Calculator Inputs
Formula Used
This calculator models a quadratic trial path and a quadratic perturbation.
Trial path: y(x) = ax2 + bx + c
Perturbation: η(x) = px2 + qx + r
Functional density: F(x, y, y′) = αy2 + β(y′)2 + γxy
First variation: δJ = ∫[(2αy + γx)η + 2βy′η′] dx
The page evaluates this integral numerically with Simpson’s rule across the chosen interval.
How to Use This Calculator
- Enter coefficients for the quadratic trial path y(x).
- Enter coefficients for the perturbation function η(x).
- Set weights for y2, y′2, and xy.
- Choose the interval start, interval end, and sample count.
- Press the calculate button to place the result below the header.
- Review the sampled table, graph, and exported report files.
Example Data Table
This example uses y(x) = x2 - x + 2, η(x) = 0.5x2 + x, α = 2, β = 1, γ = 1, and interval [0, 2].
| x | y(x) | y′(x) | η(x) | η′(x) | Variation Density |
|---|---|---|---|---|---|
| 0.00 | 2.00 | -1.00 | 0.00 | 1.00 | -2.00 |
| 0.50 | 1.75 | 0.00 | 0.63 | 1.50 | 4.69 |
| 1.00 | 2.00 | 1.00 | 1.50 | 2.00 | 17.50 |
| 1.50 | 2.75 | 2.00 | 2.63 | 2.50 | 42.81 |
| 2.00 | 4.00 | 3.00 | 4.00 | 3.00 | 90.00 |
Expected example first variation: 52.133333
FAQs
1. What does first variation measure?
It measures how a functional changes under a small perturbation of the trial path. A zero first variation often signals a stationary path or candidate extremum in variational problems.
2. Why does this calculator use quadratic functions?
Quadratic trial and perturbation functions keep input simple while still showing how coefficients, derivatives, and interval choice affect the first variation numerically and visually.
3. What do the weights α, β, and γ do?
They scale the y2, y′2, and xy parts of the modeled functional density. Changing them alters both the density curve and the integrated first variation value.
4. Why must the sample count be even?
Simpson’s rule works with an even number of subintervals. The calculator automatically adjusts odd values so the numerical integration remains valid.
5. Is this an exact symbolic solver?
No. It is a numerical learning tool. It evaluates the chosen model across the interval and estimates the first variation with Simpson integration.
6. What does the variation density graph show?
The graph shows how y(x), the perturbation, and the local first variation density change through the interval. It helps identify where the integral receives its strongest contributions.
7. When is the result especially useful?
It is useful for coursework, quick experiments, method demonstrations, and checking whether a selected trial path appears closer to a stationary solution under a chosen perturbation.
8. What do the CSV and PDF exports contain?
The CSV export contains the sampled table. The PDF export includes the main result values and the displayed sample rows for reporting or revision notes.