Can I always consider velocities and coordinates to be independent?

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Discussion Overview

The discussion revolves around the independence of velocities and coordinates, particularly in the context of both Cartesian and generalized coordinates. Participants explore whether this independence holds universally and its implications in Lagrangian mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants express uncertainty about whether velocities and coordinates can always be considered independent, questioning this in both Cartesian and generalized contexts.
  • One participant connects this inquiry to Lagrangian mechanics, suggesting that the independence of variables is a fundamental aspect of the functional form of the Lagrangian.
  • Another participant argues that velocities depend on a specific frame of reference, implying that this dependence complicates the notion of independence from coordinates.
  • The discussion includes a mention of the Euler-Lagrange equations as an alternative to Newton's second law, indicating a shift in analysis when solving physical problems.
  • Concerns are raised about the clarity of explanations in textbooks regarding the relationship between coordinates and velocities in Lagrangian mechanics.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the independence of velocities and coordinates, with multiple competing views presented regarding their relationship and dependence on reference frames.

Contextual Notes

Participants highlight the potential limitations in understanding the independence of velocities and coordinates, particularly in relation to the choice of reference frames and the implications for Lagrangian mechanics.

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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