Classical Mechanics (Goldstein) · Problem 39

Canonical Transformations — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · poisson-brackets

Problem

(a) Show from the Poisson bracket condition for conserved quantities that the Laplace--Runge--Lenz vector

\mathbf{A}

\mathbf{A} = \mathbf{p}\times\mathbf{L} - \frac{mk\mathbf{r}}{r},

is a constant of the motion for the Kepler problem. (b) Verify the Poisson bracket relations for the components of

\mathbf{A}

as given by Eq. (9.131).