This might be a very basic question about the Lagrangian, but I honestly do not know what L actually represents. Let E be a fixed immutable quantity. E can be freely exchanged between T and V, as long as

$$T + V = E$$

1. What does the quantity $$\int_x T - V $$ signify? What is the importance of this quantity?

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Let E now be the budget of a factory. E can either be spent on T or V in any proportion on any given day x.

2. What does the quantity $$\int_x T - V $$ signify for the factory?

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Let E now be the total energy of a system. E can be freely exchanged between the kinetic energy T or the potential energy V as particle moves from point A to point B.

3. What does the quantity $$\int_x T - V $$ signify for the choice of motion of the system?

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Suppose a particle wants to take all possible paths s connecting A to B. Does a weighted quantity of the integral of T - V make sense over infinitely many paths?

4. What does the quantity $$\int_{all} \int_s \frac {T - V}{weight}$$ signify?