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Finding generalized coordinates for implicit constraints

Given some coordinates $(x_1,\dots,x_N)$ and $h$ holonomic constraints, it should always be possible to reduce the coordinates to $n=N-h$ generalized coordinates $(q_1,\dots,q_n)$.

Let's say I have some constraint $f(x_1, x_2) = 0$. All excercises in my textbook made it possible to find some function $g$ wherefore $x_1 = g(x_2)$, thus effectively reducing the number of coordinates. Yesterday however, I found an example hanging above my kitchen table where this could not be done.

Imagine two pendulums hanging from a fixed ceiling tied together by a rigid string (two-dimensional for simplicity):

example of two pendulums tied together

My initial choice for coordinates are $\alpha_1$, $\alpha_2$, $\beta_1$ and $\beta_2$. Using these coordinates, there is only one constraint: the connecting string, the strings with length $l$ and the ceiling should always form a quadrilateral with given lengths for all sides.

With some uninteresting geometry, one can find a relation between $\alpha_1$ and $\alpha_2$ (you can skip to the answer just above the image):

Call the length of the diagonal connecting $\alpha_2$ with its opposite point $q$ and call the angle between the diagonal and the ceiling $\phi_1$ and the angle between the diagonal and the string with length $l$ $\phi_2$ as illustrated below. By the cosine rule we have \begin{align} q^2 &= l^2 + D^2 - 2lD\cos(\alpha_1) \\ l^2 &= q^2 + D^2 - 2qD\cos(\phi_1) \\ d^2 &= q^2 + l^2 - 2ql\cos(\phi_2)\text{.} \end{align} This already gives the length $q$ as a function of $\alpha_1$. Now, reasonably assuming $\phi_1$ and $\phi_2$ are positive and smaller than $\pi$, we have \begin{align} \phi_1 &= \arccos\left(\frac{q^2+D^2-l^2}{2qD}\right) \\ \phi_2 &= \arccos\left(\frac{q^2+l^2-d^2}{2ql}\right) \end{align} Now, given some $\alpha_1$, this gives two solutions for $\alpha_2$: either $\alpha_2 = \phi_1 + \phi_2$ or $\alpha_2 = \phi_1 - \phi_2$. two possibilities illustrated

Just to show this constraint is indeed holonomic, it can be written as $|\alpha_2 - \phi_1(\alpha_1)| - \phi_2(\alpha_1) = 0$.

This poses a problem: $\alpha_2$ is not completely fixed by $\alpha_1$. I'm not able to reduce the count of coordinates, as eliminating $\alpha_2$ as coordinate gives rise to two possible configurations for a given $(\alpha_1, \beta_1, \beta_2)$.

Questions:

  • Given this kind of constraint (constraints where one coordinate cannot be written explicitely as a function of the others), what is the general approach of transforming dependent coordinates into independent generalized coordinates?
  • How does one proceed to find the Lagrangian of such systems?