.. include:: ../references.txt
.. _gala-integrate:
************************************
Integration (`gala.integrate`)
************************************
Introduction
============
Scipy provides numerical ODE integration functions but they aren't very
user friendly or object oriented. This subpackage implements the Leapfrog
integration scheme (not available in Scipy) and provides wrappers to
higher order integration schemes such as a 5th order Runge-Kutta and the
Dormand-Prince 85(3) method.
For code blocks below and any pages linked below, I assume the following
imports have already been executed::
>>> import astropy.units as u
>>> import numpy as np
>>> import gala.integrate as gi
>>> from gala.units import galactic
Getting Started
===============
All of the integrator classes have the same basic call structure. To create an
integrator object, you pass in a function that evaluates derivatives of, for
example, phase-space coordinates, then you call the
`~gala.integrate.Integrator.run` method while specifying timestep information.
The integration function must accept, at minimum, two arguments: the current
time, ``t``, and the current position in phase-space, ``w``. The time is a
single floating-point number and the position will have shape ``(ndim,
norbits)`` where ``ndim`` is the full dimensionality of the phase-space (e.g., 6
for a 3D coordinate system) and ``norbits`` is the number of orbits. These
inputs will *not* have units associated with them (e.g., they are not
:class:`astropy.units.Quantity` objects). An example of such a function (that
represents a simple harmonic oscillator) is::
>>> def F(t, w):
... x, x_dot = w
... return np.array([x_dot, -x])
Even though time does not explicitly enter into the equation, the function must
still accept a time argument. We will now create an instance of
`~gala.integrate.LeapfrogIntegrator` to integrate an orbit::
>>> integrator = gi.LeapfrogIntegrator(F)
To actually run the integrator, we need to specify a set of initial conditions.
The simplest way to do this is to specify an array::
>>> w0 = np.array([1.,0.])
However this then causes the integrator to work without units. The orbit object
returned by the integrator will then also have no associated units. For example,
to integrate from these initial conditions with a time step of 0.5 for 100
steps::
>>> orbit = integrator.run(w0, dt=0.5, n_steps=100)
>>> orbit.t.unit
Unit(dimensionless)
>>> orbit.pos.xyz.unit
Unit(dimensionless)
We can instead specify the unit system that the function (``F``) expects, and
then pass in a `~gala.dynamics.PhaseSpacePosition` object with arbitrary units
in as initial conditions::
>>> import gala.dynamics as gd
>>> from gala.units import UnitSystem
>>> usys = UnitSystem(u.m, u.s, u.kg, u.radian)
>>> integrator = gi.LeapfrogIntegrator(F, func_units=usys)
>>> w0 = gd.PhaseSpacePosition(pos=[100.]*u.cm, vel=[0]*u.cm/u.yr)
>>> orbit = integrator.run(w0, dt=0.5, n_steps=100)
>>> orbit.t.unit
Unit("s")
Now the orbit object will have quantities in the specified unit system.
Example: Forced pendulum
-------------------------
Here we demonstrate how to use the Dormand-Prince integrator to compute the
orbit of a forced pendulum. We will use the variable ``q`` as the angle of the
pendulum with respect to the vertical and ``p`` as the conjugate momentum. Our
Hamiltonian is
.. math::
H(q,p) = \frac{1}{2}p^2 + \cos(q) + A\sin(\omega_D t)
so that
.. math::
\dot{q} &= p\\
\dot{p} &= -\sin(q) + A\omega_D\cos(\omega_D t)
Our derivative function is then::
>>> def F(t, w, A, omega_D):
... q, p = w
... wdot = np.zeros_like(w)
... wdot[0] = p
... wdot[1] = -np.sin(q) + A*omega_D*np.cos(omega_D*t)
... return wdot
This function has two arguments -- :math:`A` (``A``), the amplitude of the
forcing, and :math:`\omega_D` (``omega_D``), the driving frequency. We define an
integrator object by specifying this function, along with values for the
function arguments::
>>> integrator = gi.DOPRI853Integrator(F, func_args=(0.07, 0.75))
To integrate an orbit, we use the `~gala.integrate.Integrator.run` method. We
have to specify the initial conditions along with information about how long to
integrate and with what stepsize. There are several options for how to specify
the time step information. We could pre-generate an array of times and pass that
in, or pass in an initial time, end time, and timestep. Or, we could simply pass
in the number of steps to run for and a timestep. For this example, we will use
the last option. See the API below under *"Other Parameters"* for more
information.::
>>> orbit = integrator.run([3., 0.], dt=0.1, n_steps=10000)
We can plot the integrated (chaotic) orbit::
>>> fig = orbit.plot(subplots_kwargs=dict(figsize=(8,4))) # doctest: +SKIP
.. plot::
:align: center
:context: close-figs
import astropy.units as u
import matplotlib.pyplot as pl
import numpy as np
import gala.integrate as gi
def F(t, w, A, omega_D):
q, p = w
wdot = np.zeros_like(w)
wdot[0] = p
wdot[1] = -np.sin(q) + A*omega_D*np.cos(omega_D*t)
return wdot
integrator = gi.DOPRI853Integrator(F, func_args=(0.07, 0.75))
orbit = integrator.run([3., 0.], dt=0.1, n_steps=10000)
fig = orbit.plot(subplots_kwargs=dict(figsize=(8,4)))
Example: Lorenz equations
-------------------------
Here's another example of integrating the
`Lorenz equations `_, a 3D
nonlinear system::
>>> def F(t, w, sigma, rho, beta):
... x, y, z, *_ = w
... wdot = np.zeros_like(w)
... wdot[0] = sigma * (y - x)
... wdot[1] = x * (rho-z) - y
... wdot[2] = x*y - beta*z
... return wdot
>>> sigma, rho, beta = 10., 28., 8/3.
>>> integrator = gi.DOPRI853Integrator(F, func_args=(sigma, rho, beta))
>>> orbit = integrator.run([0.5, 0.5, 0.5, 0, 0, 0], dt=1E-2, n_steps=1E4)
>>> fig = orbit.plot() # doctest: +SKIP
.. plot::
:align: center
:context: close-figs
def F(t, w, sigma, rho, beta):
x, y, z, *_ = w
wdot = np.zeros_like(w)
wdot[0] = sigma * (y - x)
wdot[1] = x * (rho-z) - y
wdot[2] = x*y - beta*z
return wdot
sigma, rho, beta = 10., 28., 8/3.
integrator = gi.DOPRI853Integrator(F, func_args=(sigma, rho, beta))
orbit = integrator.run([0.5,0.5,0.5,0,0,0], dt=1E-2, n_steps=1E4)
fig = orbit.plot()
API
===
.. automodapi:: gala.integrate