Formula used
For a generalized coordinate q and velocity q̇, the Euler–Lagrange equation is:
This calculator applies that rule to selected Lagrangians and then integrates the resulting second‑order equation using a 4th‑order Runge–Kutta method.
How to use this calculator
- Select a system (oscillator, pendulum, quartic, or custom potential).
- Enter parameters (mass and any system coefficients).
- Set t₀, t₁, and dt, then choose initial conditions.
- Click Solve. The results section appears above the form.
- Use CSV/PDF buttons to export the computed trajectory and summary.
Example data table
Example: harmonic oscillator with m=1, k=4, q(0)=1, q̇(0)=0, dt=0.1.
| t | q | q̇ | E |
|---|---|---|---|
| 0.0 | 1.0000 | 0.0000 | 2.0000 |
| 0.1 | 0.9801 | -0.1987 | 2.0000 |
| 0.2 | 0.9211 | -0.3894 | 1.9999 |
| 0.3 | 0.8253 | -0.5646 | 1.9999 |
| 0.4 | 0.6967 | -0.7170 | 2.0000 |
Article: Interpreting results from an Euler-Lagrange solver
1) Modeling motion with energy terms
The Euler-Lagrange framework starts from energy. You describe a system with a kinetic energy T and a potential energy V, then form the Lagrangian L = T - V. This calculator uses that description to build a second-order equation for a generalized coordinate q, which may represent displacement or an angle.
2) From L(q,qdot) to an equation of motion
After selecting a model, the tool applies d/dt(dL/dqdot) - dL/dq = 0 and simplifies it to a standard form. For the harmonic oscillator, it returns m qddot + k q = 0. For the pendulum, it returns theta_ddot + (g/l) sin(theta) = 0. These expressions are shown in the Results panel.
3) Built-in systems and what they represent
The preset options cover common teaching and lab cases. The harmonic oscillator matches springs, small vibrations, and linear circuits with an equivalent k/m ratio. The simple pendulum captures rotational motion under gravity, including nonlinear behavior at larger angles. The quartic option demonstrates strongly nonlinear restoring forces where frequency depends on amplitude.
4) Numerical solving with a fourth-order method
Most Euler-Lagrange equations do not have a convenient closed-form solution for arbitrary initial conditions. The calculator integrates q(t) using a fourth-order Runge-Kutta (RK4) stepper, which samples the acceleration multiple times per step to improve accuracy. For smooth dynamics, RK4 typically reduces error much faster than basic Euler stepping at the same dt.
5) Energy diagnostics as a quality check
For conservative systems, total energy E = T + V should remain nearly constant. The trajectory table includes T, V, and E at each time. If E drifts steadily, your time step is likely too large, or the motion is stiff or highly nonlinear. Reduce dt and compare runs to confirm convergence.
6) Choosing dt, time range, and initial conditions
dt controls stability and resolution. Smaller dt captures fast oscillations and reduces numerical energy error, but increases row count. A practical approach is to run once, then halve dt and check whether q(t) and E(t) change noticeably. For pendulum motion, remember q is an angle in radians; values like 0.2 to 1.2 radians are common test cases.
7) Custom polynomial potentials for experiments
The custom potential option implements V(q) = (c2/2) q^2 + (c4/4) q^4 + (c6/6) q^6, useful for approximating real systems near equilibrium or fitting a measured restoring curve. Positive c2 provides a basic well, while c4 and c6 add hardening or softening behavior. Exported CSV/PDF output makes it easy to document parameter sweeps.
FAQs
1) What does q represent in this calculator?
q is the generalized coordinate. It is displacement for oscillator models and the angle (theta) in radians for the pendulum model. The labels in Results indicate which interpretation applies.
2) Why does the energy E(t) drift in my output?
Energy drift usually means dt is too large for your dynamics. Reduce dt, shorten the time range, or lower the initial amplitude. For stiff or highly nonlinear motion, smaller steps are essential.
3) How do I know if my solution is accurate enough?
Run the same case with a smaller dt (for example, half the original). If the trajectories and energy curves change only slightly, your original step size is likely acceptable.
4) Can I solve multi-coordinate Lagrangians here?
This version solves one generalized coordinate q. For coupled coordinates, you would extend the model to a vector state and apply Euler-Lagrange to each coordinate, then integrate the full system.
5) What units should I use for inputs?
Use consistent units throughout. For example, SI units (kg, m, s) work well. The equations are unit-agnostic as long as m, k, g, l, and coefficients match the chosen unit system.
6) Why is the pendulum equation nonlinear?
The exact restoring term is sin(theta), not theta. The small-angle approximation sin(theta) ~= theta linearizes the motion, but larger angles require the nonlinear form shown in Results.
7) What do the CSV and PDF exports include?
CSV includes the trajectory columns (t, q, qdot, T, V, E). PDF includes a summary, the displayed equation, a data table, and a plot image when available.