I guess this question has been asked before, but I'm looking at a slightly different aspect.
I'm reading Landau's book on classical mechanics. In deriving the lagrangian for a free particle, I understand the significance of the homogeneity of space and time, and isotropic nature of space with respect to time evolution of the system. But I am a little unclear as to how this affects the lagrangian.
For example, in an inertial frame, the homogeneity of space means that if you translate positions of all particles at an instant of time, the future positions are also translated by that same amount. But what's the mathematical logic behind arriving at the conclusion that the lagrangian does not depend on the trajectories $x(t),y(t),z(t)$?
In the lagrangian formalism, $S = \int_{t_1}^{t_2} L(q,\dot{q},t)\,dt$ , the boundary condition is on the initial and final positions.
Using this view or the Euler-Lagrange equations, could someone explain the mathematics used to arrive at the conclusion here as well as the other two cases - independency of lagrangian on time and direction of velocity?