While solving a particular classical mechanics problem , I was told that for a system of particles to be bound under their mutual forces, their initial energy (With Respect To the COM) must be less than 0. An example of this is maybe when we derive the expression for the escape velocity of a body from a given planet. However in that case we don't explicitly use the COM frame, as such can someone provide a mathematical proof for the above statement? Let me write it out again: Statement: A (closed)system of particles is bound under their mutual forces if and only if their initial energy WRT center of mass is less than 0.
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$\begingroup$ Whoever told you that must have the proof. If not, ignore. $\endgroup$– my2ctsCommented Jan 24 at 17:12
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$\begingroup$ Classical mechanics is invariant to an offset in energy. This question makes no sense $\endgroup$– basicsCommented Jan 24 at 17:28
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$\begingroup$ How do you define "energy w.r.t. COM"? Do you mean that the COM is the datum (reference level) for GPE? $\endgroup$– StutiCommented Jan 24 at 17:44
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$\begingroup$ The designation of zero total energy is arbitrary. Due to that, it was noted that the current designation (e.g., negative total energy means a bound system) is more convenient and intuitive than other designations. Also note that differences in energy are normally what people are concerned with, so defining something as having negative total energy does not affect the results of most calculations. $\endgroup$– David WhiteCommented Jan 25 at 0:29
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2 Answers
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A classical two body system is bound if in the rest frame the sum of potential and kinetic energy is negative. For an n>2 body system stability is not guaranteed by this condition. (Gut feeling, I have no proof). Up to n-2 bodies can in principle escape. The solar system for example is not guaranteed to be stable.
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The question doesn't make too much sense. but if I were to work with the scenario provided in the question, I would say that if the only forces on the system were internal forces(mutual forces), then the initial momentum of the system would be equal to the final momentum.
And how exactly do you intend to explain the negative energy of the system?
Statement: A (closed)system of particles is bound under their mutual forces if and only if their initial energy WRT center of mass is less than 0.
This is 110% wrong. If we were to consider a system of 2 identical blocks connected by a unstretched spring and 1 block has a velocity $v$ towards the other. This is a closed system and the blocks are only bound by internal forces. here w.r.t center of mass, the total energy of system $= \frac{1}{2}mv^2 > 0$.
Now what you're doing is going to that person and asking a clarity or a proof for the statement.
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$\begingroup$ I asked for the case when the total energy is less than 0, in your case it's greater ig? $\endgroup$ Commented Jan 25 at 13:45
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$\begingroup$ If you take the internal energy of the spring into account this example fails. $\endgroup$– my2ctsCommented Jan 25 at 15:55