2019/05/27 by Toshiaki Fujiwara, Hiroshi Fukuda, Fujiwara, Toshiaki +4
Engineering · Mathematics · Physics and Astronomy · #Action (physics) #Applied mathematics #Bifurcation #Bifurcation diagram #Bifurcation theory #Classical Physics (physics.class-ph) #Classical mechanics #Control and Stability of Dynamical Systems #Dynamics and Control of Mechanical Systems #Eigenvalues and eigenvectors #FOS: Physical sciences #Hessian matrix #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Nonlinear system #Numerical methods for differential equations #Physics #Principle of least action #Quantum mechanics #Saddle-node bifurcation #Transcritical bifurcation #Variational principle #Zero (linguistics) #math-ph #math.MP #physics.class-ph
paper · pdf · doi:10.48550/arxiv.1905.11073
arxiv created 2019/05/27 · openalex publication_date 2019/05/27 · arxiv updated 2019/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.