Model trial functions for variational optimization problems. Inspect actions, gradients, convergence, and neighboring path behavior. Use practical inputs to study stability and endpoint effects.
The solver minimizes an action integral over a bounded interval.
Lagrangian: L = 0.5ay′² + 0.5by² + cy + d
Action: J[y] = ∫ from x0 to x1 of L(x, y, y′) dx
Trial path: y(x) = (1-s)y0 + sy1 + αs(1-s) + βs(1-s)(1-2s)
Here, s = (x - x0) / (x1 - x0).
The added terms keep the endpoint values fixed.
The script evaluates the action with Simpson integration.
It searches the alpha beta grid for the lowest action.
It then refines the best area with local passes.
The residual metric uses the Euler equation form a y″ - b y - c.
A lower action and smaller residual usually indicate a better stationary path.
Enter the interval start and end values first.
Set the fixed boundary values for y.
Choose the Lagrangian coefficients a, b, c, and d.
Set alpha and beta search ranges.
Increase grid points for a finer search.
Increase integration steps for smoother action estimates.
Use refinement rounds to improve the final local minimum.
Press the solve button to generate the result block.
Review action, gradients, curvature, and residual data.
Download CSV or PDF reports when needed.
| Case | x0 | x1 | y0 | y1 | a | b | c | d | Alpha Range | Beta Range |
|---|---|---|---|---|---|---|---|---|---|---|
| Default Demo | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | -4 to 4 | -4 to 4 |
| Stiffer Path | 0 | 2 | 1 | 0 | 3 | 2 | -1 | 0.5 | -3 to 3 | -2 to 2 |
| Weak Potential | -1 | 1 | 0 | 0 | 1 | 0.2 | 0.4 | 0 | -5 to 5 | -5 to 5 |
A variational principle solver helps you estimate stationary paths. It is useful in calculus of variations, mechanics, and optimization. Many exact solutions are hard to derive. A numerical trial function gives a fast approximation. This calculator keeps the endpoints fixed. It then searches for the lowest action over adjustable parameters.
The core output is the action value. Lower action often signals a better candidate path. The tool also reports gradients. These show whether the current point is close to stationarity. Curvature values describe local shape near the optimum. Residual data checks how well the Euler condition is satisfied along the interval.
Trial functions give structure to the search. They reduce an infinite dimensional problem to a smaller parameter set. This page uses two free coefficients. That choice balances flexibility and speed. The endpoint preserving terms are important. They let the interior bend while the boundaries stay fixed. That mirrors standard variational constraints.
Start with wide alpha and beta ranges. Use moderate grid counts first. Then refine around the best region. Raise integration steps if the action changes too much between runs. Compare neighboring actions in the sensitivity table. A clear minimum is easier to trust. Very flat results may need wider ranges or different coefficients.
This solver fits educational work, quick research checks, and classroom demonstrations. It helps compare boundary value setups. It also supports simple physical models with quadratic energies. You can export results for reports or audits. The CSV file is useful for spreadsheets. The PDF summary works well for documentation and review.
This page uses a compact two parameter trial family. That keeps runtime low. It also means the exact minimizer may lie outside the chosen shape class. Search ranges matter as well. Narrow bounds can hide better candidates. Wider bounds improve coverage, but they increase total evaluations and runtime.
Do not read one number alone. Look at action, gradients, and residual together. A low action with a small stationarity score is usually stronger. Also check midpoint behavior and slope samples. Those values reveal whether the path shape matches your expectation. This makes the calculator more reliable for careful mathematical interpretation.
It approximates a stationary path for a chosen action integral. The tool searches over trial function parameters while holding the endpoint values fixed.
They control the interior shape of the trial path. This lets the solver bend the curve without changing the boundary values.
The action is the integral of the Lagrangian over the interval. The solver looks for the parameter pair that gives the smallest action in the tested region.
It measures how closely the trial path follows the Euler equation linked to the chosen quadratic Lagrangian. Smaller values usually indicate a better fit.
Start with 200 steps for quick work. Increase the value when you need smoother action estimates or when results change noticeably between runs.
Refinement narrows the search around the best coarse result. This often improves accuracy without forcing a huge initial grid.
Yes. The calculator accepts negative values. Still, interpret results carefully because some coefficient choices can create unstable or nonphysical action landscapes.
The CSV file contains key metrics and the local sensitivity table. The PDF file gives a compact summary for sharing or archiving.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.