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fuserofworlds
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I'm currently going over some mechanics notes and am confused about the following situation:
In the book I'm looking at, it describes two particles absent of external forces, only exerting a force on each other. In deriving a potential energy equation for the two, it goes on to say that if the force is conservative and the second particle placed at the origin, one can say $$\vec{F}_{12} = -\vec{\nabla}_1 U(r_1) = -\frac {\partial U(r_1)}{\partial x_1} \hat{x} - \frac {\partial U(r_1)}{\partial y_1} \hat{y} - \frac {\partial U(r_1)}{\partial z_1} \hat{z} $$ where ##x_1, y_1, z_1## are the coordinates of particle 1, and ##\vec{F}_{12}## is the force on particle 1 due to particle 2.
My problem is, conceptually, I don't understand what ##\partial U/\partial x_1## is supposed to mean. What's the difference between ##\partial U/\partial x_1## and ## \partial U/\partial x##? Is there any?
In the book I'm looking at, it describes two particles absent of external forces, only exerting a force on each other. In deriving a potential energy equation for the two, it goes on to say that if the force is conservative and the second particle placed at the origin, one can say $$\vec{F}_{12} = -\vec{\nabla}_1 U(r_1) = -\frac {\partial U(r_1)}{\partial x_1} \hat{x} - \frac {\partial U(r_1)}{\partial y_1} \hat{y} - \frac {\partial U(r_1)}{\partial z_1} \hat{z} $$ where ##x_1, y_1, z_1## are the coordinates of particle 1, and ##\vec{F}_{12}## is the force on particle 1 due to particle 2.
My problem is, conceptually, I don't understand what ##\partial U/\partial x_1## is supposed to mean. What's the difference between ##\partial U/\partial x_1## and ## \partial U/\partial x##? Is there any?
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