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Physics · lesson 2 of 3

Hamilton equations, phase space and Poisson brackets

For a regular Lagrangian, p=∂L/∂qdot and H=p qdot−L after expressing qdot in terms of q,p,t.

Advanced university8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Lagrangian mechanics, derivatives and invertible Legendre transforms.

Legendre transform

For a regular Lagrangian, p=∂L/∂qdot and H=p qdot−L after expressing qdot in terms of q,p,t.

Canonical evolution

Hamilton equations are qdot=∂H/∂p and pdot=−∂H/∂q. Phase space records coordinates and momenta, not just position.

Brackets

In one canonical pair, {f,g}=fq gp−fp gq. In particular {q,p}=1 under this convention.

Conservation

With no explicit time dependence, H is conserved along Hamiltonian evolution. Identifying H with physical energy requires the appropriate system assumptions.

Worked examples

Example 1

For H=p²/6+2q² and p=9, find qdot.

∂H/∂p=p/3=3.

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Example 2

For H=p²/6+2q² and q=2, find pdot.

pdot=−∂H/∂q=−4q=−8.

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Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 For H=p²/6+2q² and p=9, find qdot.

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3

∂H/∂p=p/3=3.

2 For H=p²/6+2q² and q=2, find pdot.

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-8

pdot=−∂H/∂q=−4q=−8.

3 What variables specify a point in canonical phase space?

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Coordinates and conjugate momenta

A canonical state uses q and p for each degree of freedom.

4 Using {f,g}=fq gp−fp gq, find {q,p}.

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1

The relevant derivatives are 1,1,0,0.

5 What is required for the usual local velocity-to-momentum inversion?

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A nonsingular velocity Hessian

Regularity of the Legendre transform enables inversion.

6 Derive H=p²/(2m)+V(q) from L=mqdot²/2−V(q).

Sample answer and marking guidance

p=mqdot, so qdot=p/m. Substitute into p qdot−L to obtain p²/m−p²/(2m)+V.

Credit elimination of velocity and correct signs.

7 Evaluate {q²,p²} at q=2,p=3 in the stated convention.

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24

The bracket is (2q)(2p)=4qp=24.

8 Why can changing a Poisson-bracket sign convention cause an apparent disagreement?

Sample answer and marking guidance

Reversing the definition reverses bracket signs; evolution equations must change consistently. State the convention before comparing results.

Credit consistency rather than declaring both isolated formulas correct.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.