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{{AFC submission|d|reason|What is the ''topic'' of your draft? |
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And also, you have two copies on this same page, please delete the extra material so we can focus on your current draft.|u=Mikerollem|ns=2|decliner=MatthewVanitas|declinets=20170225205843|ts=20170225132112}} <!-- Do not remove this line! --> |
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{{AFC submission|d|reason|This does not appear to be a draft article but a class exercise based on Newton's Principia. If it is meant to be an article draft, it needs a [[WP:LEDE|lede paragraph]] and to focus on a topic.|u=Mikerollem|ns=2|decliner=Robert McClenon|declinets=20170223180848|small=yes|ts=20170223150302}} <!-- Do not remove this line! --> |
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[[File:Principia Revolving Orbits.jpg|Principia Revolving Orbits]] |
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== To Determine the Central Force for a Particular Orbit and Revolving Orbits == |
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== Introduction == |
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Sorry I really can't understand what lede-paragraph means. Can someone please show a bit of pity? |
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In Section 9 Propositions 43 and 44 of Book 1 of the Principia, Newton considers revolving orbits. He proves the force that makes a body move in a particular orbit, increased or decreased by an additional force will make that body’s orbit rotate. He derives the form that the additional force must take. His proof is exceedingly complicated. To give some idea, here is a sentence from it. |
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“And therefore, if with centre C and any radius CP or Cp a circular sector is described equal to the total area VPC which the body P revolving in an immobile orbit has described in any time by a radius drawn to the centre, the difference between the forces by which the body P in an immobile orbit and body p in a mobile orbit revolve will be to the centripetal force by which some body, by a radius drawn to the centre, would have been able to describe that sector uniformly in the same time in which the area VPC was described as G<sup>2</sup> - F<sup>2</sup> to F<sup>2</sup>.” |
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What follows is a much simplified proof, containing the essentials of his proof but without the complications. |
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It is preceded by Proposition 6 from Section 2 of Book 1 of the Principia which is used to effect the proof. |
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[[File:Principia Revolving Orbits.jpg|Principia Revolving Orbits]] |
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== Proposition 6 == |
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In Fig. 1, a body is moving along a specific curve MN acted on by a (centripetal) force, towards the fixed point S. The force depends only of the distance of the point from S. The aim of this proposition is to determine how the force varies with the radius, SP. The method applies equally to the case where the force is centrifugal. |
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In a small time, τ (= tau), the body moves from P to the nearby point Q. Draw QR parallel to SP meeting the tangent at R, and QT perpendicular to SP meeting it at T. |
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If there was no force present it would have moved along the tangent at P with the speed that it had at |
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P, arriving at the point, R. If the force on the body moving from P to Q was constant in magnitude and parallel to the direction SP, the arc PQ would be parabolic with PR as its tangent and QR would be proportional to that constant force and the square of the time, τ. Conversely, if instead of arriving at R, the body was deflected to Q then the force along PQ, assuming the body has unit mass so the force is the same as the acceleration, |
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:<math> |
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F \propto \frac{QR}{\tau^2} |
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</math> |
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However, since the direction of the radius from S to points on the arc PQ and also the magnitude of the force towards S will change along PQ, the above relation will not give the exact acceleration at P. If Q is sufficiently close to P, the direction of force will be almost parallel to SP all along PQ and if the force changes little, PQ can be assumed to be approximated by a parabolic arc with the force given as above in terms of QR and τ. |
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The time, τ is proportional to the area of the sector SPQ. This is Kepler’s Second Law. A proof is demonstrated in Proposition 1, Book 1 in the Principia. Since the arc PQ can be approximated by a straight line, the area of the sector SPQ and the area of the triangle SPQ can be taken as equal so: |
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:<math> |
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F \propto \frac{QR}{\tau^2} = \frac{k.QR}{(SP^2.QT^2) } |
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</math> |
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Where k is constant. |
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Again, this is not exact for finite lengths PQ. However, if the limit of the above expression exists as PQ approaches zero, it will give the exact acceleration at P. If for a particular curve the expression above can be found at each point P on the curve as a function of SP, then the force law required to make the body follow that curve has been found. |
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This proposition is based on Galileo’s analysis of a body following a parabolic trajectory under the action of a constant acceleration. Newton mentions Galileo several times in relation to it in the Principia. Combining it with Kepler's Second Law gives the very simple and elegant method. |
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In the historically very important case where MN in Fig. 1 was part of an ellipse and S was one of its foci, Newton showed in Proposition 11 that the limit QR / QT<sup>2</sup> was constant at each point on the curve so that the force on the body directed towards the fixed point S varied inversely as the square of the distance SP. |
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Besides the ellipse with the centre at the focus, Newton also applied Proposition 6 to the hyperbola (Proposition 12), and the parabola (Proposition 13). Also, to the ellipse with the centre of force at the centre of the ellipse (Proposition 10), to the equiangular spiral (Proposition 9), and to the circle, with the centre of force not coinciding with the centre, and even on the circumference (Proposition 7). |
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Notes: |
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In time τ, the body with no acceleration would have reached point, W, further from P than R. However, in the limit QW becomes parallel to SP. The point W is completely ignored in Newton’s proof. |
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This method only allows the force law to be found for a specific orbit. It cannot determine the orbit of a body under the action of a specified central force. |
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== Propositions 43 and 44 == |
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Propositions 43 defines Revolving Orbits as follows: |
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“It is to be effected that a body may be able to move in any trajectory that is revolving about the centre of forces exactly as another body moves in that same trajectory at rest.” |
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It then explains what this means in more detail. |
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A body moves in an orbit about a centre of force, and this will be referred to as the static orbit. |
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A second body moves in an orbit around a different centre of force, the revolving orbit. The latter is related to the first orbit by being at the same distance from its centre, and making an angle from the starting direction which is a multiple of the corresponding angle, after the same length of time has elapsed for both orbits. |
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Proposition 44 shows how this can be achieved. |
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This proposition uses Proposition 6 to prove a result about revolving orbits. Instead of applying it to a specific type of curve as with the other cases mentioned above, the body moves in an orbit under the action of an arbitrary force directed towards a fixed point, as in Fig. 1. MN is part of that orbit. At point P, the body moves to Q under the action of a force directed towards S, as before. The force, F(SP) is defined at each point P on the curve. |
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In Fig.2, the corresponding part of the revolving orbit is mn with s as its centre of force. Assume that initially, the body in the static orbit starts out at right angles to the radius with speed V. The body in the revolving orbit must also start at right angles and assume its speed is v, with v > V. If SA is the initial direction of the static orbit, and sa, that of the revolving orbit. If after a certain time the bodies in the respective orbits are at P and p, then the ratio of the angles |
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:<math> |
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\frac{asp}{ASP} = \frac{psq}{PSQ} = \frac{v}{V }. |
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</math> |
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and similarly the areas |
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:<math> |
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\frac{asp}{ASP} = \frac{psq}{PSQ} = \frac{v}{V }. |
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</math> |
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The radii: |
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:<math> |
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SP = sp |
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</math> |
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:<math> |
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SQ = sq |
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</math> |
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The figure pryx and the arc py in Fig. 2 is the figure PRQT and the arc PQ in Fig. 1, expanded linearly in the horizontal direction in the ratio v / V, so that |
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:<math> |
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PU = pu, |
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</math> |
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:<math> |
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UT = ux |
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</math> |
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:<math> |
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qt = \frac{ v.QT }{V} |
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</math> |
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The straight lines qt and QT should really be circular arcs with centres s and S and radii sq and SQ respectively. In the limit, their ratio becomes v /V whether they are straight lines or arcs. |
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Since in the limit the force is parallel to SP and sp, if the same force acted on the body in Fig. 2 as in Fig. 1, the body would arrive at y, since ry = RQ. The difference in horizontal speed does not affect the vertical distances. Newton refers to Corollary 2 of the Laws of Motion where the motion of the bodies is resolved into a component in the radial direction acted on by the whole force, and the other component transverse to it, acted on by no force. |
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However, the distance from y to the centre, s is now greater than SQ, so an additional force is required to move the body to q such that sq = SQ. The extra force is represented by yq, and f is proportional to ry + yq, just as F is to RQ. |
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:<math> |
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RQ = SP - PU - ST |
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</math> |
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:<math> |
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rq = sp - pu - st |
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</math> |
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:<math> |
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rq = RQ + ST - st |
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</math> |
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:<math> |
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ST - st = yq |
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</math> |
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and can be found as follows: |
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:<math> |
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ST^2 = SQ^2 - QT^2 |
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</math> |
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:<math> |
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st^2 = sq^2 - qt^2 |
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</math> |
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:<math> |
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(ST - st)(ST + st) = qt^2 - QT^2 |
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</math> |
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And in the limit, as QT and qt approach zero, (ST + st) becomes equal to SQ + sq or 2SP so |
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:<math> |
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(ST - st).2SP = QT^2(v^2 - V^2) / V^2 |
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</math> |
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Therefore |
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:<math> |
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rq = RQ + ST - st = RQ + \frac{QT^2(v^2 - V^2) }{2SP.V^2} |
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</math> |
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Dividing by |
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:<math> |
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\tau^2 = (sp.QT)^2 |
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</math> |
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to obtain the forces: |
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:<math> |
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f(sp) = F(sp) + \frac{k(v^2 - V^2) }{2sp^3.V^2} |
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</math> |
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In Fig. 3 at the initial point A of the static curve, draw the tangent AR, which is perpendicular to SA and the circle AQD which just touches the curve at A. Let ρ be the radius of that circle. Since angle SAR is a right-angle, the centre of the circle lies on SA. From the property of a circle: |
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:<math> |
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QT^2 = AT.(2\rho - AT) = RQ.(2\rho - RQ) |
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</math> |
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and in the limit as Q approaches A, this becomes: |
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:<math> |
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\rho = QT^2 / 2RQ |
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</math> |
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Hence |
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:<math> |
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F(SA) = \frac{k}{(2\rho.SA^2) } |
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</math> |
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And since F(SA) is given, this determines the constant k. However, Newton wants the force at A, to be of the form c.V<sup>2</sup> / SA<sup>2</sup> where c is a constant so that |
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:<math> |
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f(sp) = F(sp) + \frac{c\rho (v^2 - V^2) }{sp^3} |
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</math> |
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Where |
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:<math> |
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c = \frac{F(SA). SA^2}{V^2} |
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</math> |
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The expression for f(sp) above is the same as Newton’s in Corollary 4 of Proposition 44, except that he uses different letters. |
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Notes |
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The above proof uses Proposition 6 directly, it is simple and appears to be perfectly satisfactory. |
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Newton’s proof of Proposition 44 is considerably more complicated and very difficult to understand. Although he states that the problem was to be solved by Proposition 6, he does not use it explicitly but his proof does employ the principle. |
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Newton’s approach shows very clearly why the additional force is required to make the orbit revolve with respect to the static orbit. |
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==Bibliography== |
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{{More footnotes|date=February 2017}} |
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* [[Isaac Newton]] (1726) Principia, A New Translation, by I. Bernard Cohen and Anne Whitman – University of California Press, 1999, Book I, §2, Proposition 6, and §9, Propositions 43 & 44 |
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* {{cite book | author = Samuel Earnshaw| year = 1832 | title = Dynamics, Or an Elementary Treatise on Motion and a Short Treatise on Attractions | edition = 1st | publisher = J. & J. J. Deighton; and Whittaker, Treacher & Arnot| pages = 52 }} |
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*Brougham and Vaux, Henry Brougham, Baron & Routh, E. J. (ed. I. B. Cohen) [1855] (1972) ''Analytical View of Sir Isaac Newton's Principia'', New York: Johnson Reprint Corp, Pages 87 – 89 |
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