Goldstein says that when a system of $$N$$ particles is subject to $$k$$ holonomic constraints, the positions $$\mathbf{r}_1, \dots, \mathbf{r}_N$$ can be parameterized by $$3N - k$$ independent coordinates $$q_1, \dots, q_{3N - k}$$ and time. He then says that:

It is always assumed that we can also transform back from the ($$q_i$$) to the ($$\mathbf{r}_l$$) set, i.e., that [the parameterizations] combined with the $$k$$ equations of constraint can be inverted to obtain any $$q_i$$ as a function of the ($$\mathbf{r}_l$$) variable and time.

My question: Why would we need the $$k$$ equations of constraint? It seems to me that all of the constraint information is stored in the parameterizations of $$\mathbf{r}_1, \dots, \mathbf{r}_N$$. No?

• Commented Aug 21, 2021 at 10:12

It is a fundamental result$$^1$$ in the theory of embedded differentiable submanifolds that they can equivalently be described

• locally$$^2$$ as a parametrized submanifold/graph,

• or locally as a constrained submanifold.

Example: An ellipse in the 2D plane can either be described by a parametrization $$(x,y)=(a\cos\theta,b\sin\theta)$$ or via a constraint $$(x/a)^2+(y/b)^2=1$$.

Depending on application, both descriptions can be useful. Often it is simplest to use the description with as few variables as possible. If one of the descriptions fails, it means that some of the technical regularity conditions (which are mostly implicitly assumed in Goldstein) are not fulfilled, cf. e.g. this & this Phys.SE posts.

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$$^1$$ This result is included in any decent textbook on differential geometry (DG). (See e.g. Proposition 3.2.1 in Ben Andrews, Lectures on DG.) The main tool in its proof is the inverse function theorem.

$$^2$$ The word "locally" here means "in an open neighborhood".

I understand it like this:

Example: one particle with sphere coordinates (parameter $$r\,,\theta\,,\varphi$$)

$$\vec{R}=\left[ \begin {array}{c} x\\ y\\ z\end {array} \right] =\left[ \begin {array}{c} r\cos \left( \theta \right) \sin \left( \varphi \right) \\ r\sin \left( \theta \right) \sin \left( \varphi \right) \\ r\cos \left( \varphi \right) \end {array} \right]\tag 1$$

we can solve equation (1) for $$r\,,\theta\,,\varphi$$

$$r=\sqrt{x^2+y^2+z^2}\tag 2$$ $$\theta=\arctan(x/y)\tag 3$$ $$\varphi=\frac{z}{x^2+y^2}\tag 4$$

the constraint equation

$$r^2-l^2\cos^2(\varphi)\cos(\theta)=0\tag 5$$

now we can choose the generalized coordinates:

if we solve equation (5) for $$r$$ then we get (3) $$\quad q_1(x,y)=\theta$$ and (4) $$\quad q_2(x,y,z)=\varphi$$

if we solve equation (5) for $$\theta$$ then we get (2) $$\quad q_1(x,y,z)=r$$ and (4) $$\quad q_2(x,y,z)=\varphi$$

and

if we solve equation (5) for $$\varphi$$ then we get (2) $$\quad q_1(x,y,z)=r$$ and (3) $$\quad q_2(x,y)=\theta$$

we get always unique result for $$q_1(x,y,z)$$ and $$q_2(x,y,z)$$ and we used the "inverse" of the position vector and the constrained equation.

I think that the author is just referring to "they" as the numbers that you have for the different variables $$Q=\{q_i\}$$ whereas you are correct that these numbers only make sense in this context via their parameterizations $$\mathbf r_i(Q)$$ and if you understand the sentence in that way it is expressing a tautology and no further information is needed.

With that said if you had a really simple system there are probably some degenerate-ish cases where just the equations for those parameters are not invertible without fully knowing the constraints. For example we might have two particles living in 2D, one is constrained to live on the line $$x=0$$ and the other is constrained to live on the line $$y=0$$, and let's say that they are joined by a spring with rest length $$\ell$$. We know that we can describe this system with two variables $$(x, y)$$ and the mapping from $$(\mathbf r_1, \mathbf r_2) \mapsto (x, y)$$ is going to be $$\big([x_1, y_1], [x_2, y_2]\big) \mapsto (x_1, y_2)$$ but discovering that $$x_2=0, y_2=0$$ is not directly possible just given this function; it is not invertible.

But with that said this is indeed somewhat of an unnatural way to describe the description. The more natural $$\mathbf r_i(Q)$$ way is indeed to specify $$\mathbf r_1(x, y) = [x, 0]\\\mathbf r_2(x, y) = [0, y]$$and this indeed does embody the constraints and therefore no further reference to the constraints is needed to use the $$x,y$$ to determine the positions.