cognitive/knowledge_base/mathematics/generalized_coordinates.md

Generalized Coordinates in Active Inference

Overview

Generalized coordinates are a fundamental concept in continuous-time active inference that allows for a richer representation of dynamical systems by explicitly incorporating higher-order temporal derivatives into the state space.

Mathematical Foundation

Basic Definition

A state x in generalized coordinates is represented as a vector of temporal derivatives:

x̃ = [x, x', x'', ..., x^(n)]

where:

• x is the state value
• x' is the first temporal derivative (velocity)
• x'' is the second temporal derivative (acceleration)
• etc.

Shift Operator

The shift operator D maps between orders of motion:

D[x, x', x''] = [x', x'', 0]

With factorial scaling for Taylor series:

D[x, x', x''] = [1!x', 2!x'', 0]

Role in Active Inference

1. Belief Representation

Beliefs about states are represented in generalized coordinates:

q(x̃) = N(μ̃, Σ̃)

where:

• μ̃ is the vector of means across orders
• Σ̃ is the precision (inverse covariance) matrix

2. Dynamics

The generalized motion of states follows:

dx̃/dt = Dx̃ - ∂F/∂x̃

where:

• D is the shift operator
• F is the variational free energy
• ∂F/∂x̃ are the gradients in generalized coordinates

3. Prediction

Predictions in generalized coordinates allow for:

• Smooth trajectories
• Velocity matching
• Acceleration matching
• Higher-order consistency

Implementation Details

1. State Representation

class ContinuousState:
    belief_means: np.ndarray      # Shape: [n_states, n_orders]
    belief_precisions: np.ndarray # Shape: [n_states, n_orders]

2. Shift Operator

def create_shift_operator(n_orders):
    D = np.zeros((n_orders, n_orders))
    for i in range(n_orders - 1):
        D[i, i+1] = factorial(i+1) / factorial(i)
    return D

3. Free Energy

The free energy includes terms for all orders:

F = Σᵢ (prediction_errorᵢ)²/2σᵢ²

where i runs over all orders of motion.

Advantages

1. Smooth Dynamics: Natural handling of continuous trajectories
2. Rich Predictions: Incorporation of velocity and acceleration
3. Temporal Consistency: Enforced across multiple orders
4. Uncertainty Propagation: Through all orders of motion

Visualization

1. State Space: Plot of position vs. velocity
2. Generalized Coordinates: Multiple plots for each order
3. Prediction Errors: Across all orders of motion
4. Taylor Expansions: Showing predictive power

References

1. Friston, K. J., et al. (2008). DEM: A variational treatment of dynamic systems.
2. Buckley, C. L., et al. (2017). The free energy principle for action and perception: A mathematical review.
3. Baltieri, M., & Buckley, C. L. (2019). Generalized synchronization through learning in coupled dynamical systems.