How to formally show that the choice of a coordinate system in a physical system can be arbitrary?

Question

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1. What the question is really asking

“Why does it not matter which coordinate system I use to describe a physical system? How can we prove, in a clean and rigorous way, that the conclusions we draw (the equations of motion, measured quantities, etc.) are the same no matter what coordinates we pick?”

In plain language the problem is asking for a coordinate‑independent (or covariant) formulation of physics and a proof that any two admissible coordinate charts give the same physical predictions.
The proof must work for arbitrary smooth changes of coordinates – not only rotations, but translations, scalings, curvilinear maps, etc.

2. Step‑by‑step proof

The proof is most transparent in the language of differential geometry.
We will treat a classical mechanical system; the same ideas carry over to field theory, relativity, and quantum mechanics.

2.1 The setting

Let (\mathcal{M}) be the configuration space of the system (e.g. (\mathbb R^{3N}) for (N) point particles).
A coordinate chart is a smooth bijection

[ \phi : U\subset\mathcal{M}\longrightarrow V\subset \mathbb R^{n},\qquad x =\phi(p) ]

that assigns to each point (p\in U) an ordered (n)-tuple of numbers (x^{i}) (the coordinates).

Any other admissible chart (\tilde\phi) is related to (\phi) by a smooth change of coordinates

[ \tilde{x}^{i}= \tilde{\phi}^{i}\bigl(\phi^{-1}(x)\bigr)\equiv f^{i}(x),\qquad \text{with } f:V\to\tilde V\text{ a diffeomorphism.} ]

Thus the Jacobian matrix

[ J^{i}{}_{j}(x)=\frac{\partial \tilde{x}^{i}}{\partial x^{j}} ]

is invertible everywhere on the overlap of the two charts.

2.2 Physical quantities are geometric objects

The key physical principle is:

Principle of Coordinate Independence – All measurable (or “real”) quantities are geometric objects on (\mathcal{M}) that do not depend on the choice of coordinates.

Geometric objects are precisely those that transform under a change of coordinates according to a tensorial rule.
Examples:

Geometric object	Coordinate components	Transformation rule
Scalar (S)	(S(x))	( \tilde S(\tilde x)=S(x) ) (no change)
Vector (V)	(V^{i}(x))	(\tilde V^{i}(\tilde x)=J^{i}{}_{j}(x)\,V^{j}(x))
Covector (1‑form) (\alpha)	(\alpha_{i}(x))	(\tilde\alpha_{i}(\tilde x)= (J^{-1})^{j}{}{i} \,\alpha{j}(x))
Rank‑(k) tensor ({T^{i_{1}\dots i_{p}}{j{1}\dots j_{q}}})	…	product of (J)’s and ((J^{-1}))’s

Anything that can be written only with tensors (contractions, sums, exterior derivatives, etc.) has the same value in any chart because the Jacobian factors cancel.

2.3 The action is a scalar

In Lagrangian mechanics the dynamics is encoded in the action functional

[ S[\gamma]=\int_{t_{1}}^{t_{2}} L\bigl(q(t),\dot q(t),t\bigr)\,dt . ]

(q(t)) are the coordinates of a path (\gamma) in (\mathcal{M}).
The Lagrangian (L) is required to be a scalar function on the tangent bundle (T\mathcal{M}\times\mathbb R).

If we change coordinates (q\to\tilde q=f(q)) then

[ \dot{\tilde q}^{\,i}= \frac{d}{dt}\tilde q^{i}=J^{i}{}_{j}(q)\,\dot q^{j}, ]

and the new Lagrangian (\tilde L) is defined by

[ \tilde L(\tilde q,\dot{\tilde q},t)\equiv L\bigl(q,\dot q,t\bigr). ]

Because (L) is a scalar, the numerical value of the integrand does not change:

[ L\bigl(q,\dot q,t\bigr)=\tilde L\bigl(\tilde q,\dot{\tilde q},t\bigr). ]

Hence the action functional itself is invariant under any smooth coordinate change.

2.4 Euler–Lagrange equations are covariant

Varying the action in any chart gives the Euler–Lagrange equations

[ \frac{d}{dt}\Bigl(\frac{\partial L}{\partial \dot q^{i}}\Bigr)-\frac{\partial L}{\partial q^{i}}=0 . ]

Apply a coordinate transformation. Using the chain rule and the Jacobian relations above one finds

[ \frac{d}{dt}\Bigl(\frac{\partial \tilde L}{\partial \dot{\tilde q}^{\,k}}\Bigr)-\frac{\partial \tilde L}{\partial \tilde q^{k}} = J^{i}{}_{k}\Bigl[\frac{d}{dt}\Bigl(\frac{\partial L}{\partial \dot q^{i}}\Bigr)-\frac{\partial L}{\partial q^{i}}\Bigr] . ]

Since the Jacobian matrix (J^{i}{}_{k}) is invertible, the set of equations in the new chart is exactly equivalent to the original set.

Thus the equations of motion are covariant: a solution curve (\gamma(t)) expressed in coordinates (q^{i}(t)) satisfies the EL equations in the (q)-chart iff the transformed curve (\tilde q^{k}(t)=f^{k}(q(t))) satisfies the EL equations in the (\tilde q)-chart.

2.5 Observables are coordinate‑independent

Physical observables are functions of the geometric state of the system:

Position of a particle – the point (p\in\mathcal{M}) itself, not its coordinate numbers.
Distance between two particles – a scalar built from the metric tensor (g): (d=\sqrt{g_{ij}\,\Delta q^{i}\Delta q^{j}}).
Energy – (E = \dot q^{i}p_{i} - L) is a scalar because (p_{i}=\partial L/\partial \dot q^{i}) transforms as a covector.

Because each observable is a scalar (or a contraction of tensors), its numerical value is unchanged when we replace (q) by any (\tilde q).

2.6 General statement

Let (\phi) and (\tilde\phi) be any two admissible coordinate charts on the part of configuration space where the motion occurs.

Geometric objects (scalars, vectors, tensors, differential forms, etc.) have components that are related by the appropriate Jacobian factors.
Fundamental equations (Newton’s law, Euler–Lagrange, Maxwell’s equations, Schrödinger equation written in covariant form) are tensor equations: each term is a tensor of the same type, so the whole equation is invariant under any smooth change of coordinates.
Solutions of the equations map to each other by the coordinate transformation: if (\gamma(t)) solves the equations in one chart, then (\tilde\gamma(t)=\tilde\phi!\bigl(\phi^{-1}(\gamma(t))\bigr)) solves them in the other.
Measured quantities (numbers that an experiment can read) are scalars obtained by contracting tensors; therefore their values are identical in every chart.

Consequently the choice of coordinate system is completely arbitrary: it is a matter of convenience, not of physics.

3. Final answer

The rigorous proof consists of three logical steps:

Identify physical quantities as geometric objects (tensors).
Show that the fundamental action (or field functional) is a scalar, so it is unchanged by any smooth diffeomorphism of the coordinates.
Derive the equations of motion from the invariant action; because they are tensor equations, they retain exactly the same form under any coordinate change, and solutions are carried into each other by the coordinate map.

Since all observable predictions are built from these tensors, the numerical outcomes are independent of the coordinate chart. Hence we may choose any admissible coordinate system—rotated, translated, curvilinear, or otherwise—and obtain the same physical conclusions.

4. Common Mistakes

Mistake	Why it is wrong	How to avoid it
Confusing “invariance under a specific symmetry” with “invariance under any coordinate change.”	Rotational invariance is a symmetry of a particular physical law; coordinate invariance is a general property of the mathematical description.	Emphasise the tensor nature of quantities; a symmetry is a special case where the transformation also leaves the form of the law unchanged.
Treating the coordinates themselves as physical objects.	Coordinates are merely labels; only geometric objects (points, vectors, scalars) have physical meaning.	When you write an equation, check that each term is a tensor of the same type; if a term is just a list of numbers, it is not physical.
Neglecting the Jacobian when transforming derivatives.	Forgetting the factor (J^{i}{}_{j}) leads to apparent “extra terms” that look like a change in the physics.	Use the chain rule explicitly: (\dot{\tilde q}^{\,i}=J^{i}{}_{j}\dot q^{j}), and for second derivatives include the time derivative of the Jacobian.
Assuming the Lagrangian must be unchanged point‑wise.	The Lagrangian is a scalar function; its value at a point is the same, but its expression in new coordinates may look different.	Remember: (L(q,\dot q)=\tilde L(\tilde q,\dot{\tilde q})). The functional form may change, but the numerical value does not.
Thinking that non‑linear coordinate changes break Newton’s law.	Newton’s second law written as a vector equation is covariant; the components change, but the geometric equation ( \mathbf{F}=m\mathbf{a}) stays true.	Write Newton’s law using vectors (or covariant derivatives) rather than component‑wise equations.
Forgetting about the domain of the charts (overlap).	Two charts might not cover the whole configuration space; invariance only needs to be shown on the overlap where both are defined.	State explicitly that the proof works on (U\cap\tilde U); extend by covering the whole space with a collection of overlapping charts.

By keeping these points in mind, one can avoid the usual pitfalls and present a clean, mathematically rigorous argument that the physics does not depend on the coordinate system we happen to use.

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