# Landau's argument for dependence of Lagrangian on magnitude of velocity

In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the particle.

This seemed fine. But he didn't give a reason as to why it must depend of $|v|^2$ and not on some $|v|^n$ where $n\neq2$.

• Jan 16, 2014 at 13:32

The choice of the dependence is largely arbitrary at that point. In that chapter they just choose lagrangian as $L(v^2)$. If they chose it as $L^\prime(|v|^n)$ in equation $(3.1)$, they'd just have to say $L^\prime(|v|^n)=\frac12 m\sqrt[n]{|v|^n}^2$ in equation $(4.1)$.

Also, this choice is simple enough because it's merely a dot product of velocity with itself, which is the simplest scalar function of a vector.

• Why is there a requirement for simplicity here ?
– user37026
Jan 19, 2014 at 13:22
• Lack of requirement for complexity (i.e. arbitrariness of choice) means freedom to choose simple way. Jan 19, 2014 at 14:30
• That isn't a good enough logical argument. I think.
– user37026
Jan 19, 2014 at 14:53
• @L-L I think Occam's razor is a good enough argument. Jan 19, 2014 at 19:47
• It is good enough for guessing, but once you know that the right answer has been guessed, you must back it up with a more reasonable argument.
– user37026
Jan 24, 2014 at 10:36

Firstly, this is the simplest possible dependence. Second and more important this allows to reproduce the second Newton's law from variation of the action $$m \dot{v}=-\frac{\partial V}{\partial x }.$$ Where $V(x)$ is potential in which the particle is moving.

• Define simple ?
– user37026
Jan 16, 2014 at 9:04
• Could you please reformulate your question? Jan 16, 2014 at 10:21