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

In summary, the conversation discusses the use of Hamilton's Principle and the calculus of variations in varying the position and velocity independently. It is noted that although the initial approach appears to show independent variation, it is ultimately not the case due to the shared eta function. The question asks for clarification on where the independent variation is utilized in the mathematical work out.
  • #1
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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  • #2
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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  • #3
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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1. Can position and velocity vary independently in Hamilton's Principle?

Yes, according to Hamilton's Principle, the position and velocity of a system can vary independently. This is because Hamilton's Principle considers the system's entire path in phase space, rather than just its position or velocity at a specific point in time.

2. How does Hamilton's Principle account for the independence of position and velocity?

Hamilton's Principle is based on the principle of least action, which states that the actual path taken by a system is the one that minimizes the action integral. This integral takes into account both the position and velocity of the system, allowing for their independent variations.

3. Does this mean that the position and velocity of a system can change without affecting each other?

Yes, Hamilton's Principle allows for the position and velocity of a system to change independently of each other. This is because the principle considers the entire path of the system, rather than just its initial and final positions and velocities.

4. Are there any limitations to the independence of position and velocity in Hamilton's Principle?

While Hamilton's Principle allows for the independent variation of position and velocity, it assumes that the system is conservative and that the forces acting on it are derivable from a potential energy function. In non-conservative systems, the position and velocity may not vary independently.

5. How is the concept of independence of position and velocity useful in physics?

The independence of position and velocity in Hamilton's Principle allows for a more comprehensive understanding of a system's behavior. It allows for the prediction of the system's future state based on its current state, without having to consider the specific forces acting on it. This makes it a powerful tool in analyzing complex physical systems.

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