Lagrangian mechanics -- Initial questions

Hi,

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?

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 :)

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?

That doesn't necessarily mean that interactions propagate at a finite velocity. For now you could interpret such dependence as the presence of a force which is stronger or weaker depending of the particles' velocity. When you get to non-inertial frames of reference, you will learn about Coriolis force, which is velocity-dependent and can be derivated from a potential where v appears explicitly, and there is no finite popagation velocity to speak about.

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?
The lagrangian is actually quite indetermined, so there comes a point where you can interpret its terms as you please as long as you get the correct equations of motion out of it.

Umaxo
When you get to non-inertial frames of reference, you will learn about Coriolis force, which is velocity-dependent and can be derivated from a potential where v appears explicitly, and there is no finite popagation velocity to speak about.

Make sense... But in the book, they assume to be in frame of reference in which space and time are homogeneous and isotropic. Doesnt this have any influence on your discussion?

But in the book, they assume to be in frame of reference in which space and time are homogeneous and isotropic. Doesnt this have any influence on your discussion?

Ok i thought about it and this question is no longer relevant for me. However, another quetion arose in me. Sadly i dont have a lot of time right now. but i will be back soon:)