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 Incoming links: Poisson bracket of conserved quantities

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 8B / a / Solution Created 2026-09-24 Updated 2026-10-03

The Jacobi identity for the Poisson bracket is

{F, {G, H}} + {G, {H, F}} + {H, {F, G}} = 0 .

For quantities with no explicit time dependence,

\dot{F} = {F, H}, \dot{G} = {G, H} .

If

F

and

G

are conserved, the first two terms of the Jacobi identity vanish. Hence

{H, {F, G}} = 0,

or, by antisymmetry,

{{F, G}, H} = 0 .

Thus the poisson bracket of conserved quantities is itself conserved.

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