Theorem of Newton-Hamilton

The theorem of Newton-Hamilton is a theorem of Dynamique fields with central Force, of trajectory a Conique. The presentation will be made here with a ellipse.

Statement

That is to say a trajectory (T) elliptic, described under the action of a central force resulting from a point O (obviously interior with the ellipse). That is to say (D) the polar one of O compared to the ellipse; and pH the distance from the material point running P with polar (D).

The central force is F ~ R (pH) ³.

Applications

most known is that of Proposal 11 of Principia. The theorem of Hamilton is only the generalization expressed in geometry of the polar ones. Choice of O: the sun S located at the hearth of the ellipse.

Its polar is the director (D), and thus pH ³ = E ³ .r ³. One draws the force F~ 1/r ² from it.

Another application is the degenerated position of the center of the ellipse. It is then necessary to be careful, in order to show that pH ³ must be regarded as constant: the ellipse of Hooke is found.

Is a circle of diameter " vertical" Polar OA = 2R. La of O is the axis " horizontal" x' OX. One finds Proposal 7 of Principia: the point running is attracted by F ~ 1/r^5.

Demonstration

In conical (curve of the second order) F (X, there) = X ² has + 2c xy +b there ² + 2dx +2ey +f = 0, the polar line (D) of the origin O, is dx + ey +f =0.

And pH ~ dx +ey +f, P being on the conical one.

The Accélération of Siacci gives F ~ C ² r/p ³ R and the formula of Frenet R ~ (V \ A)^3 led to p ³ R ~ (dx+ey+f) ^3

One from of deduced: F ~ C ² R/(dx+ey+f) ^3 ~ C ² R (pH) ^3.

See too

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