SPMS 06: the Brachistochrone Problem and the Calculus of Variations Zhang Chenglong Victoria Junior College

SPMS 06

SPMS 06: The Brachistochrone Problem and the Calculus of Variations Zhang Chenglong Victoria Junior College

Abstract - Fundamental theories in variational friction, using variational calculus [7] and calculus, including the Euler-Lagrange optimal control [8],[9]. This has particular equation, are revisited to solve the classical importance since certain sports such as skiing brachistochrone problem. This paper then [9] and cycling [10] may involve descending on extends the classical brachistochrone problem a curved surface, and finding the best route of to consider the more general cases on curved descent for athletes is crucial to improve their surfaces. Several common curved surfaces results. Analytical and numerical solutions are selected and analytical solutions to the have been given by Shakiban [9] and Benham brachistochrone problems on these curved et al. [10]. Nonetheless, the developments of surfaces are derived whenever feasible, using their solutions were mostly confined to the variational calculus. Graphic representations of practical aspects, and only specific types of the brachistochrone curves are also given. curved surfaces were considered in their Similarities are drawn to the classical research. Although Shakiban has brachistochrone problem whenever possible. systematically developed a numerical solution for general curved surfaces [9], the tool utilised Keywords: Brachistochrone; Constrained was optimal control instead of variational variational calculus; Curved surface calculus. 1 INTRODUCTION Therefore, this paper will consider more general and fundamental cases of the In 1696, Johann Bernoulli posed the classical brachistochrone problem on a curved surface brachistochrone problem: find the curve along using the calculus of variations, excluding the which a particle moves from point to a lower effect of frictions. In this paper, comprehensive point , initially at rest, in the shortest time analysis leading to general solutions to the possible only under the influence of a uniform brachistochrone problem on some common gravitational field. It was then proved that the curved surfaces are presented, together with brachistochrone curve is a cycloid. This the graphic representation of the problem has attracted many prominent brachistochrone curves. The exclusion of mathematicians and eventually stimulated the frictions aims to simplify the problem to the development of the mathematical tool known most fundamental case. as calculus of variations, which deals with general optimisation problems involving It is the hope of the author that future functionals, which are ‘functions’ of functions researchers could make references to this [1]. paper when encountering problems of the same nature, to directly find the curve needed After discovering the solution of the classical for their work, or make further developments brachistochrone problem, numerous more easily. researchers have laboured on its generalisations, some of which includes the 2 OBJECTIVES inclusion of frictions [2]-[5], variation of the The objectives of this study are as follows: gravitational field [5] and variation of initial velocity [6]. However, due to the complexity of • To provide a complete solution to the the extended problems, these research papers classical brachistochrone problem. tend to assume prior knowledge in their analysis, making it difficult to follow through • To obtain the solutions to various their thought processes and their results. brachistochrone problems on certain curved surfaces using variational calculus. Notably, some of the researchers deal with the brachistochrone problem on a curved surface together with the consideration of Coulomb

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3 METHODOLOGY The approach to the various brachistochrone problems is variational calculus.

3.1 FORMULATION OF THE EULER- (6) LAGRANGE EQUATION Suppose we want to find a continuously differentiable scalar function that minimises the objective functional Using integration by parts to eliminate , we (1) obtain where the integrand is called the Lagrangian, subject to the Dirichlet boundary conditions

(2) (7)

We then add a small ‘disturbance’ or ‘variation’ As a result of (4). Thus, to the minimising function , where is a small real parameter controlling the size and (8) is an arbitrary continuously differentiable function. The functional thus becomes Since equation (8) holds for all arbitrary , it follows that (3) (9) where which is the Euler-Lagrange equation. (4) To illustrate this, we denote From (2) we obtain , and equation (8)

(5) becomes since the ‘disturbance’ must go to zero at the end points to satisfy the boundary conditions. (10) Notice that is a function that depends on We then use proof by contradiction. Assume only, and as . Therefore, for some . Since is we can let . At the critical function continuous, there is a region with satisfying . , the functional is minimised and hence Therefore, it is possible to set its gradient goes to zero. Hence, the gradient of also goes to zero as , since it takes on the extreme value. Therefore, . By Leibniz Integral Rule, (11) which satisfies all the conditions. With that,

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(12)

which contradicts with (8). The same construction works for the case as well, with the integral smaller than 0. Hence, it is proven that , which verifies the validity of the Euler-Lagrange equation (9). Figure 1. The classical brachistochrone problem

A secondary conclusion can be obtained from (9). When the Lagrangian is Let the curve of descent have the function independent of , notice that by chain rule, . By conservation of energy,

(13) (17) and by product rule, where denotes the velocity. Hence,

(18) (14) We aim to find the function that minimises Combining and rearranging equations (13) and , the total time of descent. Hence, the (14), we obtain objective functional is given by

(15) (19)

Since is independent of , where denotes the arclength that changes and we have from 0 to . Substituting (18) for and using , equation (19) becomes (16) where is a constant. This equation is also (20) known as the Beltrami identity.

4 RESULTS subjected to the boundary conditions (21) 4.1 THE CLASSICAL BRACHISTOCHRONE PROBLEM Applying variational calculus with , Before the discussion of the brachistochrone problem on various curved surfaces, let us and the Lagrangian restate the classical brachistochrone problem. independent of , we can apply the Beltrami Consider a particle of mass falling from identity (16), which yields point at the origin of the Cartesian coordinate system to point at with (22) , under gravity , as illustrated in Figure 1. Take the -axis to be downward This is equivalent to positive. (23)

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where the constant . To solve this first order differential equation, we express Figure 2. Illustration of a cycloid (23) as

(24) As mentioned in [5], the constant can be determined by considering the ratio . An Using separation of variables, we obtain illustration of a particular solution is presented in Figure 3. (25)

Using the substitution , the right-hand side of equation (25) becomes

(26) and using the substitution , we obtain Figure 3. A particular solution to the classical (27) brachistochrone problem with . In this case point corresponds to and (3 s.f.)

Therefore, 4.2 BRACHISTOCHRONE ON A (28) CURVED SURFACE To generalise the classical brachistochrone The boundary conditions in (21) give problem in a vertical plane to the scenario of a curved surface, there are two key differences: and an equation for the constant . firstly, there is one more dimension, which However, this equation only works provided suggests that modifications to the expression that is a function of . In cases where this is for need to be made; secondly, the not satisfied, for example, point B at , the particle must stay in the prescribed surface, which adds a constraint to the problem. equation does not have a real solution for . We will first discuss the brachistochrone In most academic literatures [1],[5],[6],[10], the problem on a general curved surface with differential equation (24) is solved by the equation trigonometric substitution , (30) which gives the parametric equations For example, one such surface could be , in this case (29) . The brachistochrone will lie within the surface, hence it satisfies the These are parametric equations of a cycloid, equation (30). which coincide with the result obtained in (24). A cycloid is the curve traced by a point on a Consider a particle of mass falling from circle as it rolls along a straight line without point at the origin of the Cartesian slipping. An illustration of a cycloid can be found in Figure 2. coordinate system to point at with , with points and lying on the curved surface, under gravity which acts vertically downwards. Take the -

SPMS 06 axis to be downward positive. Let the curve of As the particle has to stay in the surface, the the fastest descent be described by the function has been pre-determined by functions and and it cannot be varied. . During the process, the Therefore, we only need to vary to find its particle must stay on the surface . minimising function. Applying variational From conservation of energy, equation (18) calculus with and becomes , the Euler- (31) Lagrange equation (9) is Since the gravity now acts along the axis. Equation (19) remains unchanged, but now (34)

, and thus the Notice that the Lagrangian does not depend objective functional becomes on , hence . From (32),

(32) (35) with new boundary conditions From (35) we can make the subject, (33) (36) A purely analytical solution might not be available for such a problem, especially if both where is a constant. By separation of and need to be varied. However, for variables, we have certain commonly seen curved surfaces, the problems can be simplified using methods (37) such as coordinates conversion, making analytical solutions more attainable. whose solution depends on the complexity of 4.2.1 BRACHISTOCHRONE ON A the function . CURVED SURFACE WHOSE EQUATION IS INDEPENDENT OF Y 4.2.1.1 BRACHISTOCHRONE ON AN First, we consider the brachistochrone problem INCLINED PLANE in which the curved surface is We now apply equation (37) to specific one- independent of . An example is the surface dimensional curved surfaces. First, consider the case of an inclined plane, where the , which extends indefinitely in the equation describing the surface direction, as shown in Figure 4. is simply , where here denotes the gradient, as shown in Figure 5.

Figure 4. An example of a curved surface independent of

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This brachistochrone is a cycloid on the inclined surface, as suggested by Benham et al. using rotated coordinates [10]. Although this method arrives at the same conclusion with relative ease, the method deployed in this paper stems from the more general cases for curved surfaces, and are more applicable to other surfaces.

4.2.1.2 BRACHISTOCHRONE ON A HORIZONTAL CYLINDRICAL SURFACE When it comes to a cylindrical surface, the first difficulty is that is not an explicit function of . To circumvent this, we can convert the

Cartesian coordinates to cylindrical Figure 5. Brachistochrone problem on an inclined plane coordinates to simplify the problem.

To reformulate the problem in the cylindrical Here, equation (37) becomes coordinate system, consider the cylindrical surface , with the particle falling from (38) to , as illustrated in Figure 7. Notice the similarity between (38) and (25). A similar substitution gives

(39) where , are constants. This result is illustrated in Figure 6.

Figure 7. Brachistochrone problem on a horizontal cylindrical surface

From the conservation of energy,

(40)

Therefore, with ,

the total time of descent in the cylindrical Figure 6. A particular solution to the brachistochrone coordinate system is problem on an inclined plane, with , , ranging from 0 to 1.79, constant (3 s.f.)

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(41)

Applying variational calculus with ,

, the Euler-

Lagrange equation (9) gives

(42) which can be further simplified to

Figure 8. A particular solution to the brachistochrone problem on a horizontal cylindrical surface. The approximated curves in green, red and blue use series (43) expansion of the integrand to , , respectively. where . The general solution to the integral cannot be expressed in elementary functions. However, it is possible to obtain an 4.2.2 BRACHISTOCHRONE ON A approximation to some particular solutions. CURVED SURFACE WHOSE EQUATION IS INDEPENDENT OF Z Consider a particular case with , Next, we consider the case where the equation . Assuming convergence, using of the curved surface is independent of instead of independent of . the software Mathematica to expand the This makes a critical difference as gravity acts integrand in Puiseux series, an approximation along the -axis. Consequently, instead of to the integrand is varying , we now need to vary to find the minimising function. Applying variational calculus with and

, the Euler-

Lagrange equation (9) is

(44) (45) which can then be integrated to obtain an which gives approximation to the brachistochrone curve. The resulting curves in Figure 8 are approximations using series expansion to different orders by using Mathematica to solve (46) for . As , which Applying quotient rule to the right-hand side, suggests that the approximation to is equation (46) can be further simplified to accurate. This is reflected in the close proximity of solution curves in Figure 8. (47)

4.2.2.1 BRACHISTOCHRONE ON A VERTICAL CYLINDRICAL SURFACE Owing to the symmetry of a cylinder, we set the particle to travel from where

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is the radius of the circle in the -plane, to with . Intuitively, we can conclude with . The -axis is taken to be downward positive. The that the solution is a curved cycloid. Using the equation of the surface is thus substitution , the solution to

(48) this ordinary differential equation is

From (48), (47) can be simplified to (52)

where is a parameter with its range (49) depending on the end points. Using Mathematica, it can be shown that the This differential equation is difficult to solve. parametric equation is indeed the solution to Hence, we seek to formulate the problem in the differential equation (49). Figure 10 cylindrical coordinates , with the -axis illustrates a solution to the problem. being downward positive and the particle descending from to , as illustrated in Figure 9.

Figure 10. A particular solution to the brachistochrone problem on a vertical cylindrical surface, with ,

, from 0 to 1.20, (3 s.f.) Figure 9. Brachistochrone problem on a vertical cylindrical surface

4.2.3 BRACHISTOCHRONE ON A The total time of descent is thus CURVED SURFACE WHOSE EQUATION IS DEPENDENT ON X Y (50) AND Z In the case of a more general curved surface owing to the fact that the equation of the with equation depending on , cylindrical surface is . Applying and , for example the surface variational calculus with , , an analytical solution is more difficult to obtain. However, it is still possible to with the notice that make the problems simpler through the Lagrangian is independent of , we can reformulation. use the Beltrami identity (16), and we get 4.2.3.1 BRACHISTOCHRONE ON A SPHERE (51) In the case of a sphere, it is more convenient to formulate the problem in spherical

SPMS 06 coordinates . The curved surface can surface, especially after reformulation for some then be described by . Due to the problems. However, for curved surfaces with more complicated expressions, variational symmetry of a sphere, let the particle fall from calculus could lead to differential equations to , as shown in that are not easily solvable, such as (49) and Figure 11. (54). Therefore, numerical methods might be used to obtain an approximation to the solutions, which are mostly beyond the scope of this paper. On the other hand, the study of the generalised brachistochrone problems may inspire the solution of some differential equations. For example, the analytical solution (52) could provide an insight of solving the more complicated differential equation (49). This could be a potential direction for future researchers doing relevant work.

5.2 POTENTIAL APPLICATIONS OF VARIATIONAL CALCULUS Besides solving the classical brachistochrone problem and its generalisations, variational

Figure 11. Brachistochrone problem on a spherical calculus has many other applications. For surface example, to find the shortest curve connecting two points and Energy conservation yields the same result as in a plane, we can apply variational calculus with denoting the length of the in (40), and . curve and , the Beltrami Therefore, the total time of descent is identity would give , and the equation of the curve is which is a straight line (53) connecting points and . From this seemingly trivial example, it can be seen that Applying variational calculus with variational calculus can be applied to many optimisation problems. and , the Euler- 6 CONCLUSIONS Lagrange equation for (53) yields This paper has provided a comprehensive compilation of generalised brachistochrone problems on a curved surface. Revisiting the theories in variational calculus provides the foundation for subsequent problem-solving, and a whole range of curved surfaces are (54) examined to obtain analytical solutions of which would need to be solved numerically. brachistochrone wherever possible, with some of the results being supported by previous 5 DISCUSSION studies. Future researchers can make reference to this paper when solving similar 5.1 USEFULNESS OF VARIATIONAL problems, so as to focus more on the CALCULUS IN SOLVING EXTENDED application aspect instead of the theoretical BRACHISTOCHRONE PROBLEMS aspect, which has been detailed in this paper. From the results above, it can be seen that This paper excluded the effect of frictions. In variational calculus is useful in solving reality, this should not be ignored, especially in brachistochrone problems on a curved the context of sports. Together with the several

SPMS 06 studies including frictions in their research [7]- [6] Atanackovic, T. M. (1978). The [10], a potential area for future research is to brachistochrone for a material point with consider the effect of frictions on general arbitrary initial velocity. American Journal curved surfaces. If a comprehensive database of Physics, 46(12), 1274-1275. is established, a possible application to the study is to design the most suitable curved [7] Čović, V., & Vesković, M. (2008). surfaces for various sports competitions. Brachistochrone on a surface with Coulomb friction. International Journal of In this paper, it is assumed that the particle will Non-Linear Mechanics, 43(5), 437-450. not leave the curved surface. In reality, this assumption may not hold, since the particle [8] Hennessey, M. P., & Shakiban, C. (2010). may not be closely attached to the surface. If Brachistochrone on a 1D curved surface the centripetal acceleration becomes too small using optimal control. Journal of dynamic in comparison to curvature, the particle will fall systems, measurement, and out of the surface. Therefore, a possible control, 132(3). direction for future research is to include the [9] Shakiban, C. BRACHISTOCHRONE ON A effect of centripetal acceleration to obtain more 2D CURVED SURFACE USING OPTIMAL realistic results. CONTROL Michael P. Hennessey, Ph. D. School of Engineering University of St. ACKNOWLEDGEMENTS Thomas. In Proceedings of the This research is conducted under the Conference (Vol. 665, No. 009, p. 119). supervision from Dr Tang Wee Kee from [10] Benham, G. P., Cohen, C., Brunet, E., & Nanyang Technological University (NTU). Clanet, C. (2020). Brachistochrone on a The author acknowledges the support from velodrome. Proceedings of the Royal (NTU) for conducting the Nanyang Research Society A, 476(2238), 20200153. Programme (NTU); Dr Wu Jiang and Mr Eugene Quek from Victoria Junior College for their guidance; Titus Peng for his YouTube video facilitating concept learning; Wolfram Mathematica for computation and generation of diagrams; and friends of the author who have provided genuine support along the way.

REFERENCES [1] Olver, P. J. (2016). Introduction to the Calculus of Variations. University of Minnesota. [2] Ashby, N., Brittin, W. E., Love, W. F., & Wyss, W. (1975). Brachistochrone with Coulomb friction. American Journal of Physics, 43(10), 902-906. [3] Hayen, J. C. (2005). Brachistochrone with Coulomb friction. International Journal of Non-Linear Mechanics, 40(8), 1057-1075. [4] Cherkasov, O. Y., & Zarodnyuk, A. V. (2014, December). Brachistochrone problem with linear and quadratic drag. In AIP Conference Proceedings (Vol. 1637, No. 1, pp. 195-200). American Institute of Physics. [5] Parnovsky, A. S. (1998). Some generalisations of brachistochrone problem. Acta Physica Polonica-Series A General Physics, 93(145), 55-64.