A Can position and velocity vary independently in Hamilton's Principle?

AI Thread Summary
The discussion focuses on the independence of position and velocity in Hamilton's Principle through the calculus of variations. Initially, it appears that position (q) and velocity (q-dot) can be varied independently, leading to the extraction of the Euler-Lagrange equation. However, upon closer examination, it is noted that both variables share the same eta function, suggesting they are not truly independent. The confusion arises from the interpretation of the mathematical framework used in the derivation. Ultimately, the conclusion is that position and velocity cannot vary independently due to their shared dependency in the calculations.
Trying2Learn
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TL;DR Summary
Where, in the mathematical work out, do we use the fact that position and velocity are varying independently?
To carry out the machinery of Hamilton's Principle though the calculus of variations, we desire to vary the position and velocity, independently.

We proceed by varying at action, and set the variation to zero (I will assume ONE generalized variable: q1)

1691333336661.png

In the above, I can see how we vary both q and q-dot independently: it is (if I am not mistaken) in the "machinery" of taking both partials of q and 1-dot). So far, I am fine with that: it initially appears as if position and velocity were independent.

Then we use integration by parts and obtain

1691333462304.png

And we extract the Euler Lagrange equation.

However, if I were to look more closely, I see that this work began with:

1691334513406.png


And if this is the case, I do NOT see how q and q-dot are varying independently, because both have the same eta function in their "heritage."
I can see the "intent" that they vary independently (through the "blind"-machinery of taking the partial with respect to q and q-dot, but ultimately, they are not independent, unless the two red functions were different

1691334976287.png


Could someone advise me?
 

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Trying2Learn said:
TL;DR Summary: Where, in the mathematical work out, do we use the fact that position and velocity are varying independently?

Principle though the calculus of variations, we desire to vary the position and velocity, independently.
we do not vary them independently
 
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wrobel said:
we do not vary them independently
Oh... in your simple response, I reread things and now see I misunderstood what I had read.

thank you
 
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