2.3 Stationary value of a function

In analytic mechanics, we will be interested in the stationary value of an integral. We start by considering the stationary value of a function. For example, an n-dimensional function \(F(q_1, q_2, \dots, q_n)\), can be pictured as a surface in a space with n+1 dimension (for example, imagine a 2-dimensional surface in 3-dimensional space). The stationary points of the functions are locations where the surface is “flat”, and can correspond to local “extrema” (minima or maxima) or “saddle points”. We can use the formalism of variations to find the conditions for such points.

If the point, \(P\), is a stationary value of the function, \(F(q_1, q_2, \dots, q_n)\), then, the variation of the function near \(P\) in the direction of an arbitrary virtual displacement, \(\delta\vec{q}\), is 0 (the function is flat in the infinitesimal region near the point). The variation of the function is (technically called the “first variation”, as we drop the higher order terms in the Taylor series):

\begin{align} \delta F&=\frac{\partial F}{\partial q_1}\delta q_1 + \frac{\partial F}{\partial q_2}\delta q_2 +\dots+\frac{\partial F}{\partial q_N}\delta q_N \tag{2.12}\end{align}

We can write this in terms of finite numbers if we write the virtual displacements as:

\begin{align} \delta q_i=\epsilon \alpha_i \tag{2.13}\end{align}

where \(\vec{\alpha}\) is a vector in the direction of the virtual displacement and \(\epsilon\) is a number that tends to zero. The rate of change of the function, \(F\), in the direction of \(\vec{\alpha}\) is:

\begin{align} \frac{\delta F}{\epsilon}=\frac{\partial F}{\partial q_1}\alpha_1 + \frac{\partial F}{\partial q_2}\alpha_2 +\dots+\frac{\partial F}{\partial q_N}\alpha_N \tag{2.14}\end{align}

which must vanish for a stationary point (writing the above equation as a sum):

\begin{align} \sum_i\frac{\partial F}{\partial q_i}\alpha_i =0 \tag{2.15}\end{align}

In order to have a stationary point, the rate of the change of the function must vanish for any direction, \(\vec{\alpha}\), so that each term must be equal to zero, independent of the \(\alpha_i\):

\begin{align} \frac{\partial F}{\partial q_i}\alpha_i &=0\\ \therefore \frac{\partial F}{\partial q_i}&=0 \tag{2.16}\end{align}

and we recover the familiar result from calculus that the partial derivatives must vanish at \(P\), for that point to be a stationary point of the function. The second order derivatives (“second variations”) are needed in order to know if this is a local minimum, maximum or saddle point. For analytic mechanics, we will generally only need to know if the point is stationary.