I editted this question since it was closed because it is a duplicate. However, answers in the referenced question didn't solve my question, so I am writing it again.
Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action.
To know what this function is, action needs to be defined first. Action is defined via integral.
In problems which use calculus of variation ,such as brachistochrone problem, caternary problem or finding path of least distance between two points, appropriate integral is the integral between two points of question of appropriate variable (time, potential energy, distance etc.), that is the integral of variable which is usually needed to be minimized in the problem.
When integral is defined, function is known and with Euler - Lagrange equation we get the solution to the problem. For example that can be function which defines path of least time, distance or shape of the rope as a solution of caternary problem.
What I don't understand is how to define this integral in context of Lagrangian mechanics or in context of action? In another words, how was it found out that difference in kinetic and potential energy of the system gives correct equations of motion when plugged in Euler - Lagrange equation?
