I Can I always consider velocities and coordinates to be independent?

AI Thread Summary
The discussion centers on the independence of velocities and coordinates in both Cartesian and generalized contexts, particularly in relation to Lagrangian mechanics. It highlights that the fundamental approach in Lagrangian mechanics involves analyzing the Lagrangian as a function of independent coordinates and their time derivatives, leading to the Euler-Lagrange equations. However, it emphasizes that velocities are inherently dependent on the chosen frame of reference, as they relate to a specific point that can be at rest in various coordinate systems. This dependency complicates the notion of treating velocities and coordinates as independent. Ultimately, the relationship between these quantities is more intricate than initially assumed.
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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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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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.
 
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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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).
 

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