## On approximate first integrals of Hamiltonian systems with an application to nonlinear oscillator |

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### Contents

POINCARE MAP | 23 |

BIRKHOFFGUSTAVSON NORMAL FORM | 53 |

CHAPTER k WHITTAKERS ADELPHIC INTEGRAL | 78 |

11 other sections not shown

### Common terms and phrases

action-angle variables addition angle terms approximate first integral axis B-G method bifurcation point bifurcation sub-branches Chapter coefficients const corresponds curvature degrees of freedom dimensional phase equation 7.7 equilibrium points fixed points Floquet theory given in equation H x,y Hamilton-Jacobi theory Hamilton's equations Hamiltonian becomes Hamiltonian system higher order terms homogeneous polynomials identity canonical transformation inner expansion internal resonance intersection invariant curves Lie derivative Lie transform method linear oscillator method of Lie mode is stable mode x neglect 0(h neglect terms neighborhood nonlinear oscillator nonresonance nonsimilar normal modes null space obtain outer expansions periodic motion perturb off resonance phase mode phase plane phase space Poincare map surface Poisson bracket predict stability represents saddle similar NNM's singular small energies SNM's solution stability results surface of section Table terms of 0(h trajectory transform back transition curve unstable Whittaker's adelphic integral Whittaker's method