Physics · Advanced university
Hamilton equations, phase space and Poisson brackets · revision notes
For a regular Lagrangian, p=∂L/∂qdot and H=p qdot−L after expressing qdot in terms of q,p,t.
Key ideas and conditions
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.
Explain without looking
- Evaluate {q²,p²} at q=2,p=3 in the stated convention.
- Why can changing a Poisson-bracket sign convention cause an apparent disagreement?
Further reading and review references
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