Calculator Inputs
Enter coefficients for the quadratic Lagrangian, define fixed endpoints, then solve for the extremal path.
Example Data Table
Use these sample settings to test the solver quickly.
| Example | a | b | c | m | n | x0 | x1 | y(x0) | y(x1) | Meaning |
|---|---|---|---|---|---|---|---|---|---|---|
| Hyperbolic | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | Classic fixed-endpoint quadratic functional. |
| Oscillatory | 2 | -3 | 1 | 0 | 2 | 0 | 2 | 1 | 0 | Produces a sinusoidal extremal family. |
| Parabolic | 1 | 0 | 4 | 1 | 1 | 0 | 2 | 0 | 3 | Reduces to a second-order polynomial extremal. |
Formula Used
Functional model:
J[y] = ∫ [a(y′)² + by² + cy + mx + n] dx
Euler-Lagrange equation:
d/dx(∂F/∂y′) - ∂F/∂y = 0
d/dx(2ay′) - (2by + c) = 0
2ay′′ - 2by - c = 0
Case split used by the solver:
- If b = 0, the extremal is polynomial.
- If b/a > 0, the extremal uses hyperbolic functions.
- If b/a < 0, the extremal uses sine and cosine.
How to Use This Solver
- Enter the coefficient of (y′)² as a. Keep it nonzero.
- Enter b, c, m, and n for the supported quadratic Lagrangian.
- Set the interval endpoints x0 and x1.
- Set fixed boundary values y(x0) and y(x1).
- Choose how many sample points you want for the table and graph.
- Click Solve Variational Problem to generate the Euler-Lagrange solution.
- Review the solved extremal, functional estimate, Legendre check, and plotted curves.
- Use the export buttons to download the sampled solution rows as CSV or a summary PDF.
Frequently Asked Questions
1. What kind of variational problems does this solver support?
It supports fixed-endpoint problems where the Lagrangian is quadratic in y and y′: a(y′)² + by² + cy + mx + n. That class is broad enough for many teaching examples and gives closed-form Euler-Lagrange solutions.
2. Why must a be nonzero?
The coefficient a multiplies the y′² term. If a were zero, the Euler-Lagrange equation would lose its second-order structure, and this solver’s analytic method would no longer apply correctly.
3. What does the functional value represent?
It is the estimated value of J[y] obtained by numerically integrating the Lagrangian along the computed extremal. It helps compare candidate paths inside the supported model.
4. Why are there hyperbolic, oscillatory, and parabolic cases?
Those cases come from the sign of b/a in the differential equation. Positive values lead to hyperbolic functions, negative values lead to sine and cosine, and zero gives a polynomial solution.
5. What is the Legendre condition shown in the result?
The Legendre check uses Fy′y′ = 2a. A positive value favors minimizing behavior for this quadratic model, while a negative value warns that a minimum is not supported by that condition.
6. Can this tool solve completely general Euler-Lagrange problems?
No. General nonlinear variational problems often need symbolic algebra or numerical boundary-value methods. This page is designed for a specific analytic class that remains practical and instructive.
7. What does the chart show?
The Plotly chart shows the extremal curve y(x) and the sampled integrand F on the same figure. This helps you inspect both the path shape and how the functional density changes across the interval.
8. Why might the solver say no unique extremal exists?
Some boundary conditions create a singular two-equation system for the integration constants. In those situations, the interval and coefficients do not determine one unique analytic extremal with this model.