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Physics Examples Overview

This page introduces application examples of nblade in physics.

Example List

01_rigid_body.py - Rigid Body Dynamics

Difficulty: ⭐⭐⭐ Advanced

Content: - Angular momentum as bivector - Inertia tensor - Rigid body rotation dynamics - Euler equations in geometric algebra form - Rotational kinetic energy - Simple rigid body simulation

Run: 

python examples/physics/01_rigid_body.py

Expected Output: 

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nblade Rigid Body Physics Example
Application of Geometric Algebra in Classical Mechanics
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Section 1: Angular Momentum as Bivector
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In traditional vector analysis, angular momentum L = r × p
In geometric algebra, angular momentum naturally represents as bivector L = r ∧ p

Position vector r = 3.000000e1 + 4.000000e2
Momentum vector p = 2.000000e1 + -1.000000e2 + 5.000000e3

Angular momentum bivector L = r∧p = -11.000000e1∧e2 + 15.000000e1∧e3 + 20.000000e2∧e3
...

Core Concepts

Geometric Representation of Angular Momentum

In traditional vector analysis, angular momentum is defined as cross product:

\[\mathbf{L} = \mathbf{r} \times \mathbf{p}\]

In geometric algebra, angular momentum naturally represents as bivector:

\[L = r \wedge p\]

Advantages of this representation: - Works in arbitrary dimensions (cross product only works in 3D) - Clear geometric meaning (spanned plane) - Consistent with other geometric algebra operations

Relationship with Traditional Cross Product

In 3D, the dual of angular momentum corresponds to traditional cross product:

L_bivector = r ^ p        # Geometric algebra representation
L_vector = L_bivector.dual()  # Corresponds to traditional cross product result

Rigid Body Rotation

Rigid body orientation is described by rotor:

# Angular velocity plane
omega_plane = omega.dual()

# Update rotor
delta_rotor = alg.rotor(omega_plane, omega_magnitude * dt)
rotor = delta_rotor * rotor

Theoretical Background

Euler Equations

Euler equations for rigid body without external torque:

\[\frac{d\mathbf{L}}{dt} + \boldsymbol{\omega} \times \mathbf{L} = 0\]

In geometric algebra:

\[\frac{dL}{dt} + \omega \wedge L = 0\]

Rotational Kinetic Energy

\[T = \frac{1}{2} \omega \cdot L = \frac{1}{2} \sum_i I_i \omega_i^2\]

In nblade:

T = 0.5 * (omega | L).scalar_part()

Frequently Asked Questions

Q: Why use bivector to represent angular momentum?

A: Bivectors explicitly represent the rotation plane, which is more geometrically intuitive than traditional vectors (which represent rotation axis). Additionally, bivectors work in any dimension.

Q: How to calculate rotor from angular velocity?

A: The dual of angular velocity is the rotation plane, then use that plane to create rotor:

omega = alg.vector([0, 0, 1])  # Around z-axis
omega_plane = omega.dual()      # e1∧e2
rotor = alg.rotor(omega_plane, angle)

References

Next Steps