- #1

etotheipi

*pairs*of particles, so that the total conservative force (not necessarily total) on any given particle is $$\mathbf{F}_i = \sum_{j\neq i} -\nabla_i V_{ij}$$as a sum over all other particles ##j##, evaluated at the position of ##i##. Also, ##V_{ij} = V_{ij}(\mathbf{r}_i - \mathbf{r}_j)## depends only on the relative separation of both particles.

I wondered if it were ever possible to find a total potential energy function, which would have the property that if ##V = V(\mathbf{r}_1, \mathbf{r}_2, \dots \mathbf{r}_n)##, then that same force ##\mathbf{F}_i## would be $$\mathbf{F}_i = -\nabla_i V$$ for any choice of particle ##i##? ##\nabla## is a linear operator so it seems reasonable to suggest ##V = V_{12} + V_{13} + \dots + V_{n-1, n}##, but I'm not sure if it is possible to evaluate the derivatives of e.g. the potential energy between particles 3 and 4, at the position of particle 2 (e.g. does ##\nabla_2 V_{34}## make sense?), although this would have to be zero.

Thanks!