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

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

In Hamilton's Principle, the variations of the position and velocity are not completely independent. The principle requires that the variations of the position, when substituted into the equations of motion, must satisfy the boundary conditions. Thus, while they can vary, they must do so in a way that is consistent with the constraints of the system.

How does Hamilton's Principle relate to the equations of motion?

Hamilton's Principle states that the actual path taken by a system between two points in time is the one for which the action integral is stationary (i.e., has a stationary value). From this principle, one can derive the Euler-Lagrange equations, which are the equations of motion for the system. These equations describe how the position and velocity of the system evolve over time.

What are the boundary conditions in Hamilton's Principle?

In Hamilton's Principle, the boundary conditions typically refer to the fixed initial and final positions (and sometimes velocities) of the system. These conditions must be met by any variations in the position and velocity when applying the principle. The variations are chosen such that they are zero at the boundaries, ensuring that the initial and final conditions remain satisfied.

Why are variations in position and velocity important in Hamilton's Principle?

Variations in position and velocity are crucial because they allow us to explore how small changes in the path of the system affect the action integral. By considering these variations, we can determine the path that makes the action stationary, leading to the correct equations of motion for the system. This approach provides a powerful method for deriving the dynamics of a wide range of physical systems.

Can Hamilton's Principle be applied to systems with constraints?

Yes, Hamilton's Principle can be extended to systems with constraints using techniques such as Lagrange multipliers. These constraints can be holonomic (depending only on the coordinates and time) or non-holonomic (involving velocities as well). By incorporating these constraints into the action integral, one can derive the equations of motion that respect the constraints of the system.

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