Minimising quantities YouTube

Minimising functions

Figure 2.1 - Stationary points

Figure 2.1 - Stationary points

We can find local stationary points of a function, $f(x)$, of an independent variable, $x$, by calculatiing the derivative of the function and then setting it to zero. That is, $$\frac{\mathrm{d}f}{\mathrm{d}x} = 0$$

In order to find out whether we have a maximum point, a minimum point, or a point of inflection, we have to find the second derivative and note that $$\frac{\mathrm{d}^2f }{\mathrm{d} x^2} \begin{cases}> 0 \Rightarrow & \text{minimum}\\ < 0 \Rightarrow & \text{maximum}\\ = 0 \Rightarrow & \text{inflection}\end{cases}$$

For functions of several variables, $F = F(x_1, \dots, x_N)$, we calculate the partial derivatives and set them all to zero, $$\frac{\partial F}{\partial x_i} = 0 \quad \forall i \in \{1, \dots, N\}$$

Figure 2.2 - Stationary points - two variables

Figure 2.2 - Stationary points - two variables

Example. Figure 2.2 shows the contours of the function $$F(x, y) = x^2 + xy + y^2 - 3x - 2y$$ The derivatives are $$\begin{matrix}\frac{\partial F}{\partial x} = 2x + y - 3\\ \frac{\partial F}{\partial y} = 2y + x - 2\end{matrix}$$ and setting both to zero leads to the two equations (dashed lines in the figure) $$\left. \begin{matrix}y = -2x + 3\\ y = -\tfrac{1}{2}x + 1\end{matrix} \right\} \Rightarrow P_x = \tfrac{4}{3}, P_y = \tfrac{1}{3}$$ hence $P = (P_x, P_y)$ is the only stationary point of $F$.

Nb. We would need to look at the second-order derivatives to discover which type of stationary point $P$ is.

Minimising functionals

Figure 2.3 - Global laws versus local laws

Figure 2.3 - Global laws versus local laws

Newton's formulation is a local law.

Given a position and velocity of a particle, the differential equations describe where the particle will be in the next instant in time, and indirectly, what the new velocity will be.

We can also consider motion on a global scale - calculating quantities that depend on whole trajectories, rather than specific locations on the trajectory.

In this case, we are given a initial position, and a final position after a specified length of time and we want to find the unique trajectory that minimises a certain quantities.

Nb. Specifying the position at two different times is equivalent to specifying the position and the momentum at a single time. The information content is the same.

That implies that we will need to to be able to minimise functionals (functions of functions), and in particular, integrals of functions over time.

To do this, we use the Calculus of variations, which is a very general method of minimising functionals. We will apply it in conjunction with the Principle of least action (action to be defined later), but it also applies to other integrated quantities. For example, it can be used to prove that the shortest distance between two points is a straight line, that the path that light takes between two points is the one that minimises the time taken (the Principle of least time).