PX285 - E2 - translation of space

space is homogenous if the motion of a particle is independent of its absolute position in space, ie: the lagrangian, $L (\vec{r}, \vec{v}) = L (q_{1}, \dots, {\dot{q}}_{1}, \dots)$ , is unchanged under an arbitrary translation
under a small translation of $\vec{r} \to \vec{r} + δ \vec{r}$ :

L (\vec{r}, \vec{v}) \to L (\vec{r} + δ \vec{r}, \vec{v}) = L (x + δ x, y + δ y, z + δ z, \vec{v})

using the taylor expansion:

L (\vec{r} + δ \vec{r}, \vec{v}) = L (\vec{r}, \vec{v}) + \frac{\partial L}{\partial x} δ x + \frac{\partial L}{\partial y} δ y + \frac{\partial L}{\partial z} δ z

δ L = \frac{\partial L}{\partial x} δ x + \frac{\partial L}{\partial y} δ y + \frac{\partial L}{\partial z} δ z = \vec{\nabla} L \cdot δ \vec{r}

if space is homogenous, there should be no change in the lagrangian under translations:

δ L = 0 ⟹ \frac{\partial L}{\partial x} = \frac{\partial L}{\partial y} = \frac{\partial L}{\partial z} = 0

recalling the euler-lagrange equations:

\begin{aligned} \frac{d}{d t} \frac{\partial L}{\partial \dot{x}} & = \frac{\partial L}{\partial x} = 0 \\ \frac{d}{d t} \frac{\partial L}{\partial \dot{y}} & = \frac{\partial L}{\partial y} = 0 \\ \frac{d}{d t} \frac{\partial L}{\partial \dot{z}} & = \frac{\partial L}{\partial z} = 0 \\ {\dot{p}}_{x} = {\dot{p}}_{y} & = {\dot{p}}_{z} = 0 \end{aligned}

the canonical momenta remain constant during the motion
for a single particle, this is also the total momentum of the system

generalized

for $N$ particles, each at position, ${\vec{r}}_{a}$ , with velocity, $v_{a} :$

δ L = \sum_{a} (\frac{\partial L}{\partial x_{a}} δ x + \frac{\partial L}{\partial y_{a}} δ y + \frac{\partial L}{\partial z_{a}} δ z) = \sum_{a} \frac{\partial L}{\partial {\vec{r}}_{A}} \cdot \vec{δ r}

the condition of homogeneity of space is that $δ L = 0 :$

\sum_{a} \frac{\partial L}{\partial {\vec{r}}_{a}} = 0 ⟹ \sum_{a} \frac{d}{d t} \frac{\partial L}{\partial {\vec{v}}_{a}} = 0

for any closed system, the sum of the canonical vector momenta, $\vec{P} = \sum_{a} \frac{\partial L}{\partial {\vec{v}}_{a}}$ , remains constant