Euler-Lagrange for parametric curves

...and generalizing to more coordinates (3 and more dimensions).

Previously, we assumed that the path between two points could be expressed as a function $y=y(x)$. But by limiting ourselves to functions, we rule out some possible paths, e.g. the one at the right (in 2 dimensions).

We can treat a broader class of paths between two points if we consider parametric equations for $x(u)$ and $y(u)$ as functions of a parameter $u$, where the parameter runs from $u_1$ to $u_2$ to trace out the $(x(u),y(u))$ coordinates of the arc pictured.

In our 'shortest path' problem, we'd like to minimize the integral of arc length. This could be expressed in terms of parametric equations for the path as: $$L = \int_1^2 ds = \int_1^2 \sqrt{dx^2 + dy^2} = \int_{u_1}^{u_2} \sqrt{x'(u)^2 + y'(u)^2}\,du.$$

Where the primes mean, derivatives with respect to $u$, for example $$x'(u) = \frac{dx}{du}.$$

We'd like to minimize an integral $$S = \int_{u_1}^{u_2} f( x(u), x'(u), y(u), y'(u), u )\,du$$ that depends in the most general case on

• the coordinates,$x(u),\ y(u)$,
• the parameter, $u$, and
• the first derivatives of the coordinates with respect to the parameter.

We specify a general path as $X(u)$ and $Y(u)$, which depend on a parameter $u$.

The path approaches the best path given by $x(u)$ and $y(u)$, as parameters $\alpha\to 0$ and $\beta\to 0$ according to $$X = x(u) + \alpha \xi(u);\ Y = y(u) + \beta\eta(u) $$

The stationary solution occurs when both... $$\frac{\del S(\alpha, \beta)}{\del \alpha } = 0;\ \frac{\del S(\alpha, \beta)}{\del \beta} = 0$$

And this leads to a pair of Euler-Lagrange equations, one for $x$ and one for $y$: $$\frac{\del f}{\del x}=\frac{d}{du}\frac{\del f}{\del x'};\ \frac{\del f}{\del y}=\frac{d}{du}\frac{\del f}{\del y'}$$

I think you can see how to extend this to dependence on greater numbers of variables, for example $z=z(u)$ to talk about motion in 3 dimensions.

Application to mechanics

The problem of analytical mechanics is to find the motion (path) in space as a function of the (parameter) time. Huh! You might wonder if there's some use of this formulism to find paths of particles...