differential_geometry.symbolic.compute_laplacian#

differential_geometry.symbolic.compute_laplacian(scalar_field: Basic, coordinate_axes: Sequence[Symbol], inverse_metric: MutableDenseMatrix, l_term: ImmutableDenseMatrix | MutableDenseMatrix | ImmutableDenseNDimArray | MutableDenseNDimArray | None = None, metric_density: Basic | None = None) → Basic[source]#

Compute the Laplacian \(\nabla^2 \phi\) of a scalar field in general or orthogonal curvilinear coordinates.

In a general coordinate system, the Laplacian of a scalar field \(\phi\) can be expressed as:

\[\nabla^2 \phi = \frac{1}{\rho} \,\partial_\mu \bigl(\rho \,g^{\mu\nu}\,\partial_\nu \phi\bigr) = L^\nu \,\partial_\nu \phi \;+\; g^{\mu\nu}\,\partial^2_{\mu\nu}\phi,\]

where \(\rho\) is the metric density \(\sqrt{\det g}\) and \(g^{\mu\nu}\) is the inverse metric. The L-term is defined by:

\[L^\nu = \frac{1}{\rho} \,\partial_\mu \bigl(\rho \,g^{\mu\nu}\bigr).\]

Usage:

  • If inverse_metric is a full \((n \times n)\) matrix, a fully general coordinate system is assumed.

  • If inverse_metric is a 1D array of length \(n\), an orthogonal system is assumed (the array contains the diagonal elements \(g^{\mu\mu}\)).

Parameters:

  • scalar_field (Basic) – The scalar field \(\phi\) whose Laplacian is computed.

  • coordinate_axes (list of Symbol) – The coordinate variables \(x^\mu\) with respect to which differentiation occurs.

  • inverse_metric (MutableDenseMatrix or MutableDenseNDimArray) – Either a full inverse metric \(g^{\mu\nu}\) (shape (n, n)) or a 1D diagonal array of shape (n,) for orthogonal coordinates.

  • l_term (MutableDenseNDimArray, optional) – Precomputed L-terms \(L^\nu\). If not provided, these will be derived from metric_density and inverse_metric.

  • metric_density (Basic, optional) – The metric density \(\rho = \sqrt{\det g}\). Required if l_term is not given.

Returns:

The scalar Laplacian \(\nabla^2 \phi\).

Return type:

Basic

See also

compute_Lterm

Compute the L-term from \(\rho\) and \(g^{\mu\nu}\).

compute_metric_density

For computing \(\rho\).

compute_divergence

The divergence in curvilinear coordinates.

Examples

1) Full Inverse Metric (Spherical)

>>> import sympy as sp
>>> from pymetric.differential_geometry.symbolic import compute_laplacian, compute_metric_density
>>>
>>> r, theta, phi = sp.symbols('r theta phi')
>>> scalar = r**2 * sp.sin(theta)
>>>
>>> # Full inverse metric for spherical coordinates
>>> g_inv = sp.Matrix([
...     [1, 0, 0],
...     [0, 1/r**2, 0],
...     [0, 0, 1/(r**2 * sp.sin(theta)**2)]
... ])
>>> # Metric density
>>> rho = compute_metric_density(sp.Matrix([
...     [1,      0, 0],
...     [0, r**2, 0],
...     [0,      0, (r*sp.sin(theta))**2]
... ]))
>>>
>>> # Compute Laplacian
>>> lap = compute_laplacian(scalar, [r, theta, phi], g_inv, metric_density=rho)
>>> lap
4*sin(theta) + 1/sin(theta)

2) Orthogonal Inverse Metric (Diagonal Only)

>>> # Provide just the diagonal of g^{\mu\nu} for an orthogonal system
>>> g_inv_diag = sp.Array([1, 1/r**2, 1/(r**2*sp.sin(theta)**2)])
>>> # Same metric_density as above
>>> lap_ortho = compute_laplacian(scalar, [r, theta, phi], g_inv_diag, metric_density=rho)
>>> lap_ortho
4*sin(theta) + 1/sin(theta)
On this page

This Page