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Silver2007
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- In Landau's mechanics book, I saw them argue that due to the homogeneity of time and space, the isotropy of space leads to the Lagrangian function depending only on v^2. But i want a mathematical proof for the Lagrangian function independent of position q, time t and velocity vector.
In Landau's mechanics book, I saw them argue that due to the homogeneity of time and space, the isotropy of space leads to the Lagrangian function depending only on v^2. But i want a mathematical proof for the Lagrangian function independent of position q, time t and velocity vector.
Homogeneity of space:
We have ## \mathcal{L}(\vec{r}, \vec{\dot{r}}, t) ##, and ## \vec{r} \to \vec{r}' = \vec{r} + \vec{a} ##, because homogeneity of space so equations of motion should be the same. Therefore, the Lagrangian function differs only by a total derivative with respect to time ## \Omega(\vec{r}, t) ##:
$$
\mathcal{L}' = \mathcal{L} + \frac{d}{dt}\Omega(\vec{r}, t)
$$
We assume that ## a \ll 1 ##, and we get:
$$
\vec{a} \cdot \frac{\partial \mathcal{L}}{\partial \vec{r}} = \frac{d\Omega}{dt}
$$
At this point, I argue that if the Lagrangian function depends on position, it will be impossible to find an function ##\Omega(\vec{r}, t)## that satisfies the above equation, so the Lagrangian function cannot depend on position. Is my argument above reasonable and coherent?
And I need help with similar mathematical proofs for temporal homogeneity and spatial isotropy. Thanks
Homogeneity of space:
We have ## \mathcal{L}(\vec{r}, \vec{\dot{r}}, t) ##, and ## \vec{r} \to \vec{r}' = \vec{r} + \vec{a} ##, because homogeneity of space so equations of motion should be the same. Therefore, the Lagrangian function differs only by a total derivative with respect to time ## \Omega(\vec{r}, t) ##:
$$
\mathcal{L}' = \mathcal{L} + \frac{d}{dt}\Omega(\vec{r}, t)
$$
We assume that ## a \ll 1 ##, and we get:
$$
\vec{a} \cdot \frac{\partial \mathcal{L}}{\partial \vec{r}} = \frac{d\Omega}{dt}
$$
At this point, I argue that if the Lagrangian function depends on position, it will be impossible to find an function ##\Omega(\vec{r}, t)## that satisfies the above equation, so the Lagrangian function cannot depend on position. Is my argument above reasonable and coherent?
And I need help with similar mathematical proofs for temporal homogeneity and spatial isotropy. Thanks