A When KE is a function of position

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The discussion centers on the formulation of the Lagrangian, specifically the kinetic energy (KE) as a function of generalized coordinates (q) and their time derivatives (q-dot). While KE is often presented as a function of q-dot, it can also depend on q itself, as illustrated through the example of a particle in polar coordinates. The kinetic energy is expressed as a quadratic form with coefficients that vary based on the coordinates, emphasizing the relationship between position and velocity. The conversation concludes with the realization that the general form of the Lagrangian aims to accommodate various coordinate systems and their dependencies. This highlights the flexibility and generality of the Lagrangian framework in classical mechanics.
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Question about the Lagrangian
Hi all

In the Lagrangian, we have L = KE - PE

In most cases, I have seen KE as a function of q and q-dot (using the generic symbols).

However I first learned how KE = 0.5 m * v-squared.

Later, I used generalized coordinates and THAT is when KE became a function of q.

I get all that. However, I am still wondering WHY I have the feeling that, in most cases, KE is only a function of the parameter's first time derivative and ONLY involves the parameter in certain cases.

Maybe what I am asking is: "why did they first formulate L(q, q-dot) = KE(q, q-dot) and PE(q)

(I get the P only a function of q, by the way. That is not an issue for me.)
 
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It's perhaps easiest to see through an example; consider a particle moving in ##\mathbf{R}^2##, and let the generalised co-ordinates be plane polar co-ordinates ##q = (r,\theta)##. The kinetic energy is\begin{align*}
T(q, \dot{q}) = \dfrac{1}{2} \dot{r}^2 + \dfrac{1}{2} r^2 \dot{\theta}^2
\end{align*}Not only does ##T## depend on ##\dot{r}## and ##\dot{\theta}##, but it also depends on ##r##. In the general case, the kinetic energy is a quadratic form whose coefficients ##a_{ij}(q)## depend on the co-ordinates,\begin{align*}
T(q,\dot{q}) = \dfrac{1}{2} a_{ij}(q) \dot{q}^i \dot{q}^j
\end{align*}where summation over repeated indices is implicit. For the previous example, ##a_{rr}(q) = 1## and ##a_{\theta \theta}(q) = r^2##, whilst the mixed coefficients are zero.
 
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Oh... yes Now I see. That was obvious.

Then can you say that tried to keep the form as general as possible? Is that it?

Thank you.
 
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