I am working on some practice questions and just need a bit of help with understanding the last part of the this question and the solutions.
My question is really about the last part, but here is the full question for reference:
A simple pendulum of length $ℓ$ and mass $m$ is suspended from a pivot of mass $M$ that is free to slide on a frictionless wire frame in the shape of a parabola $y = ax^2$ . The pendulum moves in the plane of the frame.
(a) Write down the cartesian coordinates of both masses in terms of $x$ and $θ$.
(b) Calculate the time derivatives of the cartesian coordinates.
(c) Write down the kinetic and potential energies using $x$ and $θ$ as generalised coordinates.
(d) Write down the Lagrangian using the approximation that $x$, $θ$ and their derivatives are small, and solve the corresponding linear Lagrange equations.
I am ok, with parts (a) through (c), and derive the following Lagrangian which is correct:
$$L = \frac{1}{2}M \dot{x}^2(1+4a^2x^2)+\frac{1}{2}m((\dot{x}+l\cos \dot{\theta})^2 + (2ax\dot{x}+l \sin \theta \dot{\theta})^2) - (M+m)gax^2-mgl\cos\theta $$
I am trying to understand the solutions for part (d):
The question says "using the approximation that $x$, $\theta$ and their derivatives are small..."
And the solution says:
The Lagrangian including only the quadratic terms is:
$$L = \frac{1}{2}(M+m)\dot{x}^2+\frac{1}{2}ml^2\dot{\theta}^2 + ml\dot{x}\dot{\theta} - (M+m)gax^2-\frac{1}{2}mgl\theta^2 $$
First of all, I am not sure how this is derived. Secondly, why does this mean take quadratic terms? From reading the question, it seems like I should be taking $x$, $\theta$ to be approximately $0$, but this doesn't give me the above lagrangian.
Also, from the first Lagrangian above, if I expand the terms, and collect quadratic terms, it doesn't seem to match up with the solutions. (For example , where is the $(1+4a^2x^2)$ in the 2nd lagrangian? Why is it excluded when it is attached to a quadratic term?
Thanks for any help you can give me.
This sounds like small oscillation approximations (at least that's what we call it in physics), where you assume any non quadratic terms in $x,\dot{x},\theta,\dot{\theta}$ are $0$.
So, terms like $4a^2x^2\dot{x}^2$, which is quartic, would be thrown out.