# Correlation between equations of elliptical orbits and pendulums

The equation for the period of a pendulum is: $$T=2π\sqrt{\frac{L}{g}}$$ Where 'g' is the acceleration due to the gravitational field and 'L' is the length.

The equation for the period in of a body travelling along an elliptical orbit is: $$T = 2π\sqrt{\frac{a^3}{GM}}$$ Where 'a' is the semi-major axis. I can see that this is derived from Kepler's 3rd law.

Is there are similar equation to Kepler's 3rd law for pendulum periods? If a pendulum string is of fixed length, does that essentially make the motion of a circular orbit? It seems like a very similar relationship between length, gravity and periods.

My question: "Is there a deeper, more fundamental relationship between these equations?

The equation for the period of a pendulum $(T=2π\sqrt{\frac{L}{g}})$ is only an approximation. That equation assumes, among other things, that gravity doesn't change with height, and that $\sin(\theta) = \theta$.
These equations correspond to the period of a harmonic motion, which generally have the form $$T=2 \pi \sqrt{\frac{m}{k}}$$