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Verlet Integration

Updates motion using current and previous positions.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Simulation
Difficulty: Intermediate
Family: Numerical Integration
Tier: Core
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 6
Code example: Not ready
Visualization: Ready

Overview

Verlet Integration estimates a new position from the current position, previous position, acceleration, and time step.

Core Concepts

position historyacceleration updatetime stepconstraint-friendly motionsimulated statephysical stateintegral approximationnumerical integration

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(t)

Average

O(t)

Worst

O(t)

Space

O(1)

When to Use

Use Verlet Integration when you need to advance position without storing velocity directly.
Use it for particle simulations with inputs like current position.
Use it when approximate behavior over time is more useful than a closed-form answer.
Use it when the Numerical Integration tradeoff is acceptable for the problem size.

When Not to Use

Avoid it when the time step, grid size, or sample count cannot support stable results.
Avoid it when the required output is not next position.

What It Does

Advances position without storing velocity directly
Uses previous position to preserve momentum-like behavior

Use Cases

Particle simulations
Cloth simulation
Constraint-based physics

Inputs

Current position
Previous position
Acceleration
Time step

Outputs

Next position

Key Formulas (1)

Verlet position update

position_next = 2 · position − position_previous + acceleration · dt²

position_next = 2 * position - position_previous + acceleration * dt^2

Uses current and previous positions to advance motion.

Pseudocode

1.Store the current position
2.Compute displacement from current and previous positions
3.Add acceleration scaled by the time step squared
4.Set the next position
5.Move the old current position into previous position

Canonical Example

Input: position = 0, velocity = 2, time step = 0.5
Output: next simulated state
Explanation: Verlet Integration: The algorithm advances the system by applying its update rule over a small step.

Implementation Notes

Keep the input contract explicit: Current position; Previous position; Acceleration; Time step.
Use this step as the implementation spine: Store the current position.
Then preserve the next state change: Compute displacement from current and previous positions.
Track units and invariants such as mass, energy, or count where they matter.

Common Pitfalls

Verlet Integration is popular for simple stable particle constraints.
Using a time step that is convenient for rendering but unstable for the simulated system.
Returning next position without handling empty, invalid, or boundary inputs.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show previous and current positions determining the next particle position.

Step 1 of 5: Store the current position

particle-history

Notes

Verlet Integration operates in O(t) time and O(1) space within the Numerical Integration family. Key governing relation: position_next = 2 * position - position_previous + acceleration * dt^2. Often compared with Euler Integration and Runge-Kutta Method.