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(This is not about independence of $q$,$\dot q$)

A system has some holonomic constraints. Using them we can have a set of coordinates ${q_i}$. Since any values for these coordinates is possible we say that these are independent coordinates.

However the system will trace a path in the configuration space ,which means that actually all our independent coordinates are in fact dependent.

Why do we then say that we have independent coordinates? For example, Goldstein

The fundamental problem of the calculus of variations is easily generalized to the case where $f$ is a function of many independent variables $y_{i}$, and their derivatives $\dot{y}_{i}$. (Of course, all these quantities are considered as functions of the parametric variable $x$.) Then a variation of the integral $J$, $$ \delta J=\delta \int_{1}^{2} f\left(y_{1}(x) ; y_{2}(x), \ldots, \dot{y}_{1}(x) ; \dot{y}_{2}(x), \ldots, x\right) d x $$

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You are confusing two different concepts. The coordinates are independent because the system can theoretically be at any point on the configuration space. On the other hand, the path taken by the system in the configuration space is determined by its equations of motion and not because the coordinates are fundamentally dependent. In other words, the labels $q^i$ and $q^i(t)$ mean different things. The former is simply a set of numbers (coordinates) which are independent, but the latter is a set of functions of time which are dependent (once you know one of the $q(t)$, you can, in principle, invert it and solve for all the other coordinates). The independence of coordinates refers to the former and not the latter.

Consider three examples of a point mass on the 2D plane:

  1. The mass is free to move.
  2. The mass is influenced by a spring force $-kr$ where $r$ is the distance from the origin.
  3. The mass is constrained to move the unit circle.

The first and second examples have the same configuration space but different equations of motion. The mass is capable of reaching anywhere on the plane in both examples. However, in the third example, the configuration space becomes the unit circle which is one dimension lower.

The point is that non-independent coordinates represent an actual reduction (confinement) of the configuration space, whereas a path is one particular function of time that satisfies certain equations of motion with certain initial conditions. Just because the system traces a path in configuration space doesn't mean that it can't trace other paths (with different initial conditions), or that it is confined to that path. It can very well be at a point outside that path and still satisfy the same equations of motion.

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  • $\begingroup$ "once you know one of the $q(t)$, you can, in principle, invert it and solve for all the other coordinates" - although in some cases you can't, depending on the coordinate system. For example, one coordinate might be constant (be it due to holonomic constraints or otherwise), or might have a period not applicable to other coordinates (as in helical motion). What you can do, however, is transform between $t$ & total traversed arclength... unless the particle is ever at rest for a finite time. $\endgroup$
    – J.G.
    Sep 11, 2021 at 17:41
  • $\begingroup$ Suppose I know one $q_1(t)$ how does this give me other coordinates? Also do you agree with Vincents answer? $\endgroup$
    – Kashmiri
    Sep 12, 2021 at 3:05
  • $\begingroup$ @Vincent Thacker, so do you mean Goldstein says that it's the coordinates $y_{i}$ that are independent and not the functions $y_(t)$ ? $\endgroup$
    – Kashmiri
    Sep 12, 2021 at 3:39
  • $\begingroup$ @Kashmiri Yes, that's what's meant by independent coordinates. $\endgroup$ Sep 12, 2021 at 5:14

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