Re: Double counting gravitational potential energy

On 14 Feb, 18:27, Jonathan Scott <jonathan_sc...@xxxxxxxxxxxx> wrote:
I thought I knew quite a lot about gravity, but I've just totally
confused myself when thinking about potential energy in good old
Newtonian gravity.

It's a standard result in Newtonian gravity that if two masses m_1 and
m_2 are moved apart from distance r_1 to distance r_2, then the total
energy transferred to the masses is -G m_1 m_2 (1/r_2 - 1/r_1). An
obvious way to see this is to integrate the force between them over
the distance moved.

However, as far as I can see, the potential energy of each mass
changes by the exactly the same amount. Each mass is effectively
moved to a different potential within the field of the other. That
means that the total change in the potential energy seems to be
exactly twice the work done to move the masses apart. This doesn't
seem to add up unless some other energy is transferred from somewhere
else at the same time, for example out of the field, but I'm not aware
of this normally being considered necessary in a Newtonian model.
Have I missed something obvious?

Hi Jonathan,

Your expression -G m_1 m_2 (1/r_2 - 1/r_1) is the potential energy
change of one particle, not the total energy change (as you are
claiming). So, assuming the kinetic energies remain constant, the
total energy change for both particles is twice this values i.e. -2*G
m_1 m_2 (1/r_2 - 1/r_1). So everything adds up as it should.

Thomas

.

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