Computable Coordinate System for Momentum Calculation

This guide introduces a novel momentum calculation method based on the Computable Coordinate System(CCS) framework. It leverages coordinate system structures and algebraic operations to provide a geometrically intuitive approach for momentum analysis in physics systems.

1. Momentum Formulation in CCS

2.1 Intrinsic Space and Momentum

• Local Coordinate System: Each object has a local frame where scaling s is linked to mass m (e.g., s = √m).
• Intrinsic Velocity (v'): $$v' = \sqrt{m} \cdot v \quad \text{(relates physical velocity `v` to intrinsic space)}$$
• Intrinsic Momentum (p'): $$p' = \sqrt{m} \cdot v' = m \cdot v = p \quad \text{(consistent with physical momentum)}$$

1.2 Momentum Conservation via Coordinate Operations

1.2.1 Elastic Collision Example

Setup:

• Object A: m1 = 1, v1 = 2 (local frame: C1)
• Object B: m2 = 4, v2 = 0 (local frame: C2)

Step 1: Transform to World Coordinate System

$$V1_{\text{world}} = v1 * C1, \quad V2_{\text{world}} = v2 * C2$$

Step 2: Collision Dynamics Using coordinate composition and division to model momentum transfer:

$$V_{\text{total}} = (V1 + V2) * (C1 + C2) \quad \text{(momentum summation)}$$

Step 3: Post-Collision Transformation After collision, transform back to local frames:

$$v1'_{\text{local}} = V_{\text{total}} / C1, \quad v2'_{\text{local}} = V_{\text{total}} / C2$$

Verification:

• Total momentum remains conserved: p1 + p2 = p1' + p2'
• Intrinsic space ensures ||v'|| maintains energy consistency (see Section 3 of the paper).

2. Advantages Over Traditional Methods

Feature	CCS Approach	Traditional Matrix/Tensor
Geometric Intuition	Directly models translation/rotation/scaling	Requires abstract tensor operations
Computational Efficiency	O(1) for basic operations (e.g., *, /)	O(n³) for matrix multiplications
Conservation Laws	Natural integration with arc length (energy)	Requires separate momentum/energy checks
Hierarchical Systems	Native support for parent-child transforms	Complex chain rule for Jacobians

3. Implementation Guide

3.1 Basic Workflow

1. Define Local Frames: Initialize coord objects for each physical entity with mass-scaled s.
2. Transform Vectors: Use * and / to convert velocities between frames.
3. Compute Momentum: Calculate p = m * v using intrinsic velocities in local frames.
4. Simulate Interactions: Update coordinates during collisions/forces using algebraic operations.

3.2 Example Code Snippet (C++)

#include <vector>
using namespace std;

// Define coordinate system with mass-scaled scaling
coord create_mass_frame(vec3 origin, vec3 mass) {
  coord c;
  c.o = origin;
  c.s = vec3(sqrt(mass.x), sqrt(mass.y), sqrt(mass.z)); // s = √m
  c.ux = vec3(1, 0, 0); // Standard basis vectors
  c.uy = vec3(0, 1, 0);
  c.uz = vec3(0, 0, 1);
  return c;
}

// Calculate intrinsic momentum in world space
vec3 momentum(coord& frame, vec3 local_velocity) {
  vec3 world_velocity = local_velocity * frame; // Local to world transform
  float mass = dot(frame.s, frame.s); // m = s²
  return mass * world_velocity;
}

4. Applications

• Classical Mechanics: Elastic/inelastic collisions, rigid body dynamics.
• Geometric Physics: Riemannian manifold-based motion.
• Computer Graphics: Hierarchical character skeletons, physics simulations.