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Overview

Forces in Particle Simulator are defined as mathematical functions that can depend on time (t) and position (x, y, z). The system evaluates these formulas in real-time to compute the acceleration and trajectory of each particle.

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Force as a Function: F(t, x, y, z)

Every force component can be written as a mathematical expression using:
  • t - Elapsed time in seconds
  • x, y, z - Current position of the particle
  • Any mathematical function from the available functions library

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Vector Force Format

Forces are always specified as three components: 
[Fx, Fy, Fz]
Each component is a string containing a mathematical formula: 
{
  "vec": ["-x", "-y", "-z"]
}

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Time-Dependent Forces

Forces that change over time use the t variable: 
{
  "vec": [
    "0",
    "0",
    "sin(t * 0.5) * 5"
  ]
}
// Oscillating force in Z direction

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Position-Dependent Forces

Forces that depend on the particle’s current position:

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Spring/Restoring Forces

{
  "vec": ["-x", "-y", "-z"]
}
// Hooke's law: F = -kx (k=1)
// Always points toward origin

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Inverse-Square Gravity

Central Force
{
  "vec": [
    "-x / (sqrt(x^2 + y^2 + z^2)^3)",
    "-y / (sqrt(x^2 + y^2 + z^2)^3)",
    "-z / (sqrt(x^2 + y^2 + z^2)^3)"
  ]
}
This creates a gravitational force:
  • Magnitude: 1/r²
  • Direction: Always toward origin
  • Note: The r³ denominator comes from normalizing the direction vector

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Combined Forces

You can combine time and position dependencies: 
{
  "vec": [
    "-x * abs(cos(t*0.2))",
    "-y * abs(cos(t*0.2))",
    "-z * 0.5"
  ]
}
// Spring constant pulses with time
// Creates expanding/contracting motion

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Force Evaluation Modes

The simulator supports two calculation modes:

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Dynamic Mode: F = ma

Forces are evaluated and Newton’s second law is applied:
// From Movimiento.ts evaluarFormula function
const Fx = evaluarFormula(force.vec[0], t, x, y, z);
const Fy = evaluarFormula(force.vec[1], t, x, y, z);
const Fz = evaluarFormula(force.vec[2], t, x, y, z);

// Acceleration: a = F/m
ax = Fx / mass;
ay = Fy / mass;
az = Fz / mass;
When to use:
  • Realistic physics simulations
  • When mass matters
  • Multiple interacting forces
Example:
{
  "mass": 5,
  "forces": [
    {"id": 1, "vec": ["-x", "-y", "0"]},
    {"id": 2, "vec": ["0", "0", "-9.8"]}
  ]
}

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Implementation Details

The force evaluation uses function caching for performance: 
// From Movimiento.ts
const formulaCache: Record<string, Function> = {};

export const evaluarFormula = (
  formula: string,
  t: number,
  x: number = 0,
  y: number = 0,
  z: number = 0
): number => {
  // Replace ^ with ** for exponentiation
  const cleanedFormula = formula.replace(/\^/g, "**");
  
  // Create function with Math context injected
  fn = new Function(
    "t", "x", "y", "z", ...mathKeys,
    `return ${cleanedFormula}`
  );
  
  // Execute with current values
  return fn(t, x, y, z, ...mathValues);
};

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Key Features:

Function Caching

Formulas are compiled once and cached to avoid repeated parsing

Math Injection

All JavaScript Math functions are automatically available

Power Operators

Both ^ and ** work for exponentiation

Error Handling

Invalid formulas return 0 instead of crashing

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Common Patterns

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Oscillators

// Simple Harmonic: F = -kx
["-x", "-y", "-z"]

// Damped: F = -kx - bv (approximated)
["-x * 1.5", "-y * 1.5", "-z * 1.5"]

// Driven: F = -kx + F₀sin(ωt)
["-x", "-y", "-z + 10*sin(2*t)"]

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Central Forces

// Attractive 1/r²
[
  "-x / (sqrt(x^2 + y^2 + z^2)^3)",
  "-y / (sqrt(x^2 + y^2 + z^2)^3)",
  "-z / (sqrt(x^2 + y^2 + z^2)^3)"
]

// Repulsive 1/r²
[
  "x / (sqrt(x^2 + y^2 + z^2)^3)",
  "y / (sqrt(x^2 + y^2 + z^2)^3)",
  "z / (sqrt(x^2 + y^2 + z^2)^3)"
]

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Rotating Forces

// Vortex (rotates around Z axis)
[
  "-y",  // Fy ∝ -y creates rotation
  "x",   // Fx ∝ x creates rotation
  "-z"   // Fz ∝ -z keeps Z bounded
]

// Pulsing Vortex
[
  "-y * abs(cos(t*0.2))",
  "x * abs(cos(t*0.2))",
  "-z"
]
Pro Tip: Start with simple formulas and add complexity gradually. Use the trajectory examples as a starting point for your own creations.

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