How Does Homogeneity of Space and Time Affect Lagrangian Mechanics?

[Andrea Vironda]

Hi,
i know that The homogeneity of space and time implies that the Lagrangian cannot contain
explicitly either the radius vector r of the particle or the time t, i.e. L must be a function of v only
but the lagrangian definition is $L=\int L(\dot q,q,t)$, so velocity appears in the definition and it's in contrast with $L=L(v^2)$
why?

 

[Chandra Prayaga]

The definition is general, and not just for homogeneous space and time. For example, in a uniform gravitational field, the Lagrangian of a particle does depend on position, through the potential energy (mgy). The Lagrangian of a free particle does not depend on position. It also does not depend on direction (isotropy) and depends only on the square of the speed.

 

[Metmann]

Why should $L(q,\dot{q},t)$ and $L(v^2)$ be in contrast at all? The latter sais, that $L$ in that specific case depends on $v=\dot{q}$ only via $v^2$ and that the dependence on $q$ and $t$ drops, so $L(v^2)$ is a specific restriction of the most general case $L(q,\dot{q},t)$, but not in contrast with it.