# Dynamics (`gala.dynamics`

)¶

For code blocks below and any pages linked below, we’ll assume the following imports have already been executed:

```
>>> import astropy.units as u
>>> import numpy as np
>>> import gala.potential as gp
>>> import gala.dynamics as gd
>>> from gala.units import galactic
```

## Introduction¶

This subpackage contains functions and classes useful for gravitational dynamics. There are utilities for transforming orbits in phase-space to action-angle coordinates, tools for visualizing and computing dynamical quantities from orbits, tools to generate mock stellar streams, and tools useful for nonlinear dynamics such as Lyapunov exponent estimation.

The fundamental objects used by many of the functions and utilities in this and
other subpackages are the `PhaseSpacePosition`

and `Orbit`

classes.

## Getting started: Working with orbits¶

As a demonstration of how to use these objects, we’ll start by integrating an
orbit using the `gala.potential`

and `gala.integrate`

subpackages:

```
>>> pot = gp.MiyamotoNagaiPotential(m=2.5E11*u.Msun, a=6.5*u.kpc,
... b=0.26*u.kpc, units=galactic)
>>> w0 = gd.PhaseSpacePosition(pos=[11., 0., 0.2]*u.kpc,
... vel=[0., 200, 100]*u.km/u.s)
>>> orbit = gp.Hamiltonian(pot).integrate_orbit(w0, dt=1., n_steps=1000)
```

This numerically integrates an orbit from the specified initial conditions,
`w0`

, and returns an `Orbit`

object. By default, the position and velocity are
assumed to be Cartesian coordinates but other coordinate systems are supported
(see the Orbit and phase-space position objects in more detail and N-dimensional representation classes pages for more
information).

The `Orbit`

object that is returned contains many useful methods, and can be
passed to many of the analysis functions implemented in Gala. For example, we
can easily visualize the orbit by plotting the time series in all Cartesian
projections using the `plot()`

method:

```
>>> fig = orbit.plot()
```

(Source code, png)

Or, we can visualize the orbit in just one projection of some transformed coordinate representation, for example, cylindrical radius \(\rho\) and \(z\):

```
>>> fig = orbit.represent_as('cylindrical').plot(['rho', 'z'])
```

(Source code, png)

From the `Orbit`

object, we can also easily compute dynamical quantities such as
the energy or angular momentum (we take the 0th element because these functions
return the quantities computed at every timestep):

```
>>> E = orbit.energy()
>>> E[0]
<Quantity −0.060740198 kpc2 / Myr2>
```

Let’s see how well the integrator conserves energy and the `z`

component of
angular momentum:

```
>>> Lz = orbit.angular_momentum()[2]
>>> np.std(E), np.std(Lz)
(<Quantity 4.654233175716351e-06 kpc2 / Myr2>,
<Quantity 9.675900603446092e-16 kpc2 / Myr>)
```

We can access the position and velocity components of the orbit separately using
attributes that map to the underlying `BaseRepresentation`

and `BaseDifferential`

subclass instances that store the
position and velocity data. The attribute names depend on the representation.
For example, for a Cartesian representation, the position components are ```
['x',
'y', 'z']
```

and the velocity components are `['v_x', 'v_y', 'v_z']`

. With a
`Orbit`

or `PhaseSpacePosition`

instance, you can check the valid compnent names using the
attributes `.pos_components`

and `.vel_components`

:

```
>>> orbit.pos_components.keys()
odict_keys(['x', 'y', 'z'])
>>> orbit.vel_components.keys()
odict_keys(['v_x', 'v_y', 'v_z'])
```

Meaning, we can access these components by doing, e.g.:

```
>>> orbit.v_x
<Quantity [ 0. ,-0.00567589,-0.01129934,..., 0.18751756,
0.18286687, 0.17812762] kpc / Myr>
```

For a Cylindrical representation, these are instead:

```
>>> cyl_orbit = orbit.represent_as('cylindrical')
>>> cyl_orbit.pos_components.keys()
odict_keys(['rho', 'phi', 'z'])
>>> cyl_orbit.vel_components.keys()
odict_keys(['v_rho', 'pm_phi', 'v_z'])
>>> cyl_orbit.v_rho
<Quantity [ 0. ,-0.00187214,-0.00369183,..., 0.01699321,
0.01930216, 0.02159477] kpc / Myr>
```

Continue to the Orbit and phase-space position objects in more detail page for more information.

## Using gala.dynamics¶

More details are provided in the linked pages below: