Question

What is dampers effect and Hamilton principle? As in, what are they used for?

Answer 1

Hamilton's Principle also known as the Principle of Least Action says that for a dynamical system it will progress from its initial state to a final state over a path such that the integral of the Lagrangian which is Kinetic Energy minus Potential Energy over this path will be minimized.
Anecdotally: it was this principle that was explained to Richard Feynman by his HS physics teacher that caused him to become a a physicist .

Answer 2

It is used in theoretical physics to gain a deeper insight into various formalisms. the Hamilton Principle led to the Hamiltonian formulation of classical mechanics an was instrumental In Schroedinger's formulation of quantum mechanics. In theoretical physics it is extremely important.