Hamiltonian mechanics

Key takeaways

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

Reference excerpt

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

Overview

Phase space coordinates (p, q) and Hamiltonian H Let ( M , L ) {\displaystyle (M,{\mathcal {L}})} be a mechanical system with configuration space M {\displaystyle M} and smooth Lagrangian L . {\displaystyle {\mathcal {L}}.} Select a standard coordinate system ( q , q ˙ ) {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})} on the tangent bundle T M . {\displaystyle TM.} The quantities p i ( q , q ˙ , t ) = def ∂ L / ∂ q ˙ i {\displaystyle \textstyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}} are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant t , {\displaystyle t,} the Legendre transformation of L {\displaystyle {\mathcal {L}}} is defined as the map ( q , q ˙ ) → ( p , q ) {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)} which is assumed to have a smooth inverse ( p , q ) → ( q , q ˙ ) . {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).} For a system with n {\displaystyle n} degrees of freedom, the Lagrangian mechanics defines the energy function

E L ( q , q ˙ , t ) = def ∑ i = 1 n q ˙ i ∂ L ∂ q ˙ i − L . {\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum _{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.}

The Legendre transform of L {\displaystyle {\mathcal {L}}} turns E L {\displaystyle E_{\mathcal {L}}} into a function H ( p , q , t ) {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)} known as the Hamiltonian. The Hamiltonian satisfies

… excerpt ends here. Continue reading the full article.

Illustrations

Hamiltonian mechanics: Sir William Rowan Hamilton — Sir William Rowan Hamilton

Worked examples

Example 1 — a first encounter with Hamiltonian mechanics

Start with the simplest possible case. Write down what Hamiltonian mechanics claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, 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 mechanics 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 mechanics 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 mechanics

In research: Hamiltonian mechanics appears in physics 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 mechanics 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 mechanics is common in secondary-school and first-year university syllabi. It links to neighbouring topics Classical mechanics, Dynamical systems, Hamiltonian mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Hamiltonian mechanics 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 mechanics in 20 minutes

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

Frequently asked questions

What is Hamiltonian mechanics in simple terms?

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta.

Why does Hamiltonian mechanics matter?

Because it connects several physics 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 mechanics?

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 mechanics.