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Physics · Advanced university

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.

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Prerequisites

Lagrangian mechanics, derivatives and invertible Legendre transforms.

Study notes

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

For H=p²/6+2q² and p=9, find qdot.

∂H/∂p=p/3=3.

3

For H=p²/6+2q² and q=2, find pdot.

pdot=−∂H/∂q=−4q=−8.

-8

Check your understanding

  1. What variables specify a point in canonical phase space?

    1. Position only
    2. Velocity only in every formulation
    3. Time and temperature only
    4. Coordinates and conjugate momenta

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

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

    1. A negative mass
    2. A nonsingular velocity Hessian
    3. A circular orbit
    4. A zero Lagrangian

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

Answers and marking guidance

Answers and marking notes
  1. Coordinates and conjugate momenta

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

  2. 1

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

  3. A nonsingular velocity Hessian

    Regularity of the Legendre transform enables inversion.

  4. 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.

Further reading and review references

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