An interesting track for the Brachistochrone
AAn interesting track for the Brachistochrone
Zafar Ahmed , and Amal Nathan Joseph , Nuclear Physics Division, Reactor Physics Design Division,Bhabha Atomic, Research Centre, Trombay, Mumbai, 400085 Homi Bhabha National Institute, Anushaktinagar, Mumbai, 400094 (Dated: October 30, 2020)
Abstract
If a particle has to fall first vertically 1 m from A and then move horizontally 1 m to B, ittakes a time t (= τ + τ = τ = √ g ) = 0 .
67 s. Under gravity and without friction, if it sidesdown on a linear track inclined at 45 between two points A and B of 1 m height, it takes time t (= τ = √ g ) = 0 .
63 s. Between these two extremes, historically, Bernoulli (1718) proved that thefastest track between these points A and B is cycloid with the least time of descent t = τ B = 0 .
58 s.Apart from other interesting cases, here we study the frictionless motion of a particle/bead on aninteresting track/wire between A and B given by y ( x ) = (1 − x ν ) /ν . For ν > t >> τ , and when ν > .
22, the motion with zero initial speed is not possible. We findthat when ν ∈ (0 . , . , τ < t < τ and when ν ∈ (0 . , , τ B < t < τ . But mostremarkably, the concave curve becomes very steep/deep if ν ∈ (0 , ν c = 0 . t = 0 . < τ B , this is as though a particle would travel 1 meter horizontally with a speed equal √ g m/secto take the time (= √ g = τ ) < τ B . The function t ( ν ) suffers a jump discontinuity at ν = ν c , weoffer some resolution. Ignoring friction and taking the accelera-tion due to gravity g = 9 . t = τ = (cid:113) g = 0 . t = τ = (cid:113) g = 0 . τ = τ + τ = √ g = 0 . , t = τ = √ g = 0 . b in Fig.1) as the fastest de-scent curve between two points and the time t = τ B = √ g (cid:82) π/ √ csc xdx = 0 . τ . The distance betweenthe end points of this cycloid is AB = √ a r X i v : . [ phy s i c s . c l a ss - ph ] O c t .2 0.4 0.6 0.8 1.00.20.40.60.81.0 A B y x a b cdef
FIG. 1: For various values of ν the tracks givenby (7). a: ν = 0 . , b : ν = 0 . , d : ν = 1 , e : ν =1 . , f : ν = 2 . The track d denotes one of thefamily of µ -curves given by (7). These µ -tracksare essentially vertical at both points A and B. equation of the energy conservation and theoptimization by variational principle. Ter-restrial [3] Brachistochrone and several vari-ants consisting in central force field [4], bentstraight wire [5] and unrestrained rollingblock [6] have been discussed. A simpli-fied method for the direct and inverse prob-lem of finding the force given the Brachis-tochrone curve has been developed [7]. Min-imum descent time along a set of connectedinclined planes has also been explored [8]. Adiscrete Brachistochrone has been proposed[9]. Quantum Brachistochrone problem hasalso been defined and formulated in terms ofHermitian, complex PT-symmmetric [10] andpseudo-Hermitian [11] Hamiltonians.The question of other types of interestingtracks (e.g., Fig.1) and their features are dis-cussed here. We study the time of descent from various tracks given by y ( x ) = 1 − x λ , y ( x ) = √ − x µ and y ( x ) = (1 − x ν ) /ν be-tween two fixed points A(0,1) and B(1,0).It is the third ν -track that gives a sur-prising result that yields t lesser than theso far acclaimed least value τ B , when ν ∈ (0 . , . m sides undergravity on the track given by y ( x ), the con-servation of energy leads to m x + ˙ y ] + mgy ( x ) = mgy ( x ) (1)Introducing differential length of the curve( ds ) = ( dx ) + ( dy ) , we can write dtdx = (cid:115) y (cid:48) ( x )[2 g ( y ( x ) − y ( x )] (2)By taking the initial point as x = 0 and thefinal point as x = 1. Here the distances arein meters and g = 9 . t = τ (cid:90) (cid:115) y (cid:48) ( x ) y (0) − y ( x ) dx, τ = 1 √ g (3)For the linear track- c (Fig.1) y = a (1 − x ),where a = tan θ >
0, we get t = 1 √ g (cid:114) a a (cid:90) dx √ x ≥ √ g = τ (4)As (1 − a ) ≥ ⇒ a ≥ a andthe equality holds for a = 1 = ⇒ θ = π/ x = 1,the one with 45 angle gives the least time ofdescent, Galileo is known to have pointed outthis fact.2 .2 0.4 0.6 0.8 1.00.20.40.60.81.0 A B y x
FIG. 2: The wavy tracks y ( x ) = 1 − xe (cid:15) cos (7 πx/ . The lower(upper) one is for (cid:15) = ± .
06. Both the tracks up to x = 0 . t = 0 . t = 0 . First, we study the curved track given by y ( x ) = 1 − x λ , λ > . (5)Using (3), we find that for this track t ( λ ) = τ (cid:90) √ λ x λ − x λ/ dx, (6) t (0) = τ . This is an improper integral asits integrand diverges at x = 0, however, itis convergent (finite) when λ <
2, like inEq. (4). Eq. (6) can be written in termsof Gamma functions or Gauss hypergeomet-ric function, nevertheless it is easily doablenumerically. We have used “NIntegrate” ofMathematica.Fig.1, presents the λ -tracks only schemati-cally. For λ ∈ (0 , . a is con-cave but deep/steep, we get τ < t ( λ ) < τ . This parametric regime in a limiting way rep-resents motion of the particle vertically downfrom point A, followed by the horizontal mo-tion with speed √ g m/s towards the pointB. When λ ∈ ( . ,
1) the track- b is concave, τ B < t ( λ ) < τ . Next, when λ ∈ (1 , e , we find that t ( λ ) >> τ .The motion is forbidden, when λ ≥ t ( λ ) (6)diverges: the convexity of the track- f does notallow the motion with zero initial velocity.We can have wavvy tracks (see Fig. 2)that are given by y ( x ) = 1 − xf ( x ) , −√ xf ( x ) , − x / f ( x ). Here we choose f ( x ) = e (cid:15) cos (7 πx/ , (cid:15) = ± .
06 to give a mild andsmooth modulation to the tracks. Thesesmall modulations in the case of concave orconvex tracks do not change the inequality ofthe time of descent t , with respect to τ . Theessence of this study is that if initial part ofa track is concave for a good length like thelower one, then t = 0 . t = 0 . t > τ The family of tracks given by y ( x ) = √ − x µ , µ > x = 0 and x = 1, seetrack- d for µ = 0 .
5. We find that when µ ∈ (0 , . τ < t ( µ ) < τ , for µ ∈ (0 . , . , t ( µ ) < τ , for µ ∈ (0 . , . t ( µ ) >> τ . For µ ∈ (1 . , . t ( µ ) >> τ with a warn-3ng of accuracy. For µ > .
61 the integral(4) starts diverging and the convexity of thetrack does not allow the motion with zeroinitial speed. For µ = 0 .
5, see the track- d , t = 0 . < τ , here the initial part of thetrack is concave followed by a convex part.We would like to remark that for both λ and µ tracks, the numerical integrationyields: τ B < t (0 + ) ≤ τ with only a warn-ing of accuracy.More interestingly, the tracks given by y ( x ) = (1 − x ν ) /ν , ν > t ( ν ) = τ I ( ν ) in terms of aninteresting integral I ( ν )= (cid:90) (cid:115) x ν − + (1 − x ν ) − ν ) /ν x − ν ) /ν [1 − (1 − x ν ) /ν ] dx (9) The simple interesting cases are I (0) =1, I (1) = 2 √ I ( ν ) = (cid:90) dxx (3 ν − / < ∞ , if ν < / . (10) NIntegrate of Mathematica gives I (0 ≤ ν <ν c ) = 1 and I ( ν c < ν < ν s ) = 3 withouta warning for any type of error and shows I ( ν > ν s ) as divergent, where ν s = 1 . I ( ν )for ν ∈ (0 , .
22) has a jump discontinuity at ν = ν c , see Fig. 2.For ν > t >> τ , and when ν > .
22, the motionwith zero initial speed is not possible as theintegral (3) diverges. We find that when ν ∈ (0 . , . , τ < t < τ andwhen ν ∈ (0 . , , τ B < t ( ν ) < τ . But Ν t (cid:72) Ν (cid:76) Τ B Τ Τ Τ FIG. 3: t ( ν ) for the track (8). In the domains ν = (0 . , . , (0 . , , (1 , . t ( ν ) is τ , ( τ , τ ) , ( τ B , τ ) , >> τ .The discontinuity is at ν = ν c = 0 . ν > . . most remarkably, if ν ∈ (0 , . ν c ),the track is concave but deep and steep then t = τ = 0 . < τ B , this is the time(= √ g = τ ) that a particle would take totravel 1 meter horizontally with a speed equal √ g m/sec which is acquired at the origin.The function t ( ν ) suffers a jump discontinu-ity at ν = ν c , see Fig. 3.Though, the numerical computation un-derlying the result t (0 < . < τ B infig. 3 have been done using “NIntegrate” ofMathematica that worked error-free yet wedecided to re-calculate t ( ν ) (6) by splitting itas t ( ν )= (cid:90) ε ( ν )0 F ( x, y, y (cid:48) ) dx + (cid:90) ε ( ν ) F ( x, y, y (cid:48) ) dx,ε ( ν ) = 10 − n ( ν ) , n ( ν ) ∈ N. (11) The first part in above, is very interesting4here the x could be extremely small in asmall domain but the range of F ( x ) consist-ing of large numbers. We find that for ev-ery ν ≥ . ε ( ν ) or n ( ν )such that I ( ν ) = 3. For instance, when ν = 0 . n = 27. For ν < . n = 98. For ν = . , n = 148. For ν = 0 . , n = 198.For ν = 0 . , n = 212. We failed to finda suitable value of n for ν < . I ( ν ) = 3, hence t = τ < τ B . We would liketo assert that ν = 0 . ν c = 0 . τ B , how-ever the discontinuity in t ( ν ) in Fig. 3 hasbeen pushed towards ν = 0. One may won-ders if I ( ν ) is discontinuous at ν = 0 suchthat I (0) = 1 , I (0 + ) = 3 or equivalently t (0) = τ , t (0 + ) = 3 τ > τ B , (12)is expected physically.Finally, let us resolve the curious domain ν ∈ (0 , . ν -track (8). In thiscase, the track becomes practically discontin-uous as y ( x = 0) = 1 , y ( x = 0 + ε ) = 0 , (13)but in the energy conservation condition (1),the particle has already been assumed to havethe initial potential energy equal to mgy (0).Next, the integral (6) determining time ofdescent, inherently starts integration from x = 0 + ε , leaving out x = 0. Consequently,the particle executes only the linear horizon-tal motion from x = 0 + ε to x = 1 with the conditioned initial speed v = (cid:112) gy (0)and takes time t = τ < τ B . One may callthis trivial case as a mathematical Brachis-tochrone (MB). In this domain of ν , the deepand steep ν -track would lie below even thetrack- f in Fig.1. When the ν > . τ < t ≤ τ , so on and so forth(see Fig. 3).The next question is how the optimiza-tion of t in the earlier treatments of Brachis-tochrone specially in textbooks [2] have ig-nored this trivial yet interesting track y ( x ) =0 (13). In this regard, our discussion differsslightly from textbooks as we take the start-ing point as (0,0) but ours the point (0,1).So in our eq. (2), 1 − y ( x ) occurs insteadof y ( x ) as in books. Optimization of t = (cid:82) F ( x, y, y (cid:48) ) dx is done using Euler-Lagrangeequation, where F = (cid:112) y (cid:48) / (1 − y ( x )). ∂F∂y − ddx ∂F∂y (cid:48) = 0 . (14)If we multiply (9) by y (cid:48) on both sides, y (cid:48) mustbe non-zero and then we get ddx (cid:18) F − y (cid:48) ∂F∂y (cid:48) (cid:19) = 0 ⇒ F − y (cid:48) ∂F∂y (cid:48) = C (15)In text books one uses (15) instead of (14), toget the ordinary differential equation (ODE)as (1 − y )(1 + y (cid:48) ) = C , solving which invarious ways one gets the acclaimed cycloid.Hence the condition that y (cid:48) ( x ) (cid:54) = 0 rules outthe trajectory y ( x ) = c from the optimization5f the time of descent, without even a men-tion, though we can see it as mathematicallyconsistent case giving t < τ B .We have discussed the time of descent of abead/particle on various tracks/wire on sev-eral tracks from point A to the point B. Wefound that even convex tracks can allow theside of the bead downwards with zero-initialspeed up to a limited initial curvature of thetrack with t >> τ . Concave tracks take timein the interval ( τ , τ ), Between two slightlywavy tracks the one that has initial part asconcave is faster. The mathematical Brachis-tochrone y = 0 pushed by an initial speed v ( x = 0) = 0 = √ g m/s is faster than eventhe cycloid track τ < τ B . The Numericalquadrature for x ∈ (0 ,
1) gives a surprisingresult wherein t turned out to be lesser than τ B , but by splitting it in two parts (11), wecould rescue this unphysical result to someextent. For both λ and µ tracks, numerically t (0 + ) is found to be in ( τ , τ ), with the warn-ing of inaccuracy. However, for the ν -track t = τ < τ B when 0 < ν < . y = 0 (13) and we have t (0 + ) = τ instead of3 τ > τ B . It could be challenging to developa numerical quadrature for the integral (9)that gives t (0 + ) = 3 τ . References
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