Today, I read a line in Goldstein Classical mechanics and got confused about one line.

To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance $|\vec{r}_i-\vec{r}_j|$ between the particles: $V_{ij} = V_{ij}(|\vec{r}_i-\vec{r}_j|)$.

What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?

I prefer to receive mathematical proof (not thorough, but provide a direction so that I can know where I'm going); yet, intuitive illustration is also welcome.

Today, I read a line in Goldstein Classical mechanics and got confused about one line.

To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the particles: $V_{ij} = V_{ij}(|{\bf r}_i-{\bf r}_j|)$.

What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?

I prefer to receive mathematical proof (not thorough, but provide a direction so that I can know where I'm going); yet, intuitive illustration is also welcome.

Today, I read a line in Goldstein Classical mechanics and got confused about one line. "To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the particles: $V_{ij} = V_{ij}(Abs[r_i-r_j])$".

What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?

I prefer to receive mathematical proof(not thorough, but provide a direction so that I can know where I'm going);yet, intuitive illustration is also welcome.

Thanks a lot!

Today, I read a line in Goldstein Classical mechanics and got confused about one line.

To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance $|\vec{r}_i-\vec{r}_j|$ between the particles: $V_{ij} = V_{ij}(|\vec{r}_i-\vec{r}_j|)$.

What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?

I prefer to receive mathematical proof  (not thorough, but provide a direction so that I can know where I'm going); yet, intuitive illustration is also welcome.

Today, I read a line in Goldstein Classical mechanics and got confused about one line. "To satisfy the strong law of action and reaction, Vij can be a function only of the distance between the particles: Vij = Vij(Abs[ri-rj])".

What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?

I prefer to receive mathematical proof(not thorough, but provide a direction so that I can know where I'm going);yet, intuitive illustration is also welcome.

Thanks a lot!

Today, I read a line in Goldstein Classical mechanics and got confused about one line. "To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the particles: $V_{ij} = V_{ij}(Abs[r_i-r_j])$".

What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances?

I prefer to receive mathematical proof(not thorough, but provide a direction so that I can know where I'm going);yet, intuitive illustration is also welcome.

Thanks a lot!