保守双摆的不可积性和混沌
Non-lntegrability and Chaos of a Conservative Compound Pendulum
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摘要: 本文用Birkhoff级数正则变换方法求出保守双摆运动方程的近似积分,并把近似积分的等值曲线与数值仿真结果作了比较.由此清楚地看出.当能级提高时,系统由近可积的成为不可积的,即其运动情况由规则的转变为混沌的.本文还介绍了演示上述性态的一个保守双摆模型.Abstract: By using a series of canonical transformations(Birkhoff's series),an approximate integral of a conservative compound pendulum is evaluated.Level lines of this approximate integral are compared with the numerical simulation results.It is seen clearly that with a raised energy level,the nearly integrable system becomes non-integrable,i.e.the regular motion pattern changes to the chaotic one.Experiments with such a pendulum device display the behavior mentioned above.
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[1] Hénon,M:and C.Héiles,The applicability of the third integral of the motion,some numerical experiment,Astron.J.,69(1964),73-79. [2] Gustavson,F.,On constructing formal integrals of a Hamiltonian system near an equilibrium point,Astron.J.,71(1966),670-686. [3] 凌复华,《非线性振动中的数值方法》,高等教育出版让(即将出版). [4] 凌复华、殷学纲、何冶奇,《常微分方程数值方法及其在力学中的应用》,重庆大学出版社(即将出版). [5] Hénon,M.,On the numerical computation of Poincare maps,Physica,50(1982),413-416. -
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