# Goldstein equation 1.33

I am trying to read from Goldstein for self-study but I am stuck on equation 1.33. Let me restate some of the lines from Goldstein (with some modification):

If $$\textbf{F}_{ij}$$ (internal force, force exerted on particle $$i$$ by particle $$j$$) depends only on the relative positions $$\textbf{r}_{ij}$$ and can be derived from a scalar potential energy function $$V_{ij}(\textbf{r}_{ij})$$ with $$V_{ij}=V_{ji}$$ then

$$\textbf{F}_{ij}=-\nabla_{i}V_{ij}$$ and

$$\textbf{F}_{ij}=-\nabla_{i}V_{ij}=+\nabla_{j}V_{ij}=+\nabla_{j}V_{ji}=-\textbf{F}_{ji}.\tag{1.33}$$

Now, I am not able to understand how I can prove $$-\nabla_{i}V_{ij}=+\nabla_{j}V_{ij}$$. After trying a lot I think the problem is that probably I didn't understand $$\textbf{F}_{ij}=-\nabla_{i}V_{ij}$$ properly. Though I do understand that for conservative forces we can express the force as negative gradient of potential. But I guess the indices here is what I don't understand. My understanding of equation $$\textbf{F}_{ij}=-\nabla_{i}V_{ij}$$ is that we are taking gradient of $$V_{ij}$$ with respect to the coordinates of the $$i$$th particle. But then I don't know how to prove $$-\nabla_{i}V_{ij}=+\nabla_{j}V_{ij}$$. Please help.

• this is newton's third axiom for pair forces en.m.wikipedia.org/wiki/Newton%27s_laws_of_motion. does this help? Jun 19, 2021 at 17:35
• I am aware of the Newton's third law, but its the other way around here, I think. If we assume the form $\textbf{F}_{ij}=-\nabla_{i}V_{ij}$ then this ensures the validity of the weak version of action-reaction law. Jun 19, 2021 at 17:40

Newton's third axiom states $$F_{ij}=-F_{ji}$$. The potential energy between the particles is the pair potential satisfying $$V(r_{ij})=V(r_{ji})$$

The force acting on particle i due to particle j is given by $$F_{ij}=-\nabla_{i}V_{ij}$$

and the force acting on particle j due to particle i is $$F_{ji}=-\nabla_{j}V(r_{ij})$$

And now use Newton's third axiom to obtain

$$F_{ij}=-\nabla_{i}V_{ij}=-F_{ji}=\nabla{j}V(r_{ij})$$

I hope this helps

• Thanks for answering, but I have found the answer, I was looking for something like this physics.stackexchange.com/a/553659/286407 Jun 19, 2021 at 18:20
• you are right this answer is much more rigorous Jun 19, 2021 at 18:25