Calculus of variations YouTube

Generalised coordinates

The quantities we are interested in minimising will be, in general, functions of both positions and velocities (which, in turn, are functions of time).

For each particle in a system, there will be three position components and three velocity components, thus for $N$ particles there will be $3N$ positions and $3N$ velocities. Alternatively, we might be considering polar coordinates and angular velocities or some other complicated set components.

A succinct notation to cover all of the possibilities is to use generalised coordinates. We use $q$ for the coordinates and $\dot{q}$ for the velocities, where the dot notation really means the derivative with respect to the independent variable. In our case, this will be time but variational analysis can be applied to functions involving any independent variable.

Suppose we have a function $L$, of $N$ generalised coordinates (and velocities), which in turn depend on time. Then we could write the function with varying levels of detail, as follows:$$\begin{align*}L&= L(q_1(t), \dots, q_N(t), \dot{q_i}(t), \dots, \dot{q_N}(t))\\&\to L(q_1, \dots, q_N, \dot{q_i}, \dots, \dot{q_N}) \textit{ time-dependence implied}\\&\to L(q_i(t), \dot{q_i}(t)) \quad i = 1, \dots, N \textit{ implied}\\&\to L(q_i, \dot{q_i}) \quad i = 1, \dots, N \textit{ implied, time-dependence implied}\\&\to L(q(t), \dot{q}(t)) \textit{ single particle or indices implied}\\&\to L(q, \dot{q}) \textit{ single particle or indices implied, time-dependence implied}\end{align*}$$

Each coordinate and each velocity is treated as a seperate variable, so we can calculate partial derivates with respect each of them (again, with varying detail): $$\begin{align*}\mathrm{d}L &= \frac{\partial L}{\partial q} \mathrm{d}q + \frac{\partial L}{\partial \dot{q}} \mathrm{d}\dot{q}\\ &= \frac{\partial L}{\partial q_i} \mathrm{d}q_i + \frac{\partial L}{\partial \dot{q_i}} \mathrm{d}\dot{q_i} \quad \textit{summation implied}\\ &= \sum_{i = 1}^{N} \left\{ \frac{\partial L}{\partial q_i} \mathrm{d}q_i + \frac{\partial L}{\partial \dot{q_i}} \mathrm{d}\dot{q_i} \right\}\end{align*}$$

Professor Susskind mostly uses the notations:$$\begin{align*}L &= L(q_i, \dot{q_i})\\ \mathrm{d}L &= \frac{\partial L}{\partial q_i} \mathrm{d}q_i + \frac{\partial L}{\partial \dot{q_i}} \mathrm{d}\dot{q_i} \quad \textit{summation implied}\end{align*}$$

Variational analysis

We consider small variations in a given quantity, $L = L(q_i, \dot{q_i})$ say, under a certain type of small change in the coordinates, $q_i$, which has the form $$q_i \to q_i + \epsilon f_i(t)$$ where $\epsilon$ is supposed to be a small (infinitesimal) number and is a new parameter. The $f_i(t)$'s are arbitrary functions of the independent variable, $t$. From this it follows that $$\dot{q_i} \to \dot{q_i} + \epsilon \dot{f_i}(t)$$

Since we want to consider the small changes with respect to $\epsilon$, and not the independent variable $t$, we use a slightly different notation for the differentials, and refer to them as variations, $$\begin{align*}\delta q_i &= \epsilon f_i(t)\\ \delta \dot{q_i}&= \epsilon \dot{f_i}(t)\end{align*}$$ and so $$\begin{align*}\delta L &= \sum_{i = 1}^{N} \left\{ \frac{\partial L}{\partial q_i} \mathrm{d}q_i + \frac{\partial L}{\partial \dot{q_i}} \mathrm{d}\dot{q_i} \right\}\\ &= \sum_{i = 1}^{N} \left\{ \frac{\partial L}{\partial q_i} \epsilon f_i(t)+ \frac{\partial L}{\partial \dot{q_i}} \epsilon \dot{f_i}(t)\right\}\\ &= \epsilon \sum_{i = 1}^{N} \left\{ \frac{\partial L}{\partial q_i} f_i(t)+ \frac{\partial L}{\partial \dot{q_i}} \dot{f_i}(t) \right\}\end{align*}$$

In fact, the interesting quantities are functionals - intregrated functions over time, $$A = \int_{t_1}^{t_2} L(q_i, \dot{q_i}) \mathrm{d}t$$

The small change (variation) in the quantity $A$, then, is $$\begin{align*}\delta A &= \int_{t_1}^{t_2} \delta L \mathrm{d}t\\ &= \int_{t_1}^{t_2} \epsilon \sum_{i = 1}^{N} \left\{ \frac{\partial L}{\partial q_i} f_i(t)+ \frac{\partial L}{\partial \dot{q_i}} \dot{f_i}(t) \right\} \mathrm{d}t\end{align*}$$

In order the minimise the quantity $A$, we set $\delta A = 0$ and see what follows. The $\epsilon$ can be factored out, and we left we trying to solve $$\int_{t_1}^{t_2} \sum_{i = 1}^{N} \left\{ \frac{\partial L}{\partial q_i} f_i(t)+ \frac{\partial L}{\partial \dot{q_i}} \dot{f_i}(t) \right\} \mathrm{d}t = 0$$

This process is called the Calculus of variations.

Different variations in different coordinate directions

We might have a set of variables $\alpha_i$, and we want to minimise a function, $F$, that depends on them. The rule would be to set $$\frac{\partial F}{\partial \alpha_i} = 0 \quad \forall i = 1, \dots, N$$ and write $$\delta F = 0$$ meaning that $F$ does not change to 1st order derivatives when the $\alpha_i$ are varied infinitesimally. That is, saying $\delta F = 0$ is the same as saying $$\sum_{i = 1}^{N} \frac{\partial F}{\partial \alpha_i} = 0$$ which, in turn, means that simply $$\frac{\partial F}{\partial \alpha_i} = 0 \quad \forall i = 1, \dots, N$$