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Noether's theorem suggests that every differential symmetry of a physical system has a corresponding conservation law. For example, a free particle in space has translational symmetry and therefore its linear momentum is conserved. Also a satellite revolving around a planet in a circular orbit has a rotational symmetry and hence its angular momentum is conserved about its centre of rotation. My question is that if we take the elliptical orbit of planetary motion instead of circular motion, its angular momentum is also conserved about an axis on one of its foci and perpendicular to the plane of motion. What is the associate continuous symmetry of this system?

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  • $\begingroup$ Angular momentum is conserved not because the orbit is circular, but because space has a continuous rotational symmetry. It doesn't matter what the shape of the orbit is. Angular momentum is conserved in any closed system around any axis with any motion of its parts. What matters is the symmetry of space, not the symmetry of the motion. For example, a curved space may not have the rotational symmetry, so proving conservation of angular momentum there may be a challenge. $\endgroup$ – safesphere Aug 3 '18 at 3:47
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In general, there are 3 types of symmetries:

  1. Symmetry of an action.

  2. Symmetry of an EOM.

  3. Symmetry of solutions to an EOM.

They are not necessarily the same cf. this Phys.SE post. E.g. 3D rotations are not a symmetry of an elliptic orbit, but they are a symmetry of the action for the Kepler problem. Noether's theorem then predicts that the angular momentum is conserved on-shell.

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If I understand your geometry properly, its that the semi-major axis can be rotated around your symmetry axis without changing $\vec L$ or $E$.

For an excellent exposition of symmetry and orbital motion, see John Baez's web page at UCR.

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