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Let's say I have two positive charges approaching one another at the same speed with only their mutual forces acting on one another. Total momentum (= 0) and energy is conserved and the charges eventually move away from one another at the same velocity.

Now suppose an external force acts on one of the charges to keep it stationary while the other one is free to move as before. Even though energy is still conserved, the total momentum only comes from one of the charges, and therefore isn't balanced by an equal and opposite momentum from the other charge to keep the total momentum zero:

What is it about momentum and energy that allows a stationary force to affect the conservation of one, but not the other?

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I think the momentum actually may be conserved still, if I'm understanding your scenario correctly.

So we have two positive charges shot towards each other (so $P_{tot}\neq 0$), and if perfectly centered and aimed, they slow down as they approach each other, and eventually halt, then turn around and accelerate away, so momentum is conserved at every point and energy is converted from kinetic to potential and back to kinetic.

You're saying that, if you "held" one of the charges with some really strong yet delicate tweezers or something so that it couldn't move, and shot the other charge at it, it will slow down as it approaches the "held" charge, then turn around and accelerate the other way. So, the shot charge will end up with the same kinetic energy it started with, but its momentum has totally reversed.

But, whatever is holding the stationary charge (I used some hypothetical mechanical tweezers here, but I suspect it's the same no matter what produces the force) isn't stationary itself; nothing is truly immovable. So, the shot charge would push the "held" charge which would push you and your tweezers back the tiniest bit (because you're so massive), but it would still be enough to conserve momentum.

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  • $\begingroup$ Also, mechanical energy is second order in $v$, mechanical momentum first order. $\endgroup$
    – Larry Harson
    Commented Sep 24, 2013 at 16:09
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Comments to the question (v1): Normally the word external implies that (whatever is external) belongs to the environment/outside and is not part of the physical system. In this context the physical system consists of two particles. The external force may change the total momentum of the physical system.

A) The reason of mechanical energy conservation (of the physical system) in the presence of an external force on particle 1, is because OP has imposed the extra constraint that particle 1 remains at a fixed position. Hence the external force does no work, and the mechanical energy is conserved.

B) On the other hand, if OP removes the constraint (that particle 1 remains at a fixed position), then the external force may produce a non-zero work, and the mechanical energy may also not be conserved.

Lagrangian of system B:

$$L~=~\frac{1}{2}m_1 \dot{\bf r}_1^2 + \frac{1}{2}m_2 \dot{\bf r}_2^2 - k_e \frac{Q_1Q_2}{|{\bf r}_1-{\bf r}_2|}+ {\bf r}_1\cdot{\bf F}_{\rm ext}, \qquad k_e~:=~\frac{1}{4\pi\varepsilon_0}. $$

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