Can I always consider velocities and coordinates to be independent?

In summary: What you're asking is whether coordinate systems and velocities can always be considered independent. The short answer is that they can, but it's not always intuitively obvious. What you're asking is whether coordinate systems and velocities can always be considered independent. The short answer is that they can, but it's not always intuitively obvious.
  • #1
Ahmed1029
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It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
 
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  • #2
Ahmed1029 said:
It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
Has this something to do with Lagrangian mechanics?
 
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  • #3
PeroK said:
Has this something to do with Lagrangian mechanics?
Studying lagrangian mechanics evoked this question in my mind, but I thought I was missing a more general case.
 
  • #4
Ahmed1029 said:
Studying lagrangian mechanics evoked this question in my mind, but I thought I was missing a more general case.
The fundamental starting point for Lagrangian mechanics is to study the functional form of the Lagrangian in terms of an abstract function of independent variables: the coordinates and their first time derivatives.

This functional analysis yields the Euler-Lagrange equations as an alternative to Newton's second law.

At this point, the quantities resume their normal role as the analysis switches to the time-based trajectories or solutions to the physical problem. I.e. when we actually solve the Euler-Lagrange equations.

This strategy is often not explained very clearly in textbooks.
 
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  • #5
Not clear on what you're asking, but here's my rather convoluted take on getting to a "no" answer to what I think you're asking.

Velocities always depend on a specific frame of reference since you HAVE to be talking about velocity in relation TO something. That something is at a minimum a point and there is a frame of reference in which that point is at rest. You can then assign an infinite number of different coordinate systems such that that point is at the origin of the system. Thus there are an infinite number of coordinate systems that are related to that velocity (or probably it would be more appropriate linguistically to say that the velocity is related to the coordinate systems).
 

1. Can I assume that velocities and coordinates are always independent in all situations?

No, velocities and coordinates may not always be independent in all situations. In some cases, they may be dependent on each other, such as in the case of circular motion where velocity is dependent on the radius and angular velocity.

2. Are there any exceptions to the assumption that velocities and coordinates are independent?

Yes, there are exceptions to this assumption. As mentioned before, in circular motion, velocity is dependent on the radius and angular velocity. Additionally, in cases of non-inertial reference frames, velocities and coordinates may also be dependent on each other.

3. Why is it important to consider the independence of velocities and coordinates?

It is important to consider the independence of velocities and coordinates because it affects the accuracy and validity of our calculations and measurements. If we assume they are independent when they are actually dependent, our results may be incorrect.

4. How can I determine if velocities and coordinates are independent in a given situation?

To determine if velocities and coordinates are independent in a given situation, you can analyze the equations that describe the relationship between them. If there are any terms that involve both velocities and coordinates, then they are not independent.

5. Are there any practical applications where considering the independence of velocities and coordinates is crucial?

Yes, there are many practical applications where considering the independence of velocities and coordinates is crucial. For example, in navigation systems, the independence of velocities and coordinates is essential for accurate tracking and positioning. In physics experiments, it is crucial for obtaining accurate results and understanding the behavior of objects in motion.

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