mathematics

Weinstein's neighbourhood theorem

Weinstein's neighbourhood theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Weinstein's neighbourhood theorem rather than just read about it. In short: In symplectic geometry, a branch of mathematics, Weinstein's neighbourhood theorem refers to a few distinct but related theorems, involving the neighbourhoods of submanifolds in symplectic manifolds and generalising the classical Darboux's theorem. They were proved by Alan Weinstein in 1971.

Key takeaways

Weinstein's neighbourhood theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Weinstein's neighbourhood theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Weinstein's neighbourhood theorem from memory before moving on to harder problems.

Reference excerpt

In symplectic geometry, a branch of mathematics, Weinstein's neighbourhood theorem refers to a few distinct but related theorems, involving the neighbourhoods of submanifolds in symplectic manifolds and generalising the classical Darboux's theorem. They were proved by Alan Weinstein in 1971.

Darboux-Moser-Weinstein theorem This statement is a direct generalisation of Darboux's theorem, which is recovered by taking a point as X {\displaystyle X} .Let M {\displaystyle M} be a smooth manifold of dimension 2 n {\displaystyle 2n} , and ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} two symplectic forms on M {\displaystyle M} . Consider a compact submanifold i : X ↪ M {\displaystyle i:X\hookrightarrow M} such that i ∗ ω 1 = i ∗ ω 2 {\displaystyle i^{*}\omega _{1}=i^{*}\omega _{2}} . Then there exist two open neighbourhoods U 1 {\displaystyle U_{1}} and U 2 {\displaystyle U_{2}} of X {\displaystyle X} in M {\displaystyle M} ; a diffeomorphism f : U 1 → U 2 {\displaystyle f:U_{1}\to U_{2}} ; such that f ∗ ω 2 = ω 1 {\displaystyle f^{*}\omega _{2}=\omega _{1}} and f | X = i d X {\displaystyle f|_{X}=\mathrm {id} _{X}} .Its proof employs Moser's trick.

Generalisation: equivariant Darboux theorem The statement (and the proof) of Darboux-Moser-Weinstein theorem can be generalised in presence of a symplectic action of a Lie group.Let M {\displaystyle M} be a smooth manifold of dimension 2 n {\displaystyle 2n} , and ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} two symplectic forms on M {\displaystyle M} . Let also G {\displaystyle G} be a compact Lie group acting on M {\displaystyle M} and leaving both ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} invariant. Consider a compact and G {\displaystyle G} -invariant submanifold i : X ↪ M {\displaystyle i:X\hookrightarrow M} such that i ∗ ω 1 = i ∗ ω 2 {\displaystyle i^{*}\omega _{1}=i^{*}\omega _{2}} . Then there exist two open G {\displaystyle G} -invariant neighbourhoods U 1 {\displaystyle U_{1}} and U 2 {\displaystyle U_{2}} of X {\displaystyle X} in M {\displaystyle M} ; a G {\displaystyle G} -equivariant diffeomorphism f : U 1 → U 2 {\displaystyle f:U_{1}\to U_{2}} ; such that f ∗ ω 2 = ω 1 {\displaystyle f^{*}\omega _{2}=\omega _{1}} and f | X = i d X {\displaystyle f|_{X}=\mathrm {id} _{X}} .In particular, taking again X {\displaystyle X} as a point, one obtains an equivariant version of the classical Darboux theorem.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Weinstein's neighbourhood theorem

Start with the simplest possible case. Write down what Weinstein's neighbourhood theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Weinstein's neighbourhood theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Weinstein's neighbourhood theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Weinstein's neighbourhood theorem

In research: Weinstein's neighbourhood theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Weinstein's neighbourhood theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Weinstein's neighbourhood theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Symplectic geometry, Theorems in differential geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Weinstein's neighbourhood theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Weinstein's neighbourhood theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Weinstein's neighbourhood theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weinstein's neighbourhood theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Weinstein's neighbourhood theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weinstein's neighbourhood theorem in simple terms?

Why does Weinstein's neighbourhood theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Weinstein's neighbourhood theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Weinstein's neighbourhood theorem.