velocity.md

Velocity

Circular trajectory

Let's define a circular trajectory:

import BrownTrack as BT

traj = BT.trajectory( X = ( 1, 0 ) )

from pylab import *

for t in linspace(0,1,15)[1:]**2 :
    point = exp(1j*t*2*pi)
    # plot( *point, 'o' )
    traj.addPoint( ( real(point), imag(point) ) )

We can plot the result:

plot( traj.x, traj.y )

This is the blue line in the figure below.

Velocity

To calculate the velocity of the above trajectory, we simply write:

traj_u = traj.diff()

The result is the orange line in the figure below. Due to the discrete differentiation, there is one less data point than in the original trajectory.

Midpoints

The discrete differentiation of a trajectory generate midpoint values. The trajectory can be mapped onto these midpoints:

traj = traj.diff_companion()

The result is the orange line on the first figure above.

Angular momentum

It is now straightforward to calculate the angular momentum of the trajectory:

angular_momentum = traj.x*traj_u.y - traj.y*traj_u.x
plot( traj.get_time_list(), angular_momentum, 'm.-')