It just is.

In classical mechanics you can think of the quantity T - V as a form of energy conservation. If the K.E increases then the P.E must decrease. That quantity is labelled L.

The key concept of Lagrangian mechanics is the Action (S). Defined by: [eqn]S = \int_{t_1}^{t_2} L dt[/eqn]It turns out that in all systems the universe tries to minimise the action - the Principle of Least Action. Why does the universe work that way? *shrug* it just does.

So once you have the form of that minimum S you have the systems equations of motion. Since T and V are themselves functions, S is therefore a function of a set of functions. This is called a 'functional'. To then minimise a functional brings you to the field of mathematics called the Calculus of Variations and the Euler–Lagrange equations (the image in your OP).

So the tl;dr is: Find the Lagrangian of a system. Plug it into the E-L equation. This will then give you systems equation of motion.

It's overkill for say Newtons laws of motion. However the Lagrangian is completely general method, works in all coordinate systems, and in highly complex systems (the Standard Model for example) it actually simplifies the calculations.

>>17040912
Because it works. Physics is experimental science. If the experiment agrees, we use the result.

Trying to understand the "why" before getting a lot of experience in math, physics, and experiments, is a brainlet behavior.

>>17040912
>What is the significance of T-V
not all lagragians are T-V
quite literally the only important quality of the lagragian is that when you shove it into the euler-lagrange equations the equations describing the physical system under study pop out
T-V shoved through the E-L equations poop out newton's laws for mechanical systems

>>17040912
T corresponds to the integral of ma over some infinitesimal displacement
U corresponds to the integral of F over some infinitesimal displacement (assuming conservative forces)
Action is the integral over time of that integral over position.

Basically, action is at a local minimum for whatever path follows F=ma.

>>17040912
Lagrangian mechanics is just a mathematical framework that makes a lot of things convenient by introducing symmetries. It just so happens that T - V is the Lagrangian that gives us good symmetries for Newtonian systems. There isn't really anything else to it, there's no "reason" that it has that form, nor a derivation of the form.

>>17040912
it is T - V for coordinate reasons (position, velocity).
If you go for coordinates of position and momentum, the relevant quantity becomes T+V.

>>17040912
>Why is it minimized or stationary?
Why is F=M A?
Why does force force things
Why does mass be mass

>>17040912
nature favors energy balance (see e.g. virial theorem, biological and ecological systems). take the difference of kinetic (motional) and potential (stationary) energies, then analyze all possible values of these. nature prefers it near zero, and the resulting equations of motion show the dynamic situation in which these become balanced.

to be more precise, these are actually energy differences (change in kinetic) minus (change in potential). only changes in energy are physically meaningful, and any changes in kinetic energy must go to changes in potential energy (ignoring other types of energy). the lagrangian is typically written in a reference frame where some reference kinetic and potential energies are zero, so you simply write the energy terms.

but the core here is lagrangian and associated hamiltonian mechanics are simply writing delta_energy = 0, tracing all possible ways this can occur, and selecting the easiest route (resulting equation of motion).

the last ingredient is why + (hamilton) vs - (lagrange) works, and this is because the reference potential energy is arbitrary. you can either take (reference - true) or (true - reference) and if reference = 0 by convention you get either +true or -true.