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In learning about Lagrange's Equations, I had always used generalized coordinates that are independent from each other.

However, in this post it was mentioned that generalized coordinates can be dependent on each other.

My question is, for generalized coordinates that are dependent on each other, does Lagrange's equations still hold for these generalized coordinates? The derivation of Lagrange Equations I learned specifically required that the generalized coordinates are independent.

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Yes, if the coordinates are related via holonomic or semi-holonomic constraints, it is still possible to modify the Lagrange equations accordingly by introducing Lagrange multipliers, see e.g. my Phys.SE answer here.

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Generalized coordinates can be dependent on one another. One example is provided by holonomic constraints, that is constraining your system to a manifold. For instance, you can define a submanifold embedded in $\mathbb{R}^n$ specifying a level set of a given function $F$: \begin{equation} M = \{(q_1, \dots, q_n) \in \mathbb{R}^n \mid F(q_1, \dots, q_n) = 0\} \end{equation} In this case, the generalized coordinates are not independent of each other. However, you can always find a set of coordinates such that $n-1$ of them are independent, while the last coordinate's value is fixed (e.g. for a point constrained on a sphere of radius $R$ in spherical coordinates you have that $r(t) = R \ \forall t \in \mathbb{R}$.). For the first $n-1$ coordinates the Lagrange equations hold.

P.S. If you have multiple holonomic constraints, then the system will move on the intersection of the level sets.

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