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

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Ahmed1029
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.

## Answers and Replies

Homework Helper
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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.
Has this something to do with Lagrangian mechanics?

Ahmed1029 and vanhees71
Ahmed1029
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.

Homework Helper
Gold Member
2022 Award
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.

Ahmed1029 and vanhees71