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1 0 . THE EFFECT OF AN OCTUPOLE This section is not merely a repetition of the previous one for a different multipole. An octupole leads to an extra term in the invariant Hamiltonian which corresponds to a nonlinear variation of frequency with amplitude. This is characteristic of even m multipoles and defines quite a different topology of phase space. We have done this for the horizontal plane where, to be precise for small radius machines, the term 1/ρ should be included. H=  px 2 1  1 1 ∂ 3 Bz 4 +  − k ( s) − k ( s) x 2 + x .

83) demands: sin3ψ = 0 , (85) so that dJ/ds = 0. Then we can write cos 3 ψ = ±1 in Eq. (84) and ask how this can be zero. Above resonance, the sign of δ is positive and provided ε is positive d ψ /ds can only be zero if: cos3ψ = −1 . These conditions on ψ define three fixed points at (86) ψ = π / 3 , 3π / 3 and 5π / 3 . (87) From Eq. (84) we can also find their amplitude: J fp = (2δ / 3ε ) . 2 (88) δ J Q (J) Jfp Q m/3 Fig. 4 The variation of Q as a function of amplitude close to a third-integer resonance Inside this amplitude the triangular trajectory becomes less and less distorted and more circular as one approaches J = 0.

2 ∂J (99) The first two terms cancel on average when J has a resonant value Jr defined by the condition: α ' ( Jr ) = − δ . (100) If we look at small changes in J about Jr by a second differentiation we find: ( J − Jr ) = − n ε ( n/ 2−1) J cosmψ . 2 α" r (101) We now have stable and unstable fixed points at cos m ψ = -1 and +1 respectively and separated in J by: nε ( n / 2 −1) Jr . α" (102) The existence of stable fixed points in the real plane is new and is only the case for evenorder resonances.