Introduction

Often I read or hear people talking about pendulums when it comes to flight stability of rockets or copters.
They try to get the center of gravity as low as possible because it will stabilize due to the pendulum effect.
But there's a very simple reason why this is not true!

The Inverted Pendulum

The most understandable example of an inverted pendulum is balancing a pencil ontop your finger. Everbody has tried it and knows that it's not that easy and you have to train it. 

Here is what I mean:

When we change our few to a mechanical engineering view we can reduce the system.
The pencil has a certain mass that's centered some distance away from the finger tip.

Our finger is a fixpoint. The pencil is connected to it via a joint.

So why is it so hard to control?

Let's have a look at the forces:
At first the pencil has a mass and earth is pulling it down: F_g

After newtons law there has to be a counterforce otherwise it would accelerate. This force (F_f) is created by our fixpoint, the finger. It has to lift the pencil. So everything is in balance -> easy.

But let's see what happens if the pencil is tilted a little bit!

The gravity still pulls vertically but the direction of the joint's force changed. A rod connected via a joint can only transduce forces in direction of the rod. So we have to split the gravity force in two smaller ones, one parallel to the rod and another one perpendicular to the first one. The parallel one gets cancelled by the joint force.
As the other one can't be cancelled it will accellerate the pencil (in the picture to the left). So the pencil starts moving away from the vertical position. It's fixed due to the joint, so it will start to rotate around the joint. The bigger the divertion from the vertical position is the bigger the force/torque that causes it to rotate. This is called unstable.

The normal pendulum

If you turn the pictures around you get the normal pendulum. The gravity acts in the other direction, so do all the other forces. The perpendicular force pulls the pencil back to the vertical position! This is a stable behaviour!

So, where's the difference to a copter or a rocket?!

The rocket/copter motors do the same: They provide thrust along the longitudinal axis. Same as the rod/joint do. This in mind the thrust axis always hits the center of gravity and therefore no torque will be provided even when the rocket/copter tilts. So there's no difference to the pendulums.

Also the gravity acts on the center of gravity -> No torque!

Long story short

The joint is missing. Yes, the rocket/copter will accelerate to the side but due to the missing joint it won't start to rotate. There's no force/moment growing. So even if the rocket/copter gets pushed (gust etc.) away from the vertical position it won't acclerate the rotation. However it will accelerate a linear motion. But it also won't come back to the vertical position. This behaviour is called neutral.

Why do rockets and copter have the same physics?

We're are looking for something that causes the vehicle to start rotating. So there has to be a torque involved. If a body can freely moove, torque can only exist when there's a force that's not hitting the center of gravity. If the force is lower or higher doesn't matter.

So in motion physics there's no difference between a rocket or a copter.

What happens if you lower the CG?

Most people who think of the inverted pendulum try to lower the center of gravity.

Copter

According to the physics nothing. Also all my copter in different configurations (CoG above or below the propellers behaved the same even when the flightcontrols were reduced to a minimum.

Rockets

In model rocketry moving the CoG does have an influence. But this influence origins from aerodynamics not propulsion. Generally moving the CoG more forward( into flight direction) makes a aerodynamic vehicle more stable. Moving it to the back makes them unstable.

Sources:

Pencil Picture: https://tradevistas.org/pencils-still-teaching-us-lessons-about-trade/
Hand: https://www.welt.de/reise/article118369800/Das-sind-die-wichtigsten-Handzeichen-weltweit.html