AniTop Physics Documentation

AniTop simulates three classical integrable rigid body systems using SO(3) integration methods for numerical stability. All simulations use Runge-Kutta 4th order or midpoint integration with periodic orthonormalization to prevent rotation matrix drift.

1. Euler Top (Free Rigid Body)

A rigid body rotating in space with no external torques. Demonstrates torque-free motion governed entirely by the body's inertia distribution.

Equations of Motion

Euler's equations in the body-fixed frame:

Rotation matrix evolution:

where [ω]ₓ is the skew-symmetric matrix of angular velocity.

Physical Quantities

I₁, I₂, I₃

Principal moments of inertia (kg·m²) - arbitrary positive values

ω = [ω₁, ω₂, ω₃]

Angular velocity in body frame (rad/s)

R ∈ SO(3)

Rotation matrix from body frame to world frame

Conservation Laws

Physical Phenomena

• Tennis Racket Effect: Instability when rotating about the intermediate principal axis (Dzhanibekov effect)
• Stable Spin: Rotation about maximum or minimum inertia axis remains stable
• Tumbling: Complex periodic motion when all axes are active

Configurable Parameters

I₁, I₂, I₃

Principal moments of inertia (0.1–10 kg·m²)

ω₀ = [ω₁, ω₂, ω₃]

Initial angular velocity (-10 to 10 rad/s)

dt

Integration timestep (0.0001–0.01 s, default: 0.001 s)

timeScale

Simulation speed multiplier (0.1–4×, default: 1.0)

2. Lagrange Top (Symmetric Top with Gravity)

A symmetric top subject to gravity with center of mass on the symmetry axis. One of three known integrable cases of the heavy top.

Symmetry Constraint

Center of mass on symmetry axis: r_com = [0, 0, h]

Equations of Motion

Modified Euler's equations with gravitational torque:

Gravitational torque in body frame:

Physical Quantities

I₁ = I₂, I₃

Principal moments with axial symmetry (kg·m²)

m

Total mass (kg)

r_com = [0, 0, h]

Center of mass position on symmetry axis (m)

g

Gravitational acceleration (m/s²)

Energy Conservation

Physical Phenomena

• Sleeping Top: Fast vertical rotation with minimal precession
• Steady Precession: Constant precession rate around vertical axis
• Nutation: Periodic wobbling superimposed on precession
• Falling Motion: Complex dynamics at low angular momentum

Configurable Parameters

I₁ = I₂, I₃

Symmetric inertia moments (0.1–10 kg·m²)

m

Mass (0.1–10 kg)

h (z-component of r_com)

Center of mass height (0.1–2 m)

g

Gravity (0–20 m/s², default: 9.81 m/s²)

ω₀, dt, timeScale

Same as Euler top

3. Kovalevskaya Top

A special integrable case discovered by Sofia Kovalevskaya (1888). Requires specific inertia ratio and center of mass position.

Special Constraints

Equations of Motion

Same form as Lagrange top:

Note: Due to I₁ = I₂, the last term vanishes.

Kovalevskaya Invariant

Additional conserved quantity (complex invariant):

Physical Quantities

I₁ = I₂ = 2I₃

Fixed inertia ratio (kg·m²)

r_com = [a, 0, 0]

Center of mass on x-axis (m)

m, g

Mass and gravity as in Lagrange top

Conservation Laws

Configurable Parameters

I₃

Determines I₁ = I₂ = 2I₃ (0.1–5 kg·m²)

a (x-component of r_com)

Center of mass position (0.1–2 m)

m, g, ω₀, dt, timeScale

Same as Lagrange top

Comparison Table

Integration Methods

Runge-Kutta 4th Order (RK4)

Fourth-order accurate integration for angular velocity evolution:

SO(3) Integration (Rodrigues Formula)

Rotation matrix updates using matrix exponential:

where θ = h|ω|. Small angle approximation used for θ < 10⁻⁸.

Numerical Stability

• Periodic Gram-Schmidt orthonormalization (every 1000 steps)
• Taylor series for small angles to avoid division by zero
• Midpoint angular velocity for rotation updates
• Energy drift typically < 0.1% over long simulations

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