Symmetries & Periodic Orbits

The time reversal symmetry across the y = 0 plane can be exploited to find certain periodic orbits. If a trajectory pierces the y = 0 plane such that it is perpendicular to the plane we know that it is "attached" to its symmetric counterpart. If the trajectory again intersects the y = 0 plane perpendicularly then it and its symmetric partner form a loop; a periodic orbit.

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