What is the meaning of the word principle in Physics? For example in the "action principle". Is it an action law, an action equation, or an unproved assumption? (I have an idea what an action is). What confuses me is when I find the expression as in: "...an action principle is defined..." when a Lagrangian is constructed for a system.
Unfortunately, there is not a standard meaning in physics. Often physicists take the term "principle" with a meaning close to that of "axiom" i.e. as a basic element which applies to any object within the range of applicability of the model/theory. The important difference between mathematicians concept of axiom and physicists concept of principle is that the principles are related to experiments whereas the axioms in math are selected only in basis to formal criteria.
An traditional example of the this meaning of the term "principle" is found in thermodynamics: E.g. first and second principles, although often they are called "laws" of "postulates". Another example is in the common phrase "first-principle computation", which means that you perform a computation in term of basic fundamental properties.
That a physical theory/model with an action functional $S[q]= \int \!dt~L $
- has an action principle,
- has a variational principle,
- fulfills Hamilton's principle,
- or fulfills the principle of stationary action,
means by definition that the classical$^1$ equations of motion for the theory are precisely the Euler-Lagrange equations.
$^1$Here the word classical is used in the sense that Planck's constant $\hbar=0$ is set to zero.
My understanding of physics is that It must order human experience into a logically coherent picture.
The "principle of extremal action" is the logical foundation to understand the behaviour of classical objects. It allows us to single out the correct or experimentally observed description of the object under investigation out of a multitude of possibilities.