# 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

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

### 2. Shift Operator

```python
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.