Calculus of Variations Physics Calculator

Calculator

Mode

Choose a closed-form template or a general numeric extremal.

x0

Start of parameter (often time).

x1

End of parameter.

y(x0)

Boundary value at x0.

y(x1)

Boundary value at x1.

Grid points N

Used for plotting and for numerical mode.

Template settings

Template

Closed-form extremals and classical action.

Mass m

Used in all templates.

Spring constant k

Used for harmonic oscillator template.

Force F

Used for uniform force template.

Reset

Example data table

Harmonic oscillator example with m=1, k=4, x0=0, x1=1, y0=1, y1=0.3 (ω=2).

x	y(x)	Comment
0.00	1.0000	Boundary value
0.25	1.2546	Intermediate point
0.50	1.2023	Intermediate point
0.75	0.8564	Intermediate point
1.00	0.3000	Boundary value

Formula used

A functional is minimized (or extremized) when the Euler–Lagrange equation holds:

d/dx(∂L/∂y\') − ∂L/∂y = 0

This calculator supports:

L = ½ m y\'^2 → m y\'' = 0
L = ½ m y\'^2 − ½ k y^2 → y\'' + (k/m) y = 0
L = ½ m y\'^2 + F y → m y\'' = F
Polynomial numerical form: L = ½ A y\'^2 + ½ B y^2 + ¼ C y^4 + D y + E x y + F0

How to use this calculator

Select a mode: template for closed-form extremals, or numerical for custom coefficients.
Set x0, x1 and the boundary values y(x0), y(x1).
Choose N for the curve resolution (higher gives smoother plots).
For template mode, enter m and optional k or F.
For numerical mode, set A…F0, iterations, relaxation, and smoothing.
Press Compute extremal. Results appear above the form.
Use Download CSV or Download PDF for exporting.

Calculus of variations in physics: practical guide

1) Why variational methods are widely used

Physics often selects the path that makes an integral stationary. In mechanics this is the action S=∫L dt, in optics it is optical path length, and in fields it is ∫ℒ d⁴x. This tool turns that idea into computable curves.

2) Building a functional with boundary conditions

Choose a function y(x) and a Lagrangian L(x,y,y′). The objective is the functional S[y]=∫_x0^x1 L dx. Fixed endpoints y(x0)=y0 and y(x1)=y1 create a two-point boundary value problem, matching the required inputs.

3) Euler–Lagrange equation as the stationarity test

Requiring δS=0 for small perturbations produces the Euler–Lagrange equation d/dx(∂L/∂y′) − ∂L/∂y = 0. It replaces “search over functions” with a differential equation plus boundary data. The result panel prints the exact form implied by your selections.

4) Template extremals: free particle

For L = ½ m y′², the equation becomes m y″ = 0, so the extremal is a straight line between endpoints. The action scales like (y1−y0)²/T with T=x1−x0, making this template a fast sanity check for endpoint entry and scaling.

5) Template extremals: harmonic and forced motion

With L = ½ m y′² − ½ k y², the solution is sinusoidal with ω=√(k/m). The classical action contains sin(ωT), so the endpoint problem can be sensitive near sin(ωT)≈0 (the tool warns you). With L = ½ m y′² + F y, the extremal is a parabola with a=F/m and endpoints fixing the initial slope.

6) Interpreting the action value

The reported S is ∫L dx along the computed extremal, evaluated by a trapezoidal rule on your grid. Absolute values depend on scaling and units, so comparisons across controlled parameter changes are most informative. Use S with the curve shape.

7) Numerical extremals for custom polynomial models

Numerical mode uses L(x,y,y′)=½A y′² + ½B y² + ¼C y⁴ + D y + E x y + F0, giving A y″ − (B y + C y³ + D + E x)=0. The solver iteratively relaxes interior points while keeping endpoints fixed, and reports a residual RMS so you can judge convergence.

8) Choosing grid, iterations, and exports

For smooth curves, N=201–801 is a practical range. If residual RMS stays above ~1e−3, increase iterations, reduce relaxation, or add mild smoothing (0.02–0.10). CSV preserves every (x,y) point for external plotting, and the PDF captures the equation and summary for documentation.

FAQs

1) What does the action value represent in this tool?

It is S = ∫ L dx evaluated along the displayed extremal path using numerical integration. Use it to compare scenarios consistently, not as a universal “energy” value.

2) Why must I enter boundary values at both ends?

This is a two-point boundary value problem. The variational principle selects an extremal curve that satisfies the Euler–Lagrange equation and matches y(x0)=y0 and y(x1)=y1.

3) When should I use numerical mode instead of templates?

Use templates for standard closed-form cases. Use numerical mode when you need custom coefficients, nonlinear terms (C y⁴), or x·y coupling, and you want a computed extremal rather than an analytic one.

4) The residual RMS is large. How can I improve it?

Increase iterations, reduce relaxation, or add smoothing. Also increase N if the curve has sharp features. Very stiff settings (large C or small |A|) may need smaller steps and more iterations.

5) How do I pick the number of grid points N?

Start with 201 for quick feedback. Raise to 401–801 for smoother curvature or stronger nonlinearities. Extremely large N can slow convergence; balance resolution against runtime and residual improvement.

6) Does the calculator enforce physical units automatically?

No. You must use a consistent unit system across inputs (for example, seconds and meters). The equations and action follow from your numbers, so unit consistency is your responsibility.

7) What is included in the CSV and PDF exports?

CSV contains the summary parameters plus all (x,y) points of the extremal. PDF includes the Euler–Lagrange form, the summary list, and a sample of points for quick documentation.