We know that a body under the action of a Newtonian/Coulomb potential $1/r$ can describe an elliptic orbit. On the other hand, we also know that a body under the action of two perpendicular Simple Harmonic Motions can also have an elliptic orbit. Hence I was wondering if we can differentiate between a body under the influence of a central potential $1/r$ and a body under the action of two perpendicular SHM’s just by observing the orbits without prior knowledge of the potential they are under. So my question is how can we differentiate between these two potentials?
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Qmechanic
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So you are essentially asking if we can determine the potential just based on the motion of a body in that potential? Or in another way, does the path taken by a body determine a unique potential, and if not then how do we differentiate between them? – BioPhysicist Feb 02 '19 at 13:40
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Yes exactly. I want to know ("does the path taken by a body determine a unique potential, and if not then how do we differentiate between them?"). – Pratik Chowdhury Feb 02 '19 at 13:48
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different but related: If the gravitational force were inversely proportional to distance (rather than distance squared), will celestial bodies fall into each other? – uhoh Feb 06 '21 at 05:05
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Your two examples are both central forces. For gravity the potential is:
$$ U_g = -\frac{k}{r} $$
while for the simple harmonic motion the potential is:
$$ U_s = kr^2 $$
Both of these allow circular orbits,and for a circular orbit you cannot tell which is which. However for an elliptical orbit you can because with gravity the origin of the force is at one focus of the ellipse with for SHM the origin of the force is at the centre of the ellipse.
As a side note: these are the only two potentials that have closed orbits. This is Bertrand's theorem. The behaviour is also different with respect to the virial theorem. For the gravitational potential the average values of the kinetic energy $T$ and the potential energy $V$ are linked by:
$$ 2T = -V $$
while for the SHM potential we get:
$$ T = V $$
John Rennie
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But if you knew the mass of the object and the value of $k$ then you could differentiate between the potentials in the circular orbits by just calculating the force from the acceleration and other relevant parameters. – BioPhysicist Feb 02 '19 at 16:34
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@AaronStevens Yes, but I'm assuming we known nothing about the potentials so we do not know the value of $k$ for either potential. – John Rennie Feb 02 '19 at 16:36
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I was wondering if we can differentiate between a body under the influence of a central potential 1/r and a body under the action of two perpendicular SHM's just by observing the orbits without prior knowledge of the potential they are under.
As a first note, you have described the two motions in different ways: the former dynamically, the latter kinematically. In the first case your description points to the kind of force acting, in the second on motion being a composition of two SHM. Of course you know the dynamics, as is shown from your title, where Hooke's law is recalled.
A second point is: what do you exactly mean by "just by observing the orbits"? If you mean simply discovering that both orbits are ellipses, obviously there's no answer - they are indistinguishable.
At the other extreme, I assume you don't think of identifying the center of force, which would give an easy solution. Yet this could be done by pure kinematics, computing the (vector) acceleration.
But there is an intermediate way, which uses Kepler's second law (the law of areas). In Newton/Coulomb case the speed has a minimum at an extreme of the major axis and a maximum at the other. In Hooke's case speed at both extremes of major axis is the same, at the minimum, and maximum is attained at minor axis' extremes. Thus a simple measurement of speed would give the answer.
Elio Fabri
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