Friday, August 21, 2015

Lagrangian Mechanics is Awesome (Double Pendulum, Part 2)

In this post, we give some calculational details of the double pendulum (introduced in Part 1). This derivation is available in several physics books at the undergraduate (upper division) level, usually fleshed out using Lagrangian mechanics, another relevant subject that is highly useful to conceptualizing these types of problems by putting them in a setting using notions of state spaces. More cool-looking demos (and also a streamlined derivation) are available in David Eberly's excellent book, Game Physics. As a warning, this will be considerably more technical than visual; I post this because I feel that some insight into the workings of these things helps us see where geometry can be useful in contexts that are not just literally pictures. Nevertheless, here's the picture for the upcoming analysis:

$\ell_i$ are the length of the pendulum rods, $m_i$ are the masses, and $\theta_i$ are their angular displacements from vertical.
The use of Lagrangian mechanics spurs on the development of thinking in terms of state spaces by considering different kinds of coordinates that may be more convenient for the problem at hand. It is easier to think about the angular positions $\theta_1$ and $\theta_2$ of a pendulum than it is to derive it using x- and y-coordinates directly (and for our problem, physical constraints in the x- and y-coordinates, here made by assuming that the rods are rigid, make it somewhat less straightforward to deal with). Of course, we might use the x- and y-coordinates in parts of our derivation, but it's not so easy to solve for them.

Lagrangian mechanics also has us think of things in terms of energy, a quantity whose properties crop up in math a lot. Our goal in this post is to derive the Lagrangian L of the system. is the difference between total kinetic energy and total potential energy V. The total kinetic energy of our system should be the sum of that of the two particles,
\[
T = \frac 1 2 m_1 v_1^2 + \frac 1 2 m_2 v_2^2,
\]
and the potential energy (assuming that our pendulums are subject to downward acceleration g) depends only on the height of the two pendulum bobs:
\[
V = m_1 g y_1 + m_2 g y_2.
\]
However, if we want to express this in terms of the angular coordinates, we find, via the usual trigonometric arguments, that $y_1 = -\ell_1\cos \theta_1$ and since $y_2$ includes the height of the first pendulum, $y_2 = -\ell_1 \cos\theta_1 - \ell_2 \cos \theta_2$. In total, this means
\[
V = -(m_1+m_2) g \ell_1 \cos \theta_1 - m_2 g \ell_2 \cos \theta_2.
\]
Now the kinetic energy is a little trickier. Here, we assume that the first pendulum bob has linear velocity $\mathbf{v}_1$, and speed $|\mathbf{v}_1| = v_1$. Now, if we let the second pendulum bob have linear velocity $\mathbf{v}_{1,2}$ relative to the first, then the total velocity of the second bob is $\mathbf{v}_2 = \mathbf{v}_1 + \mathbf{v}_{1,2}$. The reason that $\mathbf{v}_{1,2}$ is useful is that it is calculated in exactly the same way as the first pendulum (because the second pendulum is, well, a pendulum).

So the total (squared) speed of the second pendulum bob is $v_2^2= v_1^2 + 2 \mathbf{v}_1 \cdot \mathbf{v}_{1,2} + v_{1,2}^2$.

Now, to figure out $\mathbf{v}_1 \cdot \mathbf{v}_{1,2}$, we use the geometric (length and angle) formulation of the dot product:
\[
\mathbf{v}_1 \cdot \mathbf{v}_{1,2} = v_1 v_{1,2} \cos \varphi
\]
where $\varphi$ is the angle between $\mathbf{v}_1$ and $\mathbf{v}_{1,2}$. But the angle between $\mathbf{v}_1$ and $\mathbf{v}_{1,2}$, since both of them are perpendicular to their respective rods, must be the same as the angle between the rods: $\varphi=\theta_2 - \theta_1$. Finally, to express the magnitudes $v_1$ and $v_{1,2}$, we use the relations $v_1 = \dot{\theta}_1 \ell_1$ and $v_{1,2} = \dot{\theta}_2 \ell_2$. This gives
\[
v_2^2 = \ell_1^2 \dot{\theta}_1^2 + 2 \dot{\theta}_1 \dot{\theta}_2\ell_1 \ell_2 \cos(\theta_2 - \theta_1) + \ell_2^2 \dot{\theta}_2^2
\]

This finally gives
\[
T =\frac 1 2 (m_1+m_2) \ell_1^2 \dot{\theta}_1^2 +\frac 1 2 m_2\left( 2 \dot{\theta}_1 \dot{\theta}_2\ell_1 \ell_2 \cos(\theta_2 - \theta_1) + \ell_2^2 \dot{\theta}_2^2\right),
\]
or,
\[
L = T-V = \frac 1 2 (m_1+m_2) \ell_1^2 \dot{\theta}_1^2 +\frac 1 2 m_2\left( 2 \dot{\theta}_1 \dot{\theta}_2\ell_1 \ell_2 \cos(\theta_2 - \theta_1) + \ell_2^2 \dot{\theta}_2^2\right)\]\[+ (m_1+m_2) g \ell_1 \cos \theta_1 + m_2 g \ell_2 \cos \theta_2.
\]
(If you're reading this on a mobile device, please turn it to Landscape mode to see the full equations.) Actually solving these equations for $\theta_1$ and $\theta_2$ will have to wait for another entry!

As a final note, kudos to MathJax which allows rendering of math formulas using $\LaTeX$!

1 comment:

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