Holonomic constraints, independent generalized coordinates

Question

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1. What the statement means in plain language

We have a collection of (N) point particles, each with three Cartesian coordinates.
Altogether we start with a configuration space

[ \mathcal C=\mathbb R^{3N}\; . ]

The particles are not completely free: they must satisfy (p) holonomic constraints

[ g_{v}(\mathbf r_{1},\dots ,\mathbf r_{N},t)=0,\qquad v=1,\dots ,p . ]

For a fixed instant (t=t_{0}) the constraints define a subset

[ M_{t_{0}}:=\Bigl{ \mathbf r\in\mathbb R^{3N}\; \big|\; g_{v}(\mathbf r,t_{0})=0,\;v=1,\dots ,p\Bigr}. ]

The textbooks claim that the current shape of the system can be described by (3N-p) numbers (q^{1},\dots ,q^{3N-p}) (the generalised coordinates) and that these numbers are independent, i.e. there is no relation

[ F\bigl(q^{1},\dots ,q^{3N-p}\bigr)=0 ]

apart from the trivial one (F\equiv 0).

Our goal is to prove this claim rigorously.

2. Assumptions (regularity of the constraints)

The proof works under the standard regularity hypothesis for holonomic constraints:

Regularity (independence) of the constraints –
At every point (\mathbf r\in M_{t_{0}}) the Jacobian matrix

[ J(\mathbf r)=\Bigl[\;\partial_{r_{i}}g_{v}(\mathbf r,t_{0})\Bigr]_{\;v=1..p}^{\;i=1..3N} ]

has full rank (p) (i.e. the rows are linearly independent).

This is exactly the condition that the constraints are independent and holonomic; it guarantees that the constraints cut out a smooth submanifold of the expected dimension.

3. Step‑by‑step proof

3.1 The constraint surface is a ((3N-p))-dimensional manifold

Define

[ \mathbf g(\mathbf r)=\bigl(g_{1}(\mathbf r,t_{0}),\dots ,g_{p}(\mathbf r,t_{0})\bigr): \mathbb R^{3N}\longrightarrow\mathbb R^{p}. ]

Because each (g_{v}) is smooth, (\mathbf g) is a smooth map. The regularity hypothesis says that (\mathbf 0\in\mathbb R^{p}) is a regular value of (\mathbf g) :

[ \operatorname{rank}\, D\mathbf g(\mathbf r)=p \qquad\forall \mathbf r\in\mathbf g^{-1}(\mathbf 0)=M_{t_{0}} . ]

The Regular Value Theorem (also called the pre‑image theorem) then tells us that

[ M_{t_{0}}=\mathbf g^{-1}(\mathbf 0) ]

is an embedded submanifold of (\mathbb R^{3N}) whose dimension is

[ \dim M_{t_{0}}=3N-p . ]

Thus the set of admissible configurations at the instant (t_{0}) is a smooth ((3N-p))-dimensional manifold.

3.2 Existence of local coordinates on the manifold

Pick an arbitrary point (\mathbf r^{}\in M_{t_{0}}). Because the Jacobian (J(\mathbf r^{})) has rank (p), there exists a (p\times p) sub‑determinant that is non‑zero. Without loss of generality we may relabel the Cartesian coordinates so that the sub‑determinant involves the last (p) coordinates ((x^{3N-p+1},\dots ,x^{3N})).
(If the non‑zero minor involves a different set of coordinates we simply reorder the list of coordinates; the physics is unchanged.)

Write the full coordinate vector as

[ \mathbf x=(x^{1},\dots ,x^{3N-p},\,x^{3N-p+1},\dots ,x^{3N}) . ]

Now apply the Implicit Function Theorem to the equations

[ g_{v}(\mathbf x)=0,\qquad v=1,\dots ,p . ]

Because the matrix

[ \Bigl[\partial_{x^{3N-p+k}}g_{v}(\mathbf r^{*})\Bigr]_{v,k=1}^{p} ]

is invertible, the theorem guarantees that, in a neighbourhood (U) of (\mathbf r^{*}), the last (p) coordinates can be expressed uniquely as smooth functions of the first (3N-p) coordinates:

[ x^{3N-p+k}=h_{k}\bigl(x^{1},\dots ,x^{3N-p}\bigr),\qquad k=1,\dots ,p . ]

Consequently every point of (M_{t_{0}}\cap U) is uniquely determined by the tuple

[ \mathbf q:=(q^{1},\dots ,q^{3N-p})\;:=\;(x^{1},\dots ,x^{3N-p}) . ]

Define

[ \Phi:U\cap M_{t_{0}}\longrightarrow \mathbb R^{3N-p},\qquad \Phi(\mathbf r)=\bigl(q^{1},\dots ,q^{3N-p}\bigr) . ]

Because the inverse map

[ \Phi^{-1}(q^{1},\dots ,q^{3N-p})= \bigl(q^{1},\dots ,q^{3N-p},\, h_{1}(\mathbf q),\dots ,h_{p}(\mathbf q)\bigr) ]

is also smooth, (\Phi) is a diffeomorphism between the neighbourhood (U\cap M_{t_{0}}) and an open set (V\subset\mathbb R^{3N-p}).

Thus the numbers (\mathbf q=(q^{1},\dots ,q^{3N-p})) are legitimate local coordinates on the constraint manifold.

3.3 Independence of the coordinates

By definition of a coordinate system on a manifold, the coordinate functions (q^{i}:U\cap M_{t_{0}}\to\mathbb R) have linearly independent differentials. Indeed, the Jacobian matrix of the map (\Phi),

[ \frac{\partial (q^{1},\dots ,q^{3N-p})}{\partial (x^{1},\dots ,x^{3N})} = \begin{pmatrix} I_{3N-p} & 0 \end{pmatrix}, ]

has full rank (3N-p).

Suppose there existed a non‑trivial smooth function (F:\mathbb R^{3N-p}\to\mathbb R) such that

[ F\bigl(q^{1}(\mathbf r),\dots ,q^{3N-p}(\mathbf r)\bigr)=0 \quad\text{for every }\mathbf r\in U\cap M_{t_{0}} . ]

Differentiating this identity gives

[ \sum_{i=1}^{3N-p}\frac{\partial F}{\partial q^{i}}( \mathbf q )\, dq^{i}=0 . ]

Because the differentials (dq^{i}) are linearly independent, each coefficient must vanish:

[ \frac{\partial F}{\partial q^{i}}(\mathbf q)=0,\qquad i=1,\dots ,3N-p . ]

Hence the gradient of (F) is zero on the open set (V=\Phi(U\cap M_{t_{0}})). A smooth function with vanishing gradient on a connected open set is constant, and because (F) vanishes somewhere (take any point of (V)), that constant is zero. Therefore (F\equiv 0).

Thus no non‑trivial functional relation among the (q^{i}) can hold locally. The coordinates are independent in the sense required by the textbooks.

3.4 Global picture

The argument above works at every point of the constraint surface. Consequently the whole admissible configuration space

[ \mathcal Q:=\bigcup_{t}\;M_{t} ]

is a smooth manifold of dimension (3N-p).
A manifold is, by definition, covered by a collection of such local charts, each providing a set of (3N-p) independent generalized coordinates. If the constraints are simple (e.g. a single rigid body) a single global chart exists; in general several overlapping charts may be needed, but the number of independent coordinates never exceeds (3N-p).

4. Final answer

*Under the regularity (independence) assumption on the holonomic constraints, the set of admissible configurations at any fixed time is a smooth ((3N-p))-dimensional submanifold of (\mathbb R^{3N}).
The implicit‑function theorem provides, in a neighbourhood of any admissible configuration, a diffeomorphism onto an open subset of (\mathbb R^{3N-p}).
The image coordinates ((q^{1},\dots ,q^{3N-p})) are therefore independent (their differentials are linearly independent, and no non‑trivial relation (F(q^{1},\dots ,q^{3N-p})=0) can hold).

Hence a system of (3N) particles subject to (p) independent holonomic constraints can be described locally (and, after patching charts, globally) by exactly (3N-p) independent generalized coordinates.*

5. Common mistakes to avoid

Mistake	Why it is wrong	How to avoid it
Assuming the constraints are independent without checking the Jacobian rank.	If the Jacobian loses rank somewhere, the constraint surface may have a lower dimension or singular points, and the count (3N-p) fails.	Explicitly require (\operatorname{rank} D\mathbf g = p) at every point (regular value condition).
*Believing a single* global set of (q^{i}) always exists.**	Some configuration manifolds (e.g. a sphere) cannot be covered by one chart; you need an atlas of overlapping charts.	State the result locally; mention that a global chart exists only for special topologies.
Treating time as a coordinate and counting it among the (3N) variables.	The constraints are holonomic: they involve only the spatial coordinates at a fixed time; time is an external parameter.	Fix a time (t_{0}) when applying the theorem; treat (t) separately.
Confusing “no functional relation” with “no algebraic relation”.	Independence must hold for smooth functions, not only polynomials.	Use the differential argument (linear independence of (dq^{i})).
Using the implicit function theorem without verifying the invertible sub‑matrix.	The theorem requires a specific set of coordinates whose partial derivatives form an invertible matrix.	Reorder the Cartesian coordinates so that a non‑zero (p\times p) minor appears, then apply the theorem.

Keeping these points in mind yields a clean and rigorous justification of the textbook statement.

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