Euler–Lagrange Variations
Bend a fixed-endpoint path and locate the stationary value of an integral functional.
Free theory articleBend a fixed-endpoint path and locate the stationary value of an integral functional.
Concept & interpretation
The Euler-Lagrange equation determines the path that makes an integral functional stationary. Here, we bend a polynomial path with fixed endpoints to minimize a model functional, revealing the connection between the functional derivative and ordinary calculus.
Perturbing the path with real controls
Open Euler–Lagrange Variations to observe the path variations interactively.
- Set the Endpoints to "Rising" (0 to 1) and Forcing q to 2. The optimal Stationary bend c* is calculated as -1.
- Set the Bend c to 0. Notice the Functional value is 1.5, matching our manual calculation. The Euler Lagrange residual (y'' - q) is -2, indicating the path is not stationary.
- Adjust the Bend c to -1. The residual becomes 0, the Functional derivative is 0, and the Excess above minimum falls to 0.0. The path visually aligns with the dashed stationary path.
- Change Endpoints to "Shifted" (1 to 2) and Forcing q to -2. Adjust Bend c to 1 to find the new stationary configuration.