion, that is, for motion over a longer trajectory. The impelling force does not permit such retardation. And so, by reason of its speed, the object tends to move over the shortest path. But the shortest of all lines having the same end points is a straight line……Now by the same reasoning, that is, by a consideration of the speed of the incidence and the reflection, we shall prove that these rays are reflected at equal in the case of plane and spherical mirrors. For our proof must again make use of minimum lines. Proof: A Let a light ray start from point A B and reach point B after reflection. The true path is AOB such that rays AO and BO make equal O1 O C angles from the mirror.

Draw an alternate path AO1B B1

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 BO make 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 REFRACTION OF LIGHT & FERMAT’S PRINCIPLE OF LEAST TIME (1642-1727)

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

sinθ θ 1 medium 1 1 = constant sinθ2

θ 2 For medium 2 denser than medium 1 medium 2 constant > 1

Historical note: There is evidence that Thomas Hariot in England had also discovered the same law around 1600. Descartes (1596-1650) 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

v sinθ 1 1 θ = θ v1 v1 sin 1 v2 sin 2 θ1

v sinθ sinθ v 2 2 1 = constant = 2 θ 2 sinθ2 v1 v2

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

By Descartes’ explanation, light had to be moving faster in the denser medium Fermat (1601-1665) Fermat’s principle of least time (1658): Newton (1642-1727)

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; that would be a straight line. Between two points, a light ray travels in such a manner that it take the least time. For reflection this leads to equal angles of incidence and reflection

A For refraction this implies

θ1 sinθ v 1 = constant = 1 θ2 sinθ2 v2

B

By Fermat’s principle of least time, light moves slower in the denser medium. Descartes (1596-1650) Fermat (1601-1665) Descartes versus Fermat: 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 has taken only that one which serves him for his conclusion; he has therefore accommodated his means to his end, and we know as little about the subject as we did before.” 2. rejects Descartes’ assertion of infinite speed of light and therefore his illogical 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, is only a path of error and bewilderment, which Nature in no way follows and cannot intend to follow. For, as Nature is determinate in everything she does, 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 not simply suppose, that light moves more easily through dense bodies than through rare one” Fermat’s reply

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

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

But since you, Sir, assure me that she can conduct her affairs without this, and that she is satisfied with the order that M. Descartes has prescribed for her, I willingly relinquish my pretended conquest of physics and shall be content if you will leave me with a geometrial problem, quite pure and in abstracto, by means of which there can be found the path of a particle which travels through two different media and seeks to accomplish its motion as quickly as it can.” HUYGENS’ WAVE THEORY & FERMAT’S PRINCIPLE Newton (1642-1727)

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

Consider the true ray path from A to B B and also an alternate path. Because true path is perpendicular to the wavefronts, A AB (true path) < AB (alternate path) Actual principle is the principle of stationary (minimum, maximum or saddle point) time

Consider a light beam that starts in a B diverging manner from point A and then converges to point B.

All nearby paths of light should not have A different time of arrivals implying stationarity of time of travel.

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

F1

F2 Example 2: An elliptical mirror

minimum time maximum time

stationary time minimum time

45° Example 3: A circular mirror A B 60°

stationary time How does light know which path to take? Wave Theory: Light does not know which path to take. It takes all possible paths with certain and phase and these probability amplitudes interfere. The phase depends on the time of passage.

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

Light source Image Does light really go through all possible paths?

Light source No light

Experiment : Take a point different from the image. No light from the source reaches the that point.

Light source Image Now blacken parallel strips on the mirror to remove light of opposite phase. It forms a grating and light reaches many different points. PRINCIPLE OF LEAST ACTION MAUPERTUIS, LAGRANGE & EULER Newton (1642-1727) Principle of Least Action

Nature acts in a way so that it renders a quantity called action a minimum Maupertuis in “The agreement between different laws of Nature that had, until now, seemed incompatible” read on April 15, 1744 to Académie des sciences.

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

Action is defined as the product of the , the velocity and the distance. Action = m× v× s Example 1: To find the final velocity of two involved in a perfect inelastic collision.

u u v m1 1 2 m2

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

2 2 change in action = m1(u1 − v) + m2 (u2 − v)

Minimization of change in action with respect to v leads to m u + m u v = 1 1 2 2 m1 + m2 Example 2: To find the relationship between final of two masses involved in a perfect elastic collision.

u u v v2 m1 1 2 m1 1 m2 m2

2 2 change in action = m1(u1 − v1) + m2 (u2 − v2 )

In a perfect elastic collision: v1 − v2 = u2 − u1

Minimization of change in action with respect to v1 leads to

m1u1 + m2u2 = m1v1 + m2v2 Example 3: Refraction of light

A Action = v1 × AO + v2 ×OB

θ1 O Minimization of action leads to

θ2 sinθ1 v2 B = = constant sinθ2 v1

Snell’s law is verified through the principle of least action, and agrees with Descartes’ conclusions. Lagrange (1736-1813) on Maupertuis’ principle in Mécanique Analytique, 1788

“This principle, looked at analytically, consists in that, in the motion of bodies which act upon each other, the sum of the product of the masses with the velocities and with the travelled is a minimum. The author deduced from it the laws of reflection and refraction of light, as well as those of the impact of bodies. But these applications are too particular to be used for establishing the truth of a general principle. Besides, they have somewhat vague and arbitrary character, which can only render the conclusions that might have been deduced from the true correctness of the principle unsure……………… But there is another way in which it may be regarded, more general, more rigorous, and which itself merits the attention of the geometers. Euler gave the first hint of this at the end of his Traité des isopérimètres, 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 developed and given the Euler condition for making the variation of an of the form

(x2 , y2 ) ∫ f (y, y′, x)dx (x1 , y1 )

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

Euler condition d  ∂f  ∂f   − = 0 dx  ∂y′  ∂y Consider a particle moving in xy plane under the influence of a force with x-component Fx and y-component Fy

Y F y Centripetal force

F 2 x mv Fx y′ − Fy = r 1+ y′2

X

Aim is to see if the principle of least action gives the same answer Action = m∫ vds = m∫ v 1+ y′2 dx

Euler condition gives

d  ∂  ∂  (v 1+ y′2 ) − (v 1+ y′2 )= 0 dx ∂y′  ∂y

Use the relations

∂  1  ∂  1   mv2  = F ,  mv2  = F ∂x  2  x ∂y  2  y and d ∂ ∂ = + y′ dx ∂x ∂y 2 mv y′′ − Fx y′ + Fy 3 2 = (1+ y′2 ) 1+ y′2 (1+ y′2 ) Y radius of curvature r = Fy y′′

Fx 2 mv Fx y′ − Fy = r 1+ y′2

X

MAKING ACTION STATIONARY LEADS TO THE CORRECT FORCE BALANCE EQUATION What does the minimum action principle imply for one-dimensional motion?

x δx A = ∫ v(x)dx δA = ∫δv(x)dx + ∫ v(x)δ (dx)

t

 ∂ 2 δ v(x) = (E −U (x))δ x  ∂x m  1 ∂U (x) If the total energy is constant  = − δ x  2m(E −U (x)) ∂x   1  ∂U (x)  = −  δ x   mv  ∂x δ (dx) = δ (x + ∆x) −δx δ(∆x) ∆(δx) δ(x+∆x) = {(x + ∆x) − (x + ∆x)}−{x − x} ∆ x x = ∆x − ∆x δx ∆x = ∆(x − x) = d(δ x) ∆t

If the end points of the trajectory are kept fixed  1  ∂U (x) δA = −   δxdx + vd(δx) ∫  mv  ∂x ∫  1  ∂U (x) = −   δxdx + dvδx ∫  mv  ∂x ∫  1  ∂U (x) δA = −   δ xdx + dvδ x ∫  mv  ∂x ∫ Since δx(t) is arbitrary, δA=0 implies that

 1  ∂U (x) ∂U (x) dv dx −   dx = dv or − = m using = dt  mv  ∂x ∂x dt v

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

For many interacting particles also, minimization of ∑∫mivids i leads to the equation of motion (Lagrange) i  ∂U ({ri }) dviα − = mi ∂riα dt

Making Action stationary is equivalent to Newton’s second law Does the principle of least action teach us anything new?

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

This, however, changed when Hamilton (1805-1865) entered the scene in 1830’s HAMILTON’S PRINCIPLE OF VARYING ACTION PARALLEL BETWEEN GEOMETRIC OPTICS AND MECHANICS Consider the action integral = ∆s A(x) ∫ v(x1, x 2 , x3 ) ds as a 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 dA ∆A = v(x)∆s OR = v(x) ds

Can this equation be used instead to find the path taken by light? Conventional approach (Hamilton):

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

Recall that in an earlier minimization, the integrand was taken to be function of y(x) and y′(x). Thus A = ∫ v(y) 1+ y′2 dx y(x) is then found by making the variation of the action vanish with respect to variations δy(x).

Now the independent variables are taken to be {xi; αi} instead. Thus = α A ∫ v({xi ; i})ds

{αi(x)} are found by making the variation of the action vanish with respect to δ δα 2 2 2 variations { xi} and { i}. Usingα1 +α 2 +α3 =1make v({xi;αi}) homogeneous

Of degree 1 in {αi} Take the true path and a varied path around it dx+δ(dx) obtained by shifting the line element by {δxi); by changing its length by δ(ds) and by δy changing its directional cosines by {δαi}

dx δA = δ ∫ v ds = ∫δ vds + ∫ vδ (ds) δx

 ∂v ∂v  δ =  δ + δα  + δ A ∑∫ xi ds i ds ∫ v (ds) i  ∂xi ∂αi 

δαi ds +αi δ (ds) = δ (αi ds) = δ (dxi ) = d(δ xi )

⇒ δαi ds = d(δ xi ) −αi δ (ds)  ∂v ∂v   ∂v  δ =  δ + δ  +  − α δ A ∑∫ xi ds d( xi ) ∫ v ∑ i  (ds) i  ∂xi ∂αi   i ∂αi 

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

Integration by parts leads to  ∂v   ∂v   ∂v ∂v  δ =  δ 1  −  δ 0  +  − δ A ∑ xi  ∑ xi  ∫ ds d  xi i  ∂αi  i  ∂αi   ∂xi ∂αi   ∂v  +  − α δ ∫ v ∑ i  (ds)  i ∂αi  δx1(0)= variation of coordinate at the final (initial) point of the path  ∂v   ∂v   ∂v ∂v  δ =  δ 1  −  δ 0  +  − δ A ∑ xi  ∑ xi  ∫ ds d  xi i  ∂αi  i  ∂αi   ∂xi ∂αi   ∂v  +  − α δ ∫ v ∑ i  (ds)  i ∂αi 

Demand that δA vanish for arbitrary variations with the end points of the path fixed i.e. δx1/0=0. This gives ∂v = α v ∑ i since v is a homogeneous function of {αi} of degree 1 i ∂αi

2 2 2 2 2 2 Thus v(x;α) = v(x1, x2 , x3 ) α 1 +α 2+α 3 Note: α1 +α 2 +α3 =1

∂v ∂v for the path of light ray ds = d ∂xi ∂αi Now consider the integral  ∂v   ∂v  δ =  δ 1  −  δ 0  ∫ v ds ∑ xi  ∑ xi  i  ∂αi  i  ∂αi 

with the initial point fixed and δx1 non zero and in the direction of the path. 1 δ xi = αiδ s (δαi = 0)

 ∂v 1   ∂v  Then δ =  δ  =  α δ  = δ ∫ v ds ∑ xi  ∑ i s v s i  ∂αi  i  ∂αi  The action ∫ v ds is a function of the end points of the true path CONCLUSION: Stationary action implies existence of a characteristic function A(x) such that ∂A(x) ∂v = 1 ∂xi ∂αi How to find the path if A(x) and v(x,α) are given? ∂A(x) ∂v From the equation = solve for ( α , α , α ) as a function of 1 1 2 3 ∂xi ∂αi (x1,x2,x3)

Differential equation for the characteristic function A(x) ∂A(x) ∂v = = α 1 i v ∂xi ∂αi

2 2 2 α1 +α 2 +α3 =1

2 2 2  ∂A(x)   ∂A(x)   ∂A(x)  2   +   +   = v (x)  ∂x1   ∂x2   ∂x3 

v(x) is the refractive index of the medium Direction of light ray and surfaces of constant Action:

Direction of light ray α1xˆ1 +α 2 xˆ2 +α3 xˆ3 = v(α1xˆ1 +α 2 xˆ2 +α3 xˆ3 ) ∂v ∂v ∂v = xˆ1 + xˆ2 + xˆ3 ∂α1 ∂α 2 ∂α3 ∂A ∂A ∂A = xˆ + xˆ + xˆ ∂x 1 ∂x 2 ∂x 3  1 2 3 = ∇A(x) Thus light ray moves in the direction of the gradient of the characteristic function

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

A(x1, x2 , x3 ) = constant Light rays and surfaces of constant action

light rays ACTION is

surfaces of constant action EQUIVALENT to Huygens theory of light waves -PART of light rays PHASE

wavefronts Mechanical systems:

In going from point A to point B, a particle also satisfies the principle of least action δ ∫ vds = 0

2 2 2 2 2 2 v = v1 + v 2+ v 3 = v (x1, x2 , x3 ) α 1 +α 2+α 3 is the speed of the particle

Thus there exists a characteristic function A(x) for a mechanical system also such that  ∂A({xi})  vi = OR v = ∇A({xi}) ∂xi

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

2 2 2  ∂A(x)   ∂A(x)   ∂A(x)  2   +   +   = v (x)  ∂x1   ∂x2   ∂x3 

From the energy conservation equation 1 E = mv2 +U ({x }) 2 i

Thus the equation for the characteristic function is

2 2 2  ∂A(x)   ∂A(x)   ∂A(x)    +   +   + =       2mU ({xi}) 2mE  ∂x1   ∂x2   ∂x3  Example: A projectile thrown with initial speed v0 at an angle φ0 in a gravitational U (y) = mgy

The equation for the characteristic function

2 2  ∂A   ∂A    2   +   = m(v0 − 2gy)  ∂x   ∂y 

Solve the equation by separation of variables to get

1 1 A(x, y) = (v2 − k 2 )3 2 + k x − (v2 − k 2 − 2gy)3 2 3g 0 3g 0

Values of A(x,y) are obtained by substituting for x and y, the coordinates of a trajectory Action 1 1 A(x, y) = (v2 − k 2 )3 2 + k x − (v2 − k 2 − 2gy)3 2 3g 0 3g 0

Velocity of the projectile ∂A v = = k = v cosφ x ∂x 0 0 ∂A v = = v2 − k 2 − 2gy = v2 sin 2 φ − 2gy y ∂y 0 0 0

Integration of these equations leads to 1 x = v cosφ t y = v sinφ t − gt 2 0 0 0 0 2

These give the trajectory and the action for projectile motion Surfaces of constant action

Trajectories are lines perpendicular to the surfaces of constant action Trajectories of the projectile

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

Question: Can we associate the action of a particle with a phase? QUANTUM CONNECTIONS Fast forward to 1920s: When it was discovered that particles have a wave associated with them, Hamilton’s theory became the natural choice to account for it and develop the quantum-mechanical wave equation.

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

Start with the equation for the action

2 2 2  ∂A(x)   ∂A(x)   ∂A(x)    +   +   + =       2mU ({xi}) 2mE  ∂x1   ∂x2   ∂x3 

Treating action like phase, take the wavefunction Ψ as

Ψ = exp(A(x) / K) OR A(x) = K log Ψ

Substitute this wavefunction in the equation for action to obtain a quadratic form in Ψ, which is equal to zero 2 2 2  ∂Ψ(x)   ∂Ψ(x)   ∂Ψ(x)    +   +   + − Ψ 2 =       2m(U ({xi}) E) 0  ∂x1   ∂x2   ∂x3 

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

2 2 2  ∂Ψ(x)   ∂Ψ(x)   ∂Ψ(x)    δ   +   +   + 2m(U ({x }) − E)Ψ 2 dr = 0 ∫  ∂   ∂   ∂  i  x1   x2   x3  

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

K 2 − ∇2Ψ +U Ψ = E Ψ 2m Direct connection (IInd paper by Schrödinger):

Space part of the phase of matter waves = A(x) E of the waves = ; h = Planck’s constant h

Calculate the phase velocity of the wave treating surfaces of constant action as wavefronts  mA(x) − Et  φ = 2π   ; m = mass of the particle  h 

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

∆φ = m∆A(x) − E∆t = 0 ∂A ∆φ = m∆A(x) − E∆t = 0 ∆A(x) = ∆x = v ∆x ∂x particle This gives

∆x E E u phase = = = ∆t mv particle 2m(E −U )

u phase h h u phase ≠ v particle AND λ = = = E h 2m(E −U ) m v particle

Group velocity of the waves  E  ∂  ∂ω  h  ∂E ugroup = = = = v particle ∂k  1  ∂ 2m(E −U ) ∂   λ  And finally the wave equation 1 ∂ 2Ψ − ∇2Ψ = 2 2 0 u phase ∂t

Substituting E  E  u phase = AND Ψ(x;t) = Ψ(x)exp− i2π t  2(E −U )  h  leads to the Schrödinger equation

h2 − ∇2Ψ +U Ψ = EΨ 8π 2m A comparison between Classical and (Feynman):

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

Quantum Mechanics: Because of the waves associated with a particle, it does not know which path to take. It takes all possible paths with certain probability 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 largest extent possible.

Classically we see only those result when amplitudes interfere constructively giving a large final amplitude Planck on the Principle of Least Action :

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

Among the more or less general laws, the discovery of which characterize the development of physical science during the last century, the principle of Least Action is at present certainly one which, by its form and comprehensiveness, may be said to have approached most closely to the ideal aim of theoretical inquiry. Thank you According to Planck: "on this occasion everyone has to decide for himself which point of view he thinks is the basic one."

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