Given the metric, find the geodesic equation

[whatisreality]

Homework Statement

Given that $ds^2 = r^2 d\theta ^2 + dr^2$ find the geodesic equations.

Homework Equations

The Attempt at a Solution

I think the $g_{\mu\nu} = \left( \begin{array}{ccc} 1& 0 \\ 0 & r^2 \end{array} \right)$
Then I tried to use the equation
$\tau = \int_{t_1}^{t_2} \sqrt{ g_{\mu\nu}(x(t)) \frac{ dx^{\mu}}{dt}\frac{dx^{\nu}}{dt} } dt$

Which if I expand the sum gives

$\tau = \int_{t_1}^{t_2} \sqrt{ 1 + \left(\frac{ dr}{dt}\right)^2 +r^2 \left(\frac{d\theta}{dt}\right)^2 }$ $dt$

Unfortunately I don't know anything about Lagrangians, which seems to be the normal way to proceed... so I get a bit stuck here. There is an example without Langrangians in the lecture notes but I don't understand what he did to get from here to his $d\tau$. We've had two lectures on GR so far and I think I've already missed something massively important, that explains how to do this!

Thank you for any help, I really appreciate it.

 

[Orodruin]

$\tau = \int_{t_1}^{t_2} \sqrt{ g_{\mu\nu}(x(t)) \frac{ dx^{\mu}}{dt}\frac{dx^{\nu}}{dt} } dt$

Which if I expand the sum gives

$\tau = \int_{t_1}^{t_2} \sqrt{ 1 + \left(\frac{ dr}{dt}\right)^2 +r^2 \left(\frac{d\theta}{dt}\right)^2 }$ $dt$

The one in the square root should not be there. The metric only has two non-zero components and $t$ is a curve parameter, not a coordinate.

What you want to read up on is variational calculus. It will tell you the conditions for a curve to find the extreme values of integrals such as
$$
S = \int_a^b L(x,\dot x,\tau) \, d\tau.
$$
The condition that must be satisfied by $L$ is a differential equation on the form
$$
\frac{\partial L}{\partial x} - \frac{d}{d\tau} \frac{\partial L}{\partial \dot x} = 0,
$$
known as the Euler-Lagrange equation. For its derivation, you should look up any basic textbook containing variational calculus.