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 problem of stability of Hamiltonian systems is discussed.
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Notes
- 1.
We make use of the relation:
$$\displaystyle{ \sin (x)\sin (y) +\cos (x)\cos (y) =\cos (x - y)\;. }$$ - 2.
The symplectic mapping \(\varphi _{t}: x_{0}\mapsto x(t)\) from the initial conditions x 0 to the phase space point x(t) at time t is referred to as Hamiltonian flow of the system. This was discussed in Sect. 5.4
- 3.
Note that we denoted τ ≡ τ(x 0) in order to emphasize that the recurrence time τ will depend on the initial condition x 0.
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Stickler, B.A., Schachinger, E. (2016). The Double Pendulum. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_6
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DOI: https://doi.org/10.1007/978-3-319-27265-8_6
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