- #1

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I don't see why the last sentence in the attachment is true. It claims that if ##V_{ij}## or simply ##V## is also a function of the difference of velocities of particles ##i## and ##j##, then the force derived from ##V## is not central. In other words, if ##V=V(|r_i-r_j|, |v_i-v_j|)##, then (1.34) is not satisfied.

Let ##p=|r_i-r_j|## and ##q=|v_i-v_j|## and ##\partial_x=\frac{\partial}{\partial x}##.

##\nabla_iV(p, q)=\partial_{x_i}V(p, q)\vec{e_x}+\partial_{y_i}V(p, q)\vec{e_y}+\partial_{z_i}V(p, q)\vec{e_z}##. Definition of ##\nabla_i## is given by the first sentence of the attachment.

Since ##\partial_{x_i}V(p, q)=\partial_{x_i}V(p)##, the RHS above should be the same as the RHS of (1.34). Then, (1.34) will be satisfied.

##\partial_{x_i}V(p, q)=\partial_{x_i}V(p)## for the same reason as ##\partial_x(x+\dot{x}^2)=1=\partial_xx##.

EDIT: I found the mistake. ##\partial_{x_i}V(p, q)\neq\partial_{x_i}V(p)## for the same reason as ##\partial_x(x+x\dot{x})\neq\partial_xx##.

Let ##p=|r_i-r_j|## and ##q=|v_i-v_j|## and ##\partial_x=\frac{\partial}{\partial x}##.

##\nabla_iV(p, q)=\partial_{x_i}V(p, q)\vec{e_x}+\partial_{y_i}V(p, q)\vec{e_y}+\partial_{z_i}V(p, q)\vec{e_z}##. Definition of ##\nabla_i## is given by the first sentence of the attachment.

Since ##\partial_{x_i}V(p, q)=\partial_{x_i}V(p)##, the RHS above should be the same as the RHS of (1.34). Then, (1.34) will be satisfied.

##\partial_{x_i}V(p, q)=\partial_{x_i}V(p)## for the same reason as ##\partial_x(x+\dot{x}^2)=1=\partial_xx##.

EDIT: I found the mistake. ##\partial_{x_i}V(p, q)\neq\partial_{x_i}V(p)## for the same reason as ##\partial_x(x+x\dot{x})\neq\partial_xx##.

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