Test macros for
x_{1}, x_{2}, y_{1}, y_{2}
whereas these should not be changed: x_1, x_2, y_1, y_2. Here is an inline
version:
y_{2} = y_{1} + f1(x_{1}) + f2(x_{2})
of a math equation.
And again these should not be changed: x_1, x_2, y_1, y_2.

Here is an example of using the eqn_start macro:

y_{2} = y_{1} + f1(x_{1}) + f2(x_{2})

Here is a MathJax version \(y_1 + f1(x_1) + f2(x_2)\) which is inline within text.

The Euler-Lagrange Equation

The Euler-Lagrange equation is a differential equation that \(y(x)\) must satisfy in
order to minimize (or possibly maximize) an integral of the form
$$ I = \int_{x_1}^{x_2} f(x, y, y') dx $$
where also \(y(x)\) must pass through the two fixed end points \(y(x_1) = y_1\) and
\(y(x_2) = y_2\). Through reasoning about families of curves that include the
minimizing curve \(y(x)\), we can get the Euler-Lagrange equation:
$$ \frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) = 0 $$

This page is for testing and development
please see myPhysicsLab for the published
version of this simulation.