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# The averaged Kepler problem | ||
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``` | ||
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We consider the Hamiltonian | ||
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$$ | ||
H(r, \theta, p_r, p_\theta) = v p_\theta + \Vert p \Vert_{g} | ||
$$ | ||
`` | ||
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 | ||
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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$$ | ||
g = \mathrm{d}r^2 + m_\lambda^2(r)\, \mathrm{d}\theta^2, \quad | ||
m_\lambda^2(r) = \frac{\sin^2 r}{1 - \lambda \sin^2 r} | ||
$$ | ||
`` | ||
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 $\lambda = 4/5$. | ||
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). | ||
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 $\alpha_0$: | ||
$$ | ||
p_r = \sin \alpha_0, \quad p_\theta = m_\lambda(r) \cos \alpha_0. | ||
$$ | ||
**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 | ||
$$ | ||
p_\theta\, v + 1 = v\, m_\lambda(r) \cos \alpha_0 + 1 > 0. | ||
$$ | ||
``` | ||
`` | ||
p_\\theta\\, v + 1 = v\\, m_\\lambda(r) \\cos \\alpha_0 + 1 > 0. | ||
`` |