Lagrangian Mechanics

Though brachistochrone means “shortest time” in Greek, the set of problems originated with the Swiss mathematicians, John (1677-1748) and James Bernoulli.(1654-1705). It was John Bernoulli who first posed the problem of finding the shortest time of descent for a bead sliding on a wire between two fixed points with the length and the shape of the wire being variable – i.e., selected by the problem solver.  Assumed by some that a straight wire connecting the two points would be the result for this initial problem, experimenters learned instead that longer, curved paths allowed gravity to speed up the bead’s passage. With additional mathematical analysis an optimal shape for the bead’s path could be explained or found.  Early solutions to this problem paved the way for new branches of applied mathematics stemming from Newton’s calculus: the calculus of variations, optimization and Lagrangian mechanics.

Variations on Bernoulli’s problem have since abounded: more elaborate environmental models such as the gravity field and the geometry; additional obstacles in the path known in mathematics as “constraints” and real world navigation and guidance applications for vehicles such as ships, aircraft and satellite launch vehicles.  The initial solution to the brachistochrone problem involved a mathematical function or formula that can be modeled, as we shall see, by some simple mechanical behavior.  But as analysts and mathematicians built more complicated problems and systems, solutions and methods became more complex as well, relying on computer numerical techniques. 

Before mathematicians resorted to computers, however, some elegant transformations and manipulations of mathematical expressions were achieved through the aid of analysis techniques developed after the Bernoullis by mathematicians such as the French theoretical astronomer Joseph Louis Lagrange (1736-1813).  Through Lagrange’s methods  a system of particles in a physical problem  such as a bead on a wire influenced by gravity could be expressed in terms of kinetic and potential energy or work and then through a series of “derivative” operations based on time, position and velocity, these equations could be reduced to terms of forces and accelerations.

To get some grasp on the brachistochrone problem, we need to start with notions of mathematical functions and how they can be manipulated via calculus.  In basic algebra an example of a function could be a quadratic formula expressed as a function of variable “t” such as f(t)=At^2 + Bt + C with the capital letters constants ( ^ denoting exponents such as squares and cubes).  The variable t can represent time and the function can represent distance.  If this function f(t) is plotted against t as a graph, then the resulting curve has a slope that can be expressed as a function of  t as well: the velocity f ’(t).  This function, by the rules of calculus is the derivative df/dt = 2At + B.    Obtaining a derivative is a process also known as differentiation. Acceleration would be the second derivative of distance f(t) which would be f ”(t) or d2f/dt2 = 2A. In this case the second derivative is a constant. 

Now had we started our discussion of functions with a function g(t) which we denoted as an acceleration, then by a process of “integration” we could determine functional values of velocity and distance. Integration is the reverse of the derivative process.  With a constant acceleration “g”, velocity over time would increase as gt + v0, the last term a constant for the initial velocity.  Distance would be 1/2gt^2 +v0 t + d0 with the last term representing an initial distance at time t = 0.  If the initial acceleration were increasing or decreasing linearly with time for example, then the integration process would result in more complex polynomials to represent velocity and distance traveled.  For example, if the acceleration had varied linearly with time ( g(t) = At+B), then the function for distance f(t) after two integrations would look something like this: f(t)= 1/3At^3+B/2t^2 +Ct +D.

Since both calculus and the calculus of variations are often engaged in identifying maxima or minima, then it is good to know that the zero solutions of first derivatives identify extreme values of functions.  When a value of  f ’(x) =0 at a given value of x, then f(x) is either at a maximum or a minimum.  If the value of f “(x), the second derivative is negative at this value of x, the extreme is a maximum; a minimum if f ”(x) is greater than zero. 

The fact is that different polynomials or functional expressions can arise in mathematical calculations just as do different values for individual variables or coefficients. This is a point worth noting in understanding the brachistochrone problem and the mathematics it represents.  Also, it should be noted that analytical processes can be used to identify minimum and maximum values. 

To understand the relation of the brachistochrone problem to Lagrangian mechanics, let us take our review of functions to still another level.  Since we mentioned that functions or derivatives can be based on variables such as time [ f(t) and df(t)/dt] , we should also note that they can be based on other parameters such as position x [ f(x), df(x)/dx] or velocity v [ f(v), df(v)dv].  They can also be based on several variables [ f(t, x, v)] and be differentiated “partially” by one variable at a time.  When a function such as y is listed in an equation as several different derivatives of one variable such as t, the equation is known as an ordinary differential equation; for example  A y(t) + By’(t) + Cy”(t) = h(t).  Equations featuring partial derivatives are known as partial differential equations.  It was Joseph Lagrange who developed an important set of partial differential equations in which mechanical systems such as brachistochrone problems can be studied, characterized and fine tuned for desired behavior.

The shape of the wire curve that satisfies the basic brachistochrone problem is that of a point on a wheel rolled on a flat surface without slipping: a hypercycloid, which can be derived by Lagrangian methods. Isaac Newton suggested the related problem of traveling via tunnels between two cities on a spherical earth; this is solved by bead paths related to a disk rotating on the inner circumference of the earth, a hypocycloid.

Variations on the brachistochrone problem have continued perhaps to the point where it is debatable whether the name still appropriately applies.  With the calculus of variations it is difficult to solve a case with intermediate constraints; that is, a bead problem in which the wire path is constrained by an obstacle or a plateau that causes the bead path to pass along an intermediate fixed boundary.  In recent decades a number of numerical methods have been devised to address this problem which are based on segmenting the problem into components which are joined together by various curves.  Instead of fighting gravity, the problem of steering a powered river boat across a constant flow river to its opposite shore in minimum time is named for Ernst Zermelo (1871-1953).  Correspondingly, with rockets burning their propellant at a fixed rate, a minimum time solution means minimum fuel expended and maximum payload can result – subject to gravity and changing acceleration.  In such cases, the problem is often referred to as the two point boundary problem.  The initial condition might be zero velocity at a launch pad and the final condition might be an altitude or velocity which might represent delivery of a satellite into orbit.  Since delivery of a satellite would not be possible without some mathematical procedure for guidance and economy and accuracy are essential,  we can see a clear practical application here of brachistochrone problem solution methods.  With some imagination the foundational brachistochrone problem approach can be applied to other technologies and (possibly)  even business problems.