Build governing equations from configurable Lagrangian terms fast. Inspect symbolic derivatives, residuals, and acceleration estimates. Use responsive inputs, exports, examples, and clear guidance easily.
Large screens show 3 columns, medium screens show 2, and phones show 1.
This calculator evaluates the partial derivatives numerically at the chosen state and then computes the residual of the Euler–Lagrange equation.
When the effective inertial factor a + 3nq̇² is nonzero, the calculator also solves directly for q̈.
| Case | a | c | g | i | j | q | q̇ | t | Estimated q̈ |
|---|---|---|---|---|---|---|---|---|---|
| Harmonic oscillator | 1 | 4 | 0 | 0 | 0 | 0.5 | 0.2 | 0 | -2 |
| Driven linear system | 2 | 6 | 0 | 0 | 0 | 0.4 | 0.1 | 2 | -2.2 |
| Cosine potential | 1 | 0 | 9.81 | 0 | 0 | 0.2 | 0.0 | 0 | -1.949 |
| Nonlinear polynomial | 1 | 0 | 0 | 2 | 3 | 1.1 | 0.6 | 0 | -4.086 |
It builds a configurable one-coordinate Lagrangian, computes ∂L/∂q, ∂L/∂q̇, d/dt(∂L/∂q̇), the Euler–Lagrange residual, and a solved q̈ when the inertial denominator is nonzero.
For constant coefficient b, the product q·q̇ is a total derivative of q²/2. Total-derivative terms change the action by a boundary contribution and therefore cancel from the Euler–Lagrange equation.
A constant multiple of q̇ is also a total derivative. It shifts the Lagrangian numerically, but it does not alter the governing differential equation for the generalized coordinate.
They are interpreted in radians. If your source data is in degrees, convert it first by multiplying the angle by π/180 before entering it.
The residual is d/dt(∂L/∂q̇) − ∂L/∂q evaluated at your entered state. A value near zero means the supplied q, q̇, q̈, and t are consistent with the equation.
It becomes unavailable when a + 3nq̇² equals zero. In that case, the effective inertial factor vanishes and the equation cannot be solved uniquely for q̈.
Yes. The q̇⁴, q³, q⁴, sin(q), and cos(q) terms allow you to study nonlinear kinetic and potential effects within the predefined term family.
No. It is a structured advanced calculator for a rich predefined Lagrangian family. It derives the matching Euler–Lagrange form and evaluates it numerically from your coefficients and state values.
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.