# HarmonicOscillatorPotential¶

class gala.potential.potential.HarmonicOscillatorPotential(omega, *, units=None, origin=None, R=None, **kwargs)[source]

Represents an N-dimensional harmonic oscillator.

$\Phi = \frac{1}{2}\omega^2 x^2$
Parameters
omeganumeric

Frequency.

unitsiterable(optional)

Unique list of non-reducable units that specify (at minimum) the length, mass, time, and angle units.

Attributes Summary

Methods Summary

 Call self as a function. acceleration(q[, t]) Compute the acceleration due to the potential at the given position(s). Transform the input cartesian position and velocity to action-angle coordinates the Harmonic Oscillator potential. circular_velocity(q[, t]) Estimate the circular velocity at the given position assuming the potential is spherical. density(q[, t]) Compute the density value at the given position(s). energy(q[, t]) Compute the potential energy at the given position(s). gradient(q[, t]) Compute the gradient of the potential at the given position(s). hessian(q[, t]) Compute the Hessian of the potential at the given position(s). integrate_orbit(*args, **kwargs) Integrate an orbit in the current potential using the integrator class provided. mass_enclosed(q[, t]) Estimate the mass enclosed within the given position by assuming the potential is spherical. plot_contours(grid[, filled, ax, labels, ...]) Plot equipotentials contours. plot_density_contours(grid[, filled, ax, ...]) Plot density contours. replace_units(units[, copy]) Change the unit system of this potential. Save the potential to a text file. to_galpy_potential([ro, vo]) Convert a Gala potential to a Galpy potential instance Return a string LaTeX representation of this potential to_sympy(v, p) Return a representation of this potential class as a sympy expression total_energy(x, v) Compute the total energy (per unit mass) of a point in phase-space in this potential. value(*args, **kwargs)

Attributes Documentation

ndim = 3
omega = <PotentialParameter: omega [frequency]>
units

Methods Documentation

__call__(q)

Call self as a function.

acceleration(q, t=0.0)

Compute the acceleration due to the potential at the given position(s).

Parameters
q

Position to compute the acceleration at.

Returns
accQuantity

The acceleration. Will have the same shape as the input position array, q.

action_angle(w)[source]

Transform the input cartesian position and velocity to action-angle coordinates the Harmonic Oscillator potential. This transformation is analytic and can be used as a “toy potential” in the Sanders & Binney 2014 formalism for computing action-angle coordinates in _any_ potential.

Adapted from Jason Sanders’ code genfunc.

Parameters
w

The positions or orbit to compute the actions, angles, and frequencies at.

circular_velocity(q, t=0.0)

Estimate the circular velocity at the given position assuming the potential is spherical.

Parameters
qarray_like, numeric

Position(s) to estimate the circular velocity.

Returns
vcircQuantity

Circular velocity at the given position(s). If the input position has shape q.shape, the output energy will have shape q.shape[1:].

density(q, t=0.0)

Compute the density value at the given position(s).

Parameters
q

The position to compute the value of the potential. If the input position object has no units (i.e. is an ndarray), it is assumed to be in the same unit system as the potential.

Returns
densQuantity

The potential energy or value of the potential. If the input position has shape q.shape, the output energy will have shape q.shape[1:].

energy(q, t=0.0)

Compute the potential energy at the given position(s).

Parameters
q

The position to compute the value of the potential. If the input position object has no units (i.e. is an ndarray), it is assumed to be in the same unit system as the potential.

Returns
EQuantity

The potential energy per unit mass or value of the potential.

Compute the gradient of the potential at the given position(s).

Parameters
q

The position to compute the value of the potential. If the input position object has no units (i.e. is an ndarray), it is assumed to be in the same unit system as the potential.

Returns
gradQuantity

The gradient of the potential. Will have the same shape as the input position.

hessian(q, t=0.0)

Compute the Hessian of the potential at the given position(s).

Parameters
q

The position to compute the value of the potential. If the input position object has no units (i.e. is an ndarray), it is assumed to be in the same unit system as the potential.

Returns
hessQuantity

The Hessian matrix of second derivatives of the potential. If the input position has shape q.shape, the output energy will have shape (q.shape[0],q.shape[0]) + q.shape[1:]. That is, an n_dim by n_dim array (matrix) for each position.

integrate_orbit(*args, **kwargs)

Integrate an orbit in the current potential using the integrator class provided. Uses same time specification as Integrator.run() – see the documentation for gala.integrate for more information.

Parameters
w0

Initial conditions.

IntegratorIntegrator (optional)

Integrator class to use.

Integrator_kwargsdict (optional)

Any extra keyword argumets to pass to the integrator class when initializing. Only works in non-Cython mode.

cython_if_possiblebool (optional)

If there is a Cython version of the integrator implemented, and the potential object has a C instance, using Cython will be much faster.

**time_spec

Specification of how long to integrate. See documentation for parse_time_specification.

Returns
orbitOrbit
mass_enclosed(q, t=0.0)

Estimate the mass enclosed within the given position by assuming the potential is spherical.

Parameters
q

Position(s) to estimate the enclossed mass.

Returns
mencQuantity

Mass enclosed at the given position(s). If the input position has shape q.shape, the output energy will have shape q.shape[1:].

plot_contours(grid, filled=True, ax=None, labels=None, subplots_kw={}, **kwargs)

Plot equipotentials contours. Computes the potential energy on a grid (specified by the array grid).

Warning

Right now the grid input must be arrays and must already be in the unit system of the potential. Quantity support is coming…

Parameters
gridtuple

Coordinate grids or slice value for each dimension. Should be a tuple of 1D arrays or numbers.

filledbool (optional)

Use contourf() instead of contour(). Default is True.

axmatplotlib.Axes (optional)
labelsiterable (optional)

List of axis labels.

subplots_kwdict

kwargs passed to matplotlib’s subplots() function if an axes object is not specified.

kwargsdict

kwargs passed to either contourf() or plot().

Returns
figFigure
plot_density_contours(grid, filled=True, ax=None, labels=None, subplots_kw={}, **kwargs)

Plot density contours. Computes the density on a grid (specified by the array grid).

Warning

For now, the grid input must be arrays and must already be in the unit system of the potential. Quantity support is coming…

Parameters
gridtuple

Coordinate grids or slice value for each dimension. Should be a tuple of 1D arrays or numbers.

filledbool (optional)

Use contourf() instead of contour(). Default is True.

axmatplotlib.Axes (optional)
labelsiterable (optional)

List of axis labels.

subplots_kwdict

kwargs passed to matplotlib’s subplots() function if an axes object is not specified.

kwargsdict

kwargs passed to either contourf() or plot().

Returns
figFigure
replace_units(units, copy=True)

Change the unit system of this potential.

Parameters
unitsUnitSystem

Set of non-reducable units that specify (at minimum) the length, mass, time, and angle units.

copybool (optional)

If True, returns a copy, if False, changes this object.

save(f)

Save the potential to a text file. See save() for more information.

Parameters
fstr, file_like

A filename or file-like object to write the input potential object to.

to_galpy_potential(ro=None, vo=None)

Convert a Gala potential to a Galpy potential instance

Parameters
roquantity-like (optional)
voquantity-like (optional)
classmethod to_latex()

Return a string LaTeX representation of this potential

Returns
latex_strstr

The latex expression as a Python string.

classmethod to_sympy(v, p)[source]

Return a representation of this potential class as a sympy expression

Returns
exprsympy expression
varsdict

A dictionary of sympy symbols used in the expression.

total_energy(x, v)

Compute the total energy (per unit mass) of a point in phase-space in this potential. Assumes the last axis of the input position / velocity is the dimension axis, e.g., for 100 points in 3-space, the arrays should have shape (100, 3).

Parameters
xarray_like, numeric

Position.

varray_like, numeric

Velocity.

value(*args, **kwargs)