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Umaxo

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I am reading Landau-Lifshitz course in theoretical physics 1. volume, mechanics. The mechanics is derived using variatonal principle from the start.

At first they start with point particles, that do not interact with each other. Thus the equations of motions must be independent for the particles and therefore lagrangian might be written as sum of lagrangians of individual particles.

Now, if we assume homogeneity and isotropy of space and homogenity of time, then lagrangian of free particle cannot depend on spatial coordinates, time coordinate, nor can it depend on direction of velocity (independence of lagrangian on higher derivatives is assumed as experimental fact):

$$L=L\left(v^2\right)$$

Then if we use galileo relativity principle we can compare lagrangian in two systems moving relative to each other and we get usual kinetic term.

If we want mechanical system of n noninteracting free particles, we just use additivity of lagrangian. However, if we want particles to be interacting, in the book they subtract term: $$U=U\left(\vec{r}_1...\vec{r}_n\right)$$

This form is good, because it is evident that propagation velocity of interactions is infinite which is needed because of absolutness of time and galileo relativity principle.

As the book goes on, one finds out that explicit dependence on time coordinate means there is some kind of external field, or at least one can interpret it as such.

**Now, I have two questions:**

1.) Does dependence of interacting term on velocities means there is finite propagation velocity of interactions? If not, what would including such dependence mean?

1.) Does dependence of interacting term on velocities means there is finite propagation velocity of interactions? If not, what would including such dependence mean?

**2) The lagrangian in the form of kinetic (free particles) term + interaction term is in the book assumed. Obviously, if interaction term could depend on velocities, one can always write lagrangian in this form. But if not, does it make sense to have lagrangian with more general kinetic term than just free particle lagrangian?**

Thanks :)