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  "body": "\n## Example\n\nLet $k \u003e 0$.\nThe Kepler problem is the system\n$$r'' = -\\frac{k\\, r}{\\norm{r}^3}$$\nfor a position $r(t) \\in \\R^3 \\setminus \\set{0}$, with velocity $v = r'$.\nIt describes a body attracted to a fixed centre by an inverse-square force, such as a planet around the Sun.\nIt is the [[Hamiltonian system|0EDKGA2FAK]] with $H(r, v) = \\frac{1}{2} \\norm{v}^2 - \\frac{k}{\\norm{r}}$.\n\nThree quantities are [[first integrals|0262JP0NV9]]:\n\n1. the energy $E = \\frac{1}{2} \\norm{v}^2 - k / \\norm{r}$;\n2. the angular momentum $L = r \\times v$;\n3. the Laplace–Runge–Lenz vector $A = v \\times L - k\\, r / \\norm{r}$.\n\nCheck for $L$: $L' = v \\times v + r \\times r'' = 0 - \\frac{k}{\\norm{r}^3}\\, r \\times r = 0$.\n\n## Remarks\n\nSince $L$ is constant, a motion with $L \\ne 0$ stays in the plane through the origin orthogonal to $L$.\nIn polar coordinates in that plane, with $\\theta$ measured from the direction of $A$, the orbit is\n$$\\norm{r} = \\frac{\\norm{L}^2 / k}{1 + e \\cos \\theta}, \\qquad e = \\norm{A} / k.$$\nSo the orbit is a [[conic section|00HJCMC2CD]] with a [[focus|0RBKKNH4GV]] at the origin and eccentricity $e$.\nIt is an [[ellipse|0VKCCEXBJ1]] when $E \u003c 0$, a parabola when $E = 0$, and a branch of a hyperbola when $E \u003e 0$.\nFor an ellipse with semi-major axis $a$, the period $T$ satisfies $T^2 = 4 \\pi^2 a^3 / k$, which is Kepler's third law.\n\n## Explanations\n\n### L1 human\n\nThe Kepler problem asks how a body moves when it is pulled toward a fixed point with a force that falls off like the square of the distance.\nThe answer is that it moves on an ellipse, a parabola or a hyperbola, with the centre of force at a focus.\nThis explains Kepler's laws of planetary motion from Newton's law of gravitation.\n",
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