 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/9/a/solution

For fixed initial and final coordinates, the two Euler-Lagrange equations are

\frac{d}{d t} (\frac{\partial f}{\partial x ˙}) - \frac{\partial f}{\partial x} = 0, \frac{d}{d t} (\frac{\partial f}{\partial y ˙}) - \frac{\partial f}{\partial y} = 0.

δ F = \int_{0}^{T} (f_{x} δ x + f_{y} δy + f_{\overset{x}{˙}} δ \overset{x}{˙} + f_{\overset{y}{˙}} δ \overset{y}{˙}) d t .

 New to topics? Read the docs here!