A good example of a nonlinear symplectomorphism?

What is a good example of a simple, physically useful nonlinear symplectomorphism $\kappa: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked have been purely mathematical and not connected to any physical problems. Obviously linear examples are well known and useful.

Actually, it would suffice if I could see some nice example of a nonlinear diffeomorphism (i.e. change of coordinates) $\varphi: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ given by $x \mapsto y$, as one can then lift this diffeomorphism to a symplectomorphism by transforming the momenta via $p_y := (d\varphi^{-1})^{*}p_x$.

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In the second paragraph(v1), haven't you basically answered your own question? –  Qmechanic Sep 20 '12 at 21:16
The Bloch equations, viewed as a Hamiltonian flow on $S^2$. The symplectomorphism is rotation of the Bloch sphere. –  John Sidles Sep 20 '12 at 21:37
@Qmechanic, well, I couldn't think of any good diffeomorphisms $\varphi$. –  Christopher A. Wong Sep 20 '12 at 21:44
How about e.g. $\varphi(x,y,z):=(x(x^2+1),y,z)$? –  Qmechanic Sep 20 '12 at 21:57
Oh, I guess when I meant "physically useful", I meant that the coordinates were specifically used for some (hopefully easy-to-understand) physics problem. But I'll look at the Bloch equations and see if I can work out some mathematical results using them. –  Christopher A. Wong Sep 20 '12 at 22:05

Since Hamiltonian vector fields generate symplectomorphisms in $\mathbb{R}^{2n}$ (with the canonical symplectic form),one can pick any Hamiltonian,solve the Hamilton equations of motion to get a symplectomorphism. Since, a nonlinear symplectomorphism is seeked, free particle and Harmonic oscillator Hamiltonians will not be good examples, as they give linear dependence on the initial conditions. But the Hydrogen atom Hamiltonian is a good example(for $\mathbb{R}^{12}$(two bodies in three dimensions subject to an inverse square attractive cental force)). The explicitely known solutions of its equations of motion are nonlinear symplectomorphisms.