What physics describe the way a waiter/ waitress is balancing his/her serving tray? A waitress in order to balance his/her tray is continuously moving this tray up and down.
Similarly if you want to balance a pencil by its tip on your hand, you move your hand up and down.
Does anybody knows the physics and the mathematical description of this phenomenon?
 A: I think the best explanation for this can be given by the concept of center of mass.  
The center of mass is a point at which the entire mass of the object can be assumed to be concentrated, in order to simplify calculations. For example, in order to calculate the force of gravity on the earth by the moon, you would have to vectorially add all the small components of forces across the surface of the earth. But if we assume the earth to be a point object with its entire mass at its center then it's quite a simple calculation. The center of the earth is its center of mass.  
Similarly, the waiter's table, or the pencil, have their center of masses, usually at the center of symmetry. In order to balance something on a point, the center of mass must be directly above the point.  
So to balance a pencil on its tip, we would have to place it exactly vertical, (so that the center of mass is above the tip). But even the slightest variation and the location of the center of mass can change. When this happens, the pencil will start to fall.
But since you are clever, you would move your hand in the appropriate direction and for a moment again, the center of mass would be above the tip. Once again you move your hand appropriately and for this time too, will the center of mass be above the tip briefly.  
In order to balance the pencil on your hand on its tip without moving your hand,you would have to minimize any movement at all. Even the air particles in the air can knock the pencil down. Here is a useful video by minutephysics
