Cluster Analysis

PDF / 3,820,493 Bytes
208 Pages / 547.087 x 737.008 pts Page_size
24 Downloads / 188 Views

Calculus of Variations Stephen G. Nash George Mason University, Fairfax, VA, USA

Introduction The calculus of variations is the grandparent of mathematical programming. From it came such concepts as duality and Lagrange multipliers. Many central ideas in optimization were first developed for the calculus of variations, then specialized to nonlinear programming, all of this happening years before linear programming came along. The calculus of variations solves optimization problems whose parameters are not simple variables, but rather functions. For example, how should the shape of an automobile hood be chosen so as to minimize air resistance? Or, what path does a ray of light follow in an irregular medium? The calculus of variations is closely related to optimal control theory, where a set of controls are used to achieve a certain goal in an optimal way. For example, the pilot of an aircraft might wish to use the throttle and flaps to achieve a particular cruising altitude and velocity in a minimum amount of time or using a minimum amount of fuel. The modern world is full of devices designed using optimal control — in cars, elevators, heating systems, stereos, etc.

Brachistochrone Problem The calculus of variations was inspired by problems in mechanics, especially the study of three-dimensional

motion. It was used in the 18th and 19th centuries to derive many important laws of physics. This was done using the Principle of Least Action. Action is defined to be the integral of the product of mass, velocity, and distance. The Principle of Least Action asserts that nature acts so as to minimize this integral. To apply the principle, the formula for the action integral would be specialized to the setting under study, and then the calculus of variations would be used to optimize the integral. This general approach was used to derive important equations in mechanics, fluid dynamics, and other fields. The most famous problem in the calculus of variations was posed in 1696 by John Bernoulli. It is called the Brachistochrone (“least time”) problem, and asks what path a pellet should follow to drop between two points in the shortest amount of time, with gravity the only force acting on the pellet. The solution to the Brachistochrone problem can be found by solving 1 minimize pﬃﬃﬃﬃﬃ yðtÞ 2g

Z

t2 t1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ y0 ðtÞ2 dt yðtÞ

where g is the gravitational constant. If this were a finite-dimensional problem then it could be solved by setting the derivative of the objective function equal to zero, but seventeenth-century mathematics did not know how to take a derivative with respect to a function. The Brachistochrone problem was solved at the time by Newton and others, but the general techniques that inspired the name calculus of variations were not developed until several decades

S.I. Gass, M.C. Fu (eds.), Encyclopedia of Operations Research and Management Science, DOI 10.1007/978-1-4419-1153-7, # Springer Science+Business Media New York 2013

C

142

Calculus of Variations

later. The first major results w

Data Loading...