Tutorial: Numerical Geodesics

This tutorial demonstrates in brief how to use the numeric tools in pystein. For other tutorials see all tutorials.

Construct a metric from a twoform

M = Manifold('M', dim=2)
P = Patch('origin', M)

rho, phi, a = sympy.symbols('rho phi a', nonnegative=True)
cs = coords.CoordSystem('schw', P, [rho, phi])
drho, dphi = cs.base_oneforms()
ds2 = a ** 2 * ((1 / (1 - rho ** 2)) * tpow(drho, 2) + rho ** 2 * tpow(dphi, 2))
g = metric.Metric(twoform=ds2)
g

$\displaystyle a^{2} \left(\rho^{2} \operatorname{d}\phi \otimes \operatorname{d}\phi + \frac{\operatorname{d}\rho \otimes \operatorname{d}\rho}{1 - \rho^{2}}\right)$

Compute the symbolic geodesic equations

full_simplify(geodesic.geodesic_equation(0, sympy.symbols('lambda'), g))

$\displaystyle \frac{\left(\rho^{2}{\left(\lambda \right)} - 1\right) \left(\rho^{3}{\left(\lambda \right)} \left(\frac{d}{d \lambda} \phi{\left(\lambda \right)}\right)^{2} - \rho{\left(\lambda \right)} \left(\frac{d}{d \lambda} \phi{\left(\lambda \right)}\right)^{2} + \frac{d^{2}}{d \lambda^{2}} \rho{\left(\lambda \right)}\right) - \rho{\left(\lambda \right)} \left(\frac{d}{d \lambda} \rho{\left(\lambda \right)}\right)^{2}}{\rho^{2}{\left(\lambda \right)} - 1}$

Numerically integrate the geodesic equations

init = (numpy.sin(numpy.pi / 4), 0.0, numpy.cos(numpy.pi / 4), numpy.pi / 4)
lambdas = numpy.arange(0, 2.1, 0.001)
df = geodesic.numerical_geodesic(g, init, lambdas)
df.head()

	rho	phi
0	0.707107	0.000000
1	0.707814	0.000785
2	0.708520	0.001568
3	0.709226	0.002349
4	0.709931	0.003129