Hamiltonian vector field

Key takeaways

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

Reference excerpt

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f {\displaystyle f} and g {\displaystyle g} on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of f {\displaystyle f} and g {\displaystyle g} .

Definition Suppose that ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold. Since the symplectic form ω {\displaystyle \omega } is nondegenerate, it sets up a fiberwise-linear isomorphism

ω : T M → T ∗ M , {\displaystyle \omega :TM\to T^{*}M,}

between the tangent bundle T M {\displaystyle TM} and the cotangent bundle T ∗ M {\displaystyle T^{*}M} , with the inverse

Ω : T ∗ M → T M , Ω = ω − 1 . {\displaystyle \Omega :T^{*}M\to TM,\quad \Omega =\omega ^{-1}.}

Therefore, one-forms on a symplectic manifold M {\displaystyle M} may be identified with vector fields and every differentiable function H : M → R {\displaystyle H:M\rightarrow \mathbb {R} } determines a unique vector field X H {\displaystyle X_{H}} , called the Hamiltonian vector field with the Hamiltonian H {\displaystyle H} , by defining for every vector field Y {\displaystyle Y} on M {\displaystyle M} ,

d H ( Y ) = ω ( X H , Y ) . {\displaystyle \mathrm {d} H(Y)=\omega (X_{H},Y).} Or more succinctly, ι X H ω = d H {\displaystyle \iota _{X_{H}}\omega =dH} . Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.

Examples Suppose that M {\displaystyle M} is a 2 n {\displaystyle 2n} -dimensional symplectic manifold. Then locally, one may choose canonical coordinates ( q 1 , ⋯ , q n , p 1 , ⋯ , p n ) {\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})} on M {\displaystyle M} , in which the symplectic form is expressed as: ω = ∑ i d q i ∧ d p i , {\displaystyle \omega =\sum _{i}\mathrm {d} q^{i}\wedge \mathrm {d} p_{i},}

where d {\displaystyle \operatorname {d} } denotes the exterior derivative and ∧ {\displaystyle \wedge } denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian H {\displaystyle H} takes the form: X H = ( ∂ H ∂ p i , − ∂ H ∂ q i ) = Ω d H , {\displaystyle \mathrm {X} _{H}=\left({\frac {\partial H}{\partial p_{i}}},-{\frac {\partial H}{\partial q^{i}}}\right)=\Omega \,\mathrm {d} H,}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hamiltonian vector field

Start with the simplest possible case. Write down what Hamiltonian vector field 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 Hamiltonian vector field 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 Hamiltonian vector field 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 Hamiltonian vector field

In research: Hamiltonian vector field 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 Hamiltonian vector field 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: Hamiltonian vector field is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hamiltonian mechanics, Symplectic geometry, William Rowan Hamilton, so understanding it makes those chapters shorter.
In everyday life: Look for Hamiltonian vector field 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.

How to study Hamiltonian vector field in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hamiltonian vector field 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 Hamiltonian vector field out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hamiltonian vector field in simple terms?

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations i…

Why does Hamiltonian vector field 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 Hamiltonian vector field?

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 Hamiltonian vector field.