Hamiltonian objects and reference frames

For code blocks below, I assume the following imports have already been excuted:

>>> import astropy.units as u
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import gala.potential as gp
>>> import gala.dynamics as gd
>>> import gala.integrate as gi
>>> from gala.units import galactic


When integrating orbits using the potential classes directly, for example:

>>> pot = gp.HernquistPotential(m=1E10*u.Msun, c=1.*u.kpc,
...                             units=galactic)
>>> w0 = gd.PhaseSpacePosition(pos=[5.,0,0]*u.kpc,
...                            vel=[0,0,50.]*u.km/u.s)
>>> orbit = gp.Hamiltonian(pot).integrate_orbit(w0, dt=0.5, n_steps=1000)

it is implicitly assumed that the initial conditions and orbit are in an intertial (static) reference frame. In this case, the total energy or value of the Hamiltonian (per unit mass) is simply

\[H(\boldsymbol{x}, \boldsymbol{v}) = E = \frac{1}{2}\,|\boldsymbol{v}|^2 + \Phi(\boldsymbol{x})\]

It is sometimes useful to transform to alternate, non-inertial reference frames to do the numerical orbit integration. In this case, the _effective_ Hamiltonian may include other terms. For example, in the case of a rotating reference frame constantly rotating with frequency vector \(\boldsymbol{\Omega}\), the effective potential can be written

\[\Phi_{\rm eff}(\boldsymbol{x}, \boldsymbol{v}) = \Phi(\boldsymbol{x}) - \boldsymbol{\Omega} \cdot \boldsymbol{L}\]

where \(\boldsymbol{L}\) is the angular momentum. For working in non-inertial reference frames, Gala provides a way to compose potential objects (which define just the static component of the effective potential) with reference frame objects into a Hamiltonian object, which can then be used for orbit integration, evaluating the full symplectic gradient of the effective Hamiltonian, and computing the value (pseudo-energy) of the effective Hamiltonian.

Creating a Hamiltonian object with a specified reference frame

Using the potential objects and integrate_orbit() to integrate an orbit is equivalent to defining a Hamiltonian object with the potential object and a Staticframe instance:

>>> pot = gp.HernquistPotential(m=1E10*u.Msun, c=1.*u.kpc,
...                             units=galactic)
>>> frame = gp.StaticFrame(units=galactic)
>>> H = gp.Hamiltonian(potential=pot, frame=frame)
>>> w0 = gd.PhaseSpacePosition(pos=[5.,0,0]*u.kpc,
...                            vel=[0,0,50.]*u.km/u.s)
>>> orbit = H.integrate_orbit(w0, dt=0.5, n_steps=1000)

In this case, the orbit object returned from integration knows what reference frame it is in and we can therefore transform it to other reference frames. For example, we can change to a constantly rotating frame with a frequency vector that determines the axis of rotation and angular velocity of rotation around that axis:

>>> rotation_axis = np.array([8.2, -1.44, 3.25])
>>> rotation_axis /= np.linalg.norm(rotation_axis) # make a unit vector
>>> frame_freq = 42. * u.km/u.s/u.kpc
>>> rot_frame = gp.ConstantRotatingFrame(Omega=frame_freq * rotation_axis,
...                                      units=galactic)
>>> orbit_to_rot = orbit.to_frame(rot_frame)
>>> fig1 = orbit.plot(marker='') 
>>> fig1.suptitle("Static frame") 
>>> fig2 = rot_orbit.plot(marker='') 
>>> fig2.suptitle("Rotating frame") 

(Source code)

We can also integrate the orbit in the rotating frame directly by creating a Hamiltonian object with the rotating frame:

>>> H_rot = gp.Hamiltonian(potential=pot, frame=rot_frame)
>>> rot_orbit = H_rot.integrate_orbit(w0, dt=0.5, n_steps=1000)
>>> _ = rot_orbit.plot(marker='') 

(Source code, png, hires.png, pdf)


In this case, because the potential is spherical, the orbit should look the same whether we integrate it in the rotating frame or in a static frame and then transform to a rotating frame. In the example below, we consider the case of integrating orbits in an asymmetric, time-dependent bar potential.

See the Integrating an orbit in a rotating reference frame example for more information.