# 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.

Course: [Analytical and statistical mechanics](https://studybrix.com/en/courses/analytical-and-statistical-mechanics/index.md)
Level: advanced
Prerequisites: Lagrangian mechanics, derivatives and invertible Legendre transforms.

## 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.

## Practice questions

### 1. For H=p²/6+2q² and p=9, find qdot.

Answer: 3

Explanation: ∂H/∂p=p/3=3.

### 2. For H=p²/6+2q² and q=2, find pdot.

Answer: -8

Explanation: pdot=−∂H/∂q=−4q=−8.

### 3. What variables specify a point in canonical phase space?

A. Position only
B. Velocity only in every formulation
C. Time and temperature only
D. Coordinates and conjugate momenta

Answer: Coordinates and conjugate momenta

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

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

Answer: 1

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

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

A. A negative mass
B. A nonsingular velocity Hessian
C. A circular orbit
D. A zero Lagrangian

Answer: A nonsingular velocity Hessian

Explanation: Regularity of the Legendre transform enables inversion.

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

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

Explanation: Credit elimination of velocity and correct signs.

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

Answer: 24

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

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

Sample answer / marking guidance: Reversing the definition reverses bracket signs; evolution equations must change consistently. State the convention before comparing results.

Explanation: Credit consistency rather than declaring both isolated formulas correct.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.