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# Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

• * Corresponding author: José Laudelino de Menezes Neto
• We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.

Mathematics Subject Classification: Primary: 70F15, 34D20; Secondary: 70H14.

 Citation: • • Figure 1.  Double pendulum in an elliptical orbit. Orbital plane

Figure 2.  Graphics of: (1) $\beta^{(2)}(e)$, (2) $\beta^{(3)}(e)$, (3) $\beta^{(5}(e)$, (4) $\beta^{(6)}(e)$, (5) $\beta^{(7)}(e)$, (6) $\beta^{(8)}(e)$ in the plane $e\times \beta$

Figure 3.  Graphics of: (1) $\beta^{(1)}(e)$ and (2) $\beta^{(4)}(e)$ in the plane $e\times \beta$

Figure 4.  Regions of stability and instability for the equilibrium point $E_2$. Graphics of (1): $\beta^{(1)}(e)$, (2): $\beta^{(2)}(e)$, (3): $\beta^{(3)}(e)$, (4): $\beta^{(4)}(e)$, (5): $\beta^{(5)}(e)$, (6): $\beta^{(6)}(e)$, (7): $\beta^{(7)}(e)$, (8): $\beta^{(8)}(e)$

Figure 5.  Regions of stability and instability for the equilibrium point $E_3$. Graphics of (1): $\beta^{(1)}(e)$, (2): $\beta^{(2)}(e)$, (3): $\beta^{(3)}(e)$, (4): $\beta^{(4)}(e)$, (5): $\beta^{(5)}(e)$, (6): $\beta^{(6)}(e)$, (7): $\beta^{(7)}_+(e)$, (8): $\beta^{(7}_-(e)$, (9): $\beta^{(8)}(e)$, (10): $\beta^{(9)}(e)$

Figure 6.  Regions of stability and instability for the equilibrium point $E_3$. Graphics of $\beta^{(7)}_+(e)$ and $\beta^{(7)}_-(e)$. The shaded region is where the equilibrium point is unstable

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