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# Averaged Kepler problem | ||
# The averaged Kepler problem | ||
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We consider the Hamiltonian | ||
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$$ | ||
H(r, \theta, p_r, p_\theta) = v p_\theta + \Vert p \Vert_{g} | ||
$$ | ||
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where $v$ is a constant, $p = (p_r, p_\theta)$, and $\Vert \cdot \Vert_{g}$ is the norm induced by the metric | ||
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$$ | ||
g = \mathrm{d}r^2 + m_\lambda^2(r)\, \mathrm{d}\theta^2, \quad | ||
m_\lambda^2(r) = \frac{\sin^2 r}{1 - \lambda \sin^2 r} | ||
$$ | ||
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with $\lambda = 4/5$. | ||
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Along the geodesics, we have $H+p^0 = 0$. The parameter $p^0$ is constant equal to $-1$ (hyperbolic), $0$ (abnormal) or $1$ (elliptic). | ||
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**Remark.** We can parameterize the geodesics by the norm of the initial convector, setting $\Vert{p_0}\Vert_g = 1$. | ||
This amounts to parameterize by the initial angle $\alpha_0$: | ||
$$ | ||
p_r = \sin \alpha_0, \quad p_\theta = m_\lambda(r) \cos \alpha_0. | ||
$$ | ||
In that case, the hyperbolic geodeics are given by | ||
$$ | ||
p_\theta\, v + 1 = v\, m_\lambda(r) \cos \alpha_0 + 1 > 0. | ||
$$ |