Symplectic Integrator Calculator

This calculator integrates separable Hamiltonians H(p,q)=T(p)+V(q), where the symplectic update preserves the phase-space structure better than non-symplectic methods.

Symplectic Euler (kick-drift):

• p_{n+1} = p_n - dt · ∂V/∂q(q_n)
• q_{n+1} = q_n + dt · ∂T/∂p(p_{n+1})

Leapfrog / Stoermer–Verlet (kick-drift-kick):

• p_{n+1/2} = p_n - (dt/2) · ∂V/∂q(q_n)
• q_{n+1} = q_n + dt · ∂T/∂p(p_{n+1/2})
• p_{n+1} = p_{n+1/2} - (dt/2) · ∂V/∂q(q_{n+1})

Energy monitor: the calculator reports E(t)=T(p)+V(q) and the relative drift (E_f−E_0)/E_0 as a quick stability check.

1. Select a system (oscillator or pendulum) and an integrator method.
2. Enter initial state values q0 and p0.
3. Set physical parameters (mass and either k or L, g).
4. Choose a time step dt and a number of steps.
5. Press Compute to view results above the form.
6. Use Download CSV or Download PDF for exports.

1) Why symplectic time stepping matters

For many mechanical models, the equations come from a Hamiltonian split into kinetic and potential energy. Symplectic schemes are designed to preserve the geometric phase-space structure, so long simulations remain physically plausible. Instead of monotonically losing or gaining energy, the energy typically oscillates within a narrow band.

2) Systems supported in this calculator

The tool includes a harmonic oscillator and a simple pendulum, both written as separable Hamiltonians H(p,q)=T(p)+V(q). For the oscillator, V(q)=\u00bd k q\u00b2. For the pendulum, V(q)=m g L(1-\cos q). These are standard benchmarks for validating integrators.

3) Method comparison: Symplectic Euler vs Leapfrog

Symplectic Euler is first order in time, so phase error accumulates faster, yet it remains robust for conservative dynamics. Leapfrog (Stoermer\u2013Verlet) is second order and time-reversible, often producing noticeably smaller phase drift at the same step size. Both methods are explicit and efficient per step.

4) Choosing a good time step dt

A practical rule is to make dt small relative to the fastest period in the motion. For an oscillator with angular frequency \u03c9=\u221a(k/m), using 100\u2013200 steps per period usually yields smooth trajectories. If energy drift grows, reduce dt or switch to Leapfrog.

5) Energy drift as a stability diagnostic

The calculator reports initial energy, final energy, and an energy range. In symplectic methods the absolute energy is not exactly constant, but the error stays bounded for long runs. A drifting trend often indicates dt is too large, parameters are extreme, or the motion approaches a separatrix.

6) Interpreting the q(t) plot and table

The plot visualizes position or angle versus time with automatic sampling to keep rendering fast. The trajectory table can display up to your selected limit, while CSV and PDF exports include all steps. Use the table to spot numerical ringing, phase lag, or unexpected growth in amplitude.

7) Typical workflows for parameter studies

For quick sweeps, run a baseline case and note the percent energy drift. Then change one factor at a time: halve dt, double steps for the same final time, or switch methods. Because the tool caps steps at 20000, you can test long horizons without overloading a browser session.

8) Limitations and best practices

This page targets separable Hamiltonians with simple forces. For stiff systems, strongly driven motion, or discontinuous forces, you may need smaller time steps or implicit methods. Always sanity-check units, confirm initial conditions, and compare against known analytic solutions such as the small-angle pendulum limit.

1) What does “symplectic” mean in practice?

It means the update preserves phase-space geometry. Trajectories remain qualitatively correct over long times, and energy error typically stays bounded rather than drifting steadily.

2) Which method should I choose for accuracy?

Leapfrog is usually the best default. It is second order and time-reversible, often giving smaller phase and energy errors than Symplectic Euler at the same step size.

3) Why can energy change even with a symplectic method?

Symplectic schemes conserve a nearby “modified” Hamiltonian. Energy often oscillates around the true value. Large monotone drift usually signals that the time step is too large.

4) How do I set dt for the harmonic oscillator?

Estimate the period T=2\u03c0/\u03c9 with \u03c9=\u221a(k/m). Start with 100 steps per period, then reduce dt until the energy drift is acceptable for your application.

5) Does the pendulum use degrees or radians?

You can choose either. If you select degrees, the calculator converts to radians internally for integration and energy evaluation, then reports results in radians for consistency.

6) What does “energy drift percent” represent?

It is 100\u00d7(Ef\u2212E0)/E0. Use it as a quick stability indicator when comparing step sizes or methods under identical physical parameters and total simulation time.

7) Why is there a maximum step limit?

Long runs can create very large tables and heavy exports. The 20000-step cap keeps computation responsive while still supporting meaningful long-term behavior checks and parameter studies.