Principle of Least Action Manoj K. Harbola Department of physics 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 distance 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 angle between them at the point where you enter the water. Which path does a particle of total energy 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 time 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 velocity 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 Trajectory 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=area 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 speed δ (E − mgy) = (E − mgy) δ y(x) dy (ii) Change in the length 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 trajectories that the ball can take y ymax2 ymax1 a x Comparison with Newtonian approach: Given initial position 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 force 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 angles 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 Newton (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.