Here's periodic motion on a much larger scale - the oscillating Ringworld. Momentum is conserved, because the motion of the Ring is always opposite to the motion of the Sun; the lowest energy state of the system with the same (zero) total momentum is obvious the state when the Sun and Ring are stationary relative to each other. The energy is lost as the Ring oscillates because the gravity of the Sun moves the material on the Ring, which results in friction, heating, and thus energy loss. The people on the Ring could actually increase the energy loss, slowing the oscillation faster, if (for example) they set up hydroelectric plants to exploit the yearly motion of water from one side of the Ring to the other. The energy loss due to deformation of the Sun is probably much less both because the mass of the Ring is only a thousandth of the mass of the Sun, and because a gas deforms with much less energy loss than rock, soil and water do. If both the Sun and the Ring were rigid or completely elastic, no energy would be lost through deformation, and the only factor decreasing the oscillation of the Ring would be very small - gravitational waves and interaction with the cosmic background.
Below are some further details about the oscillating Ringworld, and how it would result in seasons on the Ring.
Here's how to introduce seasons to the Ring in a natural way. If we move the Ringworld so that the Sun is no longer in the plane of the Ringworld, gravity will act as a restoring force. With no friction, oscillation occurs and goes on indefinitely, changing the angle of incidence of light on the Ring - Seasons! The equations for the period of the oscillation aren't very complicated. There is a gravitational force on the Ring and an equal and opposite force on the sun. The Ring mass is about 1/1000th of the Sun, so the Sun's oscillation is a thousand times smaller than the oscillation of the Ring. The period turns out to be about 377 days, giving a reasonable length to the seasons (this analysis relies on a small angle approximation, but would be reasonably accurate for angles up to 10 or so degrees). Note that the period of 377 days is just the same as the period for a planet moving in the Ringworld's orbit. There are only so many ways to combine the gravitational constant, the mass of the sun and the mass of an object to get an answer in units of time, so it's not surprising that the same answer turns up in a few different places. By the way, the mass of the Ringworld is actually significant even compared to the huge mass of the sun, so for the calculation of the oscillation's period I need to include both masses, instead of neglecting the smaller mass, which can be done in most planetary orbit calculations.
An oscillation of 10 degrees gives significant seasonal effects, for two reasons - insolation would vary by 1.5% due to the angle (the same effect that causes seasons on Earth), and 3% due to the changing distance from the sun (the effect that students assume causes seasons on Earth), adding up to a variation of about 5%, which is comparable to the change in insolation over a year due to earth's orientation ( another exercise for the student!). Note that there are two winters and two summers in each cycle - an "up winter," a "down winter," an "up-going summer" and a "down going summer".