Variational principle for bifurcation in Lagrangian mechanics.
Variational principle for bifurcation in Lagrangian mechanics.
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发表时间：
2019-05
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通讯作者：
T. Fujiwara;H. Fukuda;H. Ozaki
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作者：
T. Fujiwara;H. Fukuda;H. Ozaki
展示了变分原理在拉格朗日力学中周期解分岔中的应用。周期解处作用积分的一些高阶导数揭示了解附近函数空间中作用的行为。然后变分原理给出了一种从解中寻找分岔的方法。作用的二阶导数（Hessian）具有重要作用。在分叉点处，Hessian 的特征值趋于零。反之，如果特征值趋于零，则零点就是分叉点。该作用的三阶及更高阶导数决定了分叉和分叉解的性质。
An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the action determine the properties of the bifurcation and bifurcated solution.