# Hamiltonian for a system of 3 interacting particles and the meaning of potential

I'd like to discuss some physics basic.

assume we have 3 particles $$\{\vec{q_i},\vec{p_i}\}, \quad i=1,2,3$$

whereas $$\vec{q_i}$$ is the position and $$\vec{p_i}$$ is the momentum.

We also have a "pair-potential" $$U(\vec{q_i},\vec{q_j})$$ between the particles $$i$$ and $$j$$.

Now the hamiltonian is given by:

$$H(\vec{q_i},\vec{q_j})=\sum_i \frac{\vec{p_i}}{2m}+\sum_{i < j}U(\vec{q_i},\vec{q_j}) \tag{1}$$

Now, I'd the second term in (2) is a sum over $$i. That's because we don't want to add up the potential between two particles twice. Which let me to think about, what the potential actually describes. I always thought of it as a "the ability to do work [work in a physics sense]". If you take 1 Liter of water at height 1m you can let it flow down and "do something with it". If you put it at 10m, you could do even more. Now what's exactly going on here? I have don't like my understanding of a situation like here.

Could anyone try to interpret the meaning of the potential as given here?

To be precise, this is not a potential, but the potential energy of two particles interacting. Let us take for example two charged particles with charges $$q_1,q_2$$, separated by vector $$\mathbf{r}_{2} - \mathbf{r}_{1}$$. The potential created by the first particle is $$\varphi_1(\mathbf{r}) = \frac{q_1}{|\mathbf{r} -\mathbf{r}_1|},$$ whereas the potential energy of the second particle in the field created by the first is $$U_2(\mathbf{r}_2) = \frac{q_1q_2}{|\mathbf{r_2} -\mathbf{r}_1|}.$$ One can repeat the same reasoning by considering the potential energy of the first particle in the field of the second with the same result. So it is reasonable to ignore which one is the first and which is the second and simply speak of the potential of their interaction: $$U_{12}(\mathbf{r}_1, \mathbf{r}_2) = U_1(\mathbf{r}_1) = U_2(\mathbf{r}_2) = \frac{q_1q_2}{|\mathbf{r_2} -\mathbf{r}_1|}.$$