2.2 Basic calculus of variations problem

The reader has probably observed that the problems of Dido, catenary, and brachistochrone, although different in their physical meaning, all take essentially the same mathematical form. We are now ready to turn to a general problem formulation that captures these examples and many related ones as special cases. For the moment we ignore the issue of constraints such as the arclength constraint (2.1), which was present in Dido's problem and the catenary problem; we will incorporate constraints of this type later (in Section 2.5).

The simplest version of the calculus of variations problem can be stated as follows. Consider a function $L:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ .

Basic Calculus of Variations Problem: Among all $\mathcal C^1$ curves $y:[a,b]\to\mathbb{R}$ satisfying given boundary conditions

$$y(a)=y_0,\qquad y(b)=y_1$$ (2.8)

find (local) minima of the cost functional

$$J(y):=\int_a^b L(x,y(x),y'(x))dx.$$ (2.9)

Since $y$ takes values in $\mathbb{R}$ , it represents a single planar curve connecting the two fixed points $(a,y_0)$ and $(b,y_1)$ . This is the single-degree-of-freedom case. In the multiple-degrees-of-freedom case, one has $y:[a,b]\to\mathbb{R}^n$ and accordingly $L:\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ . This generalization is useful for treating spatial curves ($n=3$ ) or for describing the motion of many particles; the latter setting was originally proposed by Lagrange in his 1788 monograph Mécanique Analytique. The assumption that $y\in\mathcal C^1$ is made to ensure that $J$ is well defined (of course we do not need it if $y'$ does not appear in $L$ ). We can allow $y'$ to be discontinuous at some points; we will discuss such situations and see their importance soon.

The function $L$ is called the Lagrangian, or the running cost. It is clear that a maximization problem can always be converted into a minimization problem by flipping the sign of $L$ . In the analysis that follows, it will be important to remember the following point: Even though $y$ and $y'$ are the position and velocity along the curve, $L$ is to be viewed as a function of three independent variables. To emphasize this fact, we will sometimes write $L=L(x,y,z)$ . When deriving optimality conditions, we will need to impose some differentiability assumptions on $L$ .