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I'm working through The Variational Principles of Mechanics by Cornelius Lanczos and in chapter 6, section 10, he says that the Lagrangian function is homogeneous in the velocities.

$L_1 = L(q_1,...,q_{n+1}; \frac{q_1'}{q'_{n+1}},...,\frac{q_n'}{q'_{n+1}})q'_{n+1}$

The indices go to $n+1$ because he's putting the Lagrangian into parametric form where time becomes a variable $q_{n+1}$ and the variables are all functions of an arbitrary parameter. That's also why you see the prime notation. It represents differentiation relative to this parameter.

He says that since the function is homogeneous in the velocities, it can be represented as:

$L_1 = \Sigma \frac{\partial L_1}{\partial q_i'} q_i'$

I don't see how this works because the Lagrangian could include potential energy terms that are a function only of position, not velocity, so it seems like those would be lost if you represent it this way.

I've found one other source ("On the Lagrangian being a homogeneous function of the velocity" by Gaetano Giaquinta) that also says that the Lagrangian can be represented this way, but isn't clear on how. Can someone explain this to me?

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