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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):

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 $$\alpha_2 = \phi_1 - \phi_2$$.

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?