Generalized Coordinates in the Lagrangian Formalism — What and How to Choose

Im having trouble figuring which generalized coordinates to choose; and even after having written down the Lagrangian of a system, which coordinates do I choose to write down the relevant Euler-Lagrange equation for the problem.

For example, Taylor's textbook Classical Mechanics has the following problem (7.33):

A bar of soap (mass $m$) is at rest on a frictionless rectangular plate that rests on a horizontal table. At time $t = 0$, I start raising one edge of the plate so that the plate pivots about the opposite edge with constant angular velocity $\omega$, and the soap starts to slide toward the downhill edge. Show that the equation of motion for the soap has the form $x - \omega^2 = -g \sin (\omega t)$, where $x$ is the soap's distance from the downhill edge. Solve this for $x (t)$, given that $x (0) = x_o$.

I am able to write down the Lagrangian for this problem. I set a 3-D co-ordinate system at the pivot of the plate and assume that the plate has negligible thickness (in the $y$ direction). The plate rotates about the $y$ axis. As it rotates, the $x$- component decreases in magnitude, while the $z$ component increases in magnitude. I hope I am able to explain how I set up the problem. The Lagrangian is:

$$\mathcal{L} = \frac{m}{2} \Big (\dot{x}^2 + x^2 \omega^2\Big ) - mgx \sin(\omega t).$$

Effectively, the motion of the particle can be desribed in the $x-z$ plane. Clearly by the geometry, $x$ plays the role of the radial vector and $\theta$ plays the role of the angular vector (in polar coordinates)?

So, why don't we have -- or asked -- to derive to Euler Lagrange equations? Is it because $\dot{\theta} = \omega$ is constant? I don't get: if there are two polar coordinate like variables as our generalized coordinates, should we have two equations?

I am having a hard time figuring out which generalized coordinates to use; with respect to which should I write the E-L equations and so on and so forth.

Taylor's textbook just brushes these ideas. Any recommendations where I can brush up on my concepts. I know of the following undergraduate level classical mechanics books: those by Marion and Thorton, Gregory, Tai-Chow and Morin (it cover Lagrangian Mechanics). Recommendations for the kinds of problems I am facing?

• If you've written the Lagrangian you've chosen the generalised coordinates already. Also, you can write the E-L equations for all terms in the Lagrangian, but only those associated to your generalised coordinates will be relevant - the others will give you $\text{something}=0$. – QuantumBrick Oct 5 '16 at 19:09
• @QuantumBrick Aren't there two generalize coordinates in my lagrangian. In analogy with the polar coordinates, x which plays the role of the radial vector and theta which plays the role of the angular vector, and we have theta dot is zero. If so, why does the question simply ask us to write the EL equations with respect to x, and not theta. How is the soap's/system's dynamics pinned down by 1 equation? Shouldn't we also take into account the second one? – Junaid Aftab Oct 5 '16 at 19:17
• $\omega$ is not a generalised coordinate. It is constant. You cannot have two meaningful generalised coordinates in this problem because it is one dimensional. Thus there is a single E-L equation for your only variable: $x$. – QuantumBrick Oct 5 '16 at 19:21
• Knowing which possible generalized coordinates will give you a simple (or at least tractable) result is largely a matter of experience. Start by working enough problem to be comfortable apply the E-L equation to solve problems. Then work a number of problems several times with different choices of generalized coordinates each time. – dmckee Oct 5 '16 at 19:25
• @QuantumBrick How is it one dimensional? Isn't the particle executing motion in the 2-D x-z plane? As time elapses, it's vertical height increases because I'm lifting it up and it's horizontal distance decreases. Okay, yeah, I get that the there's a constraint linking x and z. But in the lagrangian, can't I simply write phi/theta instead of omega*t and then say, look both x and phi are varying with time. So aren't there two generalized coordinates? What am I confusing? – Junaid Aftab Oct 5 '16 at 19:39