Finding the Best Function: A Way to Explain Calculus of Variations to Engineering and Science Students

In many practical problems, we need to find the most appropriate function: e.g., we need to find a control strategy u(t) that leads to the best performance of a system, we need to find the shape of the car which leads to the smallest energy losses, etc. Optimization over an unknown function can be described by the known Euler-Lagrange equations. The traditional way of deriving Euler-Lagrange equations when explaining them to the engineering and science students is, however, somewhat over-complicated. We provide a new, simpler way to deriving these equations, a way in which we directly use the fact that when the optimum is attained, all partial derivatives are equal to 0. 1 Finding the Best Function: Euler-Lagrange Equations and How They Are Derived Now Optimization is ubiquitous. In practice, we often need to make choices. For example, in engineering, when we design an object (a car, a computer) or select a control (e.g., a route for the car or for a computer message), we need to select the values of several parameters describing the object or control. Often, we can express our preferences by assigning, to each possible alternative, a numerical value describing the degree to which we are satisfied with this alternative. For example, when we design a high-performance computer, this satisfaction-describing numerical value is the throughput – the number of operations per second that this computer can perform. In such situations, we need to select the values of the corresponding parameters x1, . . . , xn for which the corresponding numerical value F (x1, . . . , xn) is the largest possible.