When studying advanced classical mechanics, we all study about Lagrangians and the Euler-Lagrange equations and their importance. Of course, the Lagrangian is calculated based on the potential and kinetic energy. But usually, this is calculated using the position function and some constants. For example, potential energy can be defined as:
$$U(x) = - \int F(x)dx$$
and
$$F = ma \, .$$
This means that potential energy can be defined with mass and acceleration, which is the second derivative of the position function. Of course kinetic energy is defined as
$$(1/2) mv^2 \, .$$
Again, kinetic energy is defined with mass and velocity, the derivative of the position function. As you can see, it is kind of repetition to use the position function to derive the Lagrangian to figure out the dynamics of a system because that is what the position function tells you. So, what is the point of the Lagrangian? I heard from @Sofia that it may be used when only the potential and kinetic energy functions are known. But in what practical experiments it that done? It id much easier to find the position function. So what uses of the Lagrangian are there that can't be done with the position function?
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