A particle of mass m moves frictionlessly under the influence of gravity on a curve defined by:
$x=a(\phi+\sin\phi)$ and $y=a(1-\cos\phi)$.
a) Set up the terms for the kinetic and potential energy.
b)Use a suitable generalized variable $q=f(\phi)$ to turn both terms into a purely quadratic form.
c)Set up the Lagrange-function $L=L(q,\dot{q})$ and derive the equations of motion from that.
We just started talking about generalized variables and Lagrange functions in class and to be honest I don't really understand the concept yet.
I mean I don't even know how to derive the kinetic and potential energy.
My only ideas for the kinetic energy were to get $\dot{x}$ and $\dot{y}$ and then going with $\frac{1}{2}m(\dot{x}^2+\dot{y}^2)$ where as $\dot{x}=a(\dot{\phi}+\dot{\phi}\sin{\phi})$ and $\dot{y}=-a\dot{\phi}\cos{\phi}$, right?
Simplification on KE: $$1/2m(a^2(\dot{\phi}^2+2\dot{\phi}^2\sin(\phi)^2+\dot{\phi}^2\sin(\phi)^2)+a^2\dot{\phi}^2\cos{\phi}^2)=1/2ma^2\dot{\phi}^2\cos(\phi)$$
Following Divergent Queries' comment the potential energy is then just $V=mga(1-\cos(\phi))$? Anyway, I hope someone with more knowledge with this than me could help me.
Your formula for kinetic energy, $E_k = \frac12 m\left(\dot{x}^2+\dot{y}^2\right)$, looks good. Expressing $\dot x$ and $\dot y$ in terms of $\phi$ is a good idea, but note that $\frac{d}{dt}\sin u = \dot u \cos u$, not $\dot u \sin u$, and $\frac{d}{dt}\cos u = -\dot u \sin u$, not $\dot u \cos u.$
You also have an error in your simplification of the kinetic energy (or what would be the kinetic energy if the formulas for $\dot x$ and $\dot y$ were what you wrote).
I find that $\dot{x} = a (\dot\phi + \dot\phi \cos\phi) = a \dot\phi(1 + \cos\phi)$ (factoring out the $\dot\phi$ since it seemed to make the formulas just a little easier to work with) and $\dot{y} = a \dot\phi \sin\phi.$ The kinetic energy then simplifies as shown below. (But you might want to try it yourself, again, with the correct formulas for $\dot x$ and $\dot y$ before you read further.)
$$\begin{eqnarray} E_k &=& \frac12 m\left(a^2 \dot\phi^2 (1 + \cos\phi)^2 + a^2 \dot\phi^2 \sin^2\phi\right) \\ &=& \frac12 m a^2 \dot\phi^2 \left((1 + \cos\phi)^2 + \sin^2\phi\right) \\ &=& \frac12 m a^2 \dot\phi^2 (1 + 2\cos\phi + \cos^2\phi + \sin^2\phi) \\ &=& \frac12 m a^2 \dot\phi^2 (2 + 2\cos\phi) \\ &=& m a^2 \dot\phi^2 (1 + \cos\phi). \\ \end{eqnarray}$$