Abstract
In preparation of the numerical treatment of the double pendulum, a well known problem of mechanics, Hamilton’s equations of motion are derived in detail. The problem is then formulated as an initial value problem and the system of ordinary differential equations is solved numerically by means of the explicit Runge-Kutta four stage method. The dynamics of the system is studied for various initial conditions and for a particular subset chaotic behavior is observed. A short discussion of the numerical analysis of chaos follows with the emphasis on Poincaré maps and plots. The notion of Lyapunov stability of a Hamiltonian system is discussed.
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Notes
- 1.
We make use of the relation:
$$\begin{aligned} \sin (x) \sin (y) + \cos (x) \cos (y) = \cos (x - y). \end{aligned}$$
References
Scheck, F.: Mechanics, 5th edn. Springer, Berlin (2010)
McCauley, J.L.: Chaos. Dynamics and Fractals. Cambridge University Press, Cambridge (1994)
Schuster, H.G., Just, W.: Deterministic Chaos, 4th edn. Wiley, New York (2006)
Lyapunov, A.M.: The General Problem of Stability of Motion. Taylor & Francis, London (1992)
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Stickler, B.A., Schachinger, E. (2014). The Double Pendulum. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-02435-6_6
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DOI: https://doi.org/10.1007/978-3-319-02435-6_6
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