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I'm currently studying rigid bodies in Goldstein's Classical Mechanics book but although I understand the theory I'm not yet understanding how to set up the Lagrangian in some situations.

One such situation, as an example, is presented in one of the exercises from the book. In that case, one considers one uniform bar of mass $M$ and of a certain length $L$ suspended from one end by a spring with constant $k$ and considers the bar can only swing in a vertical plane and the spring can only move in the vertical direction.

What confuses me here is the presence of the spring. If it were just the bar, we would have just a continuous rigid body. In that case, to describe the system one would use, as usual the configuration manifold $\mathbb{R}^2\times SO(2)$. In that case, we would need to consider a rotating system fixed on the body.

But now we have a spring together and this confuses me a lot. So going step by step, first I would need to choose generalized coordinates right? In that case, I could just consider the point of the spring fixed to the bar and use the vertical coordinate of that point as generalized coordinate for the spring.

But what about the bar then? I would need to attach a coordinate system to that point in order to describe the motion of the system?

And after that how could I properly set up the Lagrangian for this problem?

I'm a little confused yet on how to deal with this kind of problem where we have not just a rigid body, but a rigid body plus something else.

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The configuration of the system can be specified by two variables, the position of the end of the spring $z$, and the angle of the bar $\theta$. The kinetic energy can be written in terms of the time derivatives of these variables and the potential energy can be written in terms of these variables so you get your lagrangian.

I think the take home message here is that you shouldn't try to associate a coordinate with a single object like you do if you are just using the position of each object as your coordinates in newtonian mechanics. Instead, since your objects are often constrained, it makes sense to just consider the coordinates as describing the system as a whole. I think that is what threw you off.

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