pycharge.functional#

Core utility functions used in PyCharge.

Functions#

acceleration(t, charge)

Compute charge acceleration \(\mathbf{a}(t) = d^2\mathbf{r}/dt^2\) via automatic differentiation.

emission_time(r, t, charge)

Compute emission time (retarded time) \(t_r\) of a charge from an observation point.

interpolate_position(ts, position_array, ...[, t_end])

Create position interpolation function from trajectory data.

position(t, charge)

Compute charge position at time t.

velocity(t, charge)

Compute charge velocity \(\mathbf{v}(t) = d\mathbf{r}/dt\) via automatic differentiation.

Package Contents#

pycharge.functional.acceleration(t, charge)#

Compute charge acceleration \(\mathbf{a}(t) = d^2\mathbf{r}/dt^2\) via automatic differentiation.

Parameters:

  • t (ArrayLike) – Time (scalar or array).

  • charge (Charge) – Charge object.

Returns:

Acceleration \(\mathbf{a}(t) = [a_x, a_y, a_z]\).

Return type:

Array

pycharge.functional.emission_time(r, t, charge)#

Compute emission time (retarded time) \(t_r\) of a charge from an observation point.

Solves \(t_r = t - \frac{1}{c}\,|\mathbf{r} - \mathbf{r}_s(t_r)|\) at observation point \((\mathbf{r}, t)\) using fixed-point iteration followed by Newton’s method.

Parameters:

  • r (Array) – Observation point \(\mathbf{r} = [x, y, z]\).

  • t (Array) – Observation time \(t\).

  • charge (Charge) – Charge with position function and solver config.

Returns:

\(t_r\).

Return type:

Array

Note

The solver parameters (rtol, atol, max_steps, throw) are configured via charge.solver_config.

pycharge.functional.interpolate_position(ts, position_array, velocity_array, position_0_fn, t_end=None)#

Create position interpolation function from trajectory data.

Uses cubic Hermite interpolation for \(C^1\) continuity (continuous position and velocity). The interpolated position \(\mathbf{r}(t)\) for \(t_i \leq t \leq t_{i+1}\) is:

\[\mathbf{r}(t) = a\tau^3 + b\tau^2 + c\tau + d\]

where \(\tau = (t - t_i)/(t_{i+1} - t_i)\) and coefficients ensure matching positions and velocities at interval endpoints.

Parameters:

  • ts (Array) – Time points, shape (n_steps,).

  • position_array (Array) – Positions at each time, shape (n_steps, 3).

  • velocity_array (Array) – Velocities at each time, shape (n_steps, 3).

  • position_0_fn (Callable[[Scalar], Vector3]) – Original position function before simulation start.

  • t_end (Array or None) – End time for interpolation. Default: if None, uses ts[-1].

Returns:

Function returning position at time t.

For \(t \leq t_0\), returns position_0_fn(t). For \(t \geq t_{\mathrm{end}}\), returns final position. Otherwise, cubic Hermite interpolation.

Return type:

Callable[[Scalar], Array]

pycharge.functional.position(t, charge)#

Compute charge position at time t.

Parameters:

  • t (ArrayLike) – Time (scalar or array).

  • charge (Charge) – Charge object.

Returns:

Position \(\mathbf{r}(t) = [x, y, z]\).

Return type:

Array

pycharge.functional.velocity(t, charge)#

Compute charge velocity \(\mathbf{v}(t) = d\mathbf{r}/dt\) via automatic differentiation.

Parameters:

  • t (ArrayLike) – Time (scalar or array).

  • charge (Charge) – Charge object.

Returns:

Velocity \(\mathbf{v}(t) = [v_x, v_y, v_z]\).

Return type:

Array