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Physics
Classical Physics
Mechanics
Prove forces derived from a velocity-dependent potential are not central
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[QUOTE="Happiness, post: 5474284, member: 549579"] Nonetheless, I am still able to prove that the force derived from ##V=V(p, q)## satisfies (1.34), contrary to what is claimed by the book. Recall that ##p=|r_i-r_j|=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2}## and ##q=|v_i-v_j|##. Thus, ##\frac{\partial p}{\partial x_i}=\frac{1}{2p}2(x_i-x_j)=\frac{1}{p}(x_i-x_j)##. Let ##V'(p, q)=\partial_{p}V(p, q)##. ##\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}## ##=V'(p, q)\frac{\partial p}{\partial x_i}\vec{e_x}+V'(p, q)\frac{\partial p}{\partial y_i}\vec{e_y}+V'(p, q)\frac{\partial p}{\partial z_i}\vec{e_z}## ##=\frac{V'(p, q)}{p}(x_i-x_j)\vec{e_x}+\frac{V'(p, q)}{p}(y_i-y_j)\vec{e_y}+\frac{V'(p, q)}{p}(z_i-z_j)\vec{e_z}## ##=\frac{V'(p, q)}{p}(\vec{r_i}-\vec{r_j})##, which is equivalent to the RHS of (1.34). What's wrong? [/QUOTE]
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Physics
Classical Physics
Mechanics
Prove forces derived from a velocity-dependent potential are not central
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