the length of the major axis 2a k H of the ellipse E In motion for fixed energy H 0 is bounded inside a sphere. fact we think that our proof in the next section can compete with center 0 and radius k H Indeed V r H and so. both in transparency and in level of computation with the k r H or equivalently r k H. standard proof of Jakob Hermann and Johann Bernoulli Consider the following picture of the plane perpendic. making it an appropriate alternative to present in a fresh ular to L. man course on classical mechanics,We thank Alain Albouy Hans Duistermaat Ronald. Kortram Arnoud van Rooij and the referee for useful. comments on this article Note Maris van Haandel s work. was supported by NWO,A Euclidean Proof of Kepler s First Law. We shall use inner or scalar or dot products u v and. outer or vector or cross products u 9 v of vectors u and v. in R3 the compatibility conditions,u v w u v w,u v w u w v u v w. and the Leibniz product rules,u v u v u v,u v u v u v. without further explanation,For a central force field F r f r r r the angular. momentum vector L r 9 p is conserved by Newton s, law of motion F p thereby leading to Kepler s second. law For a spherically symmetric central force field The circle C with center 0 and radius k H is the. F r f r r r the energy boundary of a disc where motion with energy H 0 takes. Z place Points that fall from the circle C have the same. H p2 2m V r V r f r dr energy as the original moving point and for this reason C is. called the fall circle Let s kr rH be the projection of r. is conserved as well These are the general initial remarks from the center 0 onto the fall circle C The line L through r. From now on consider the Kepler problem f r k r2 with direction vector p is the tangent line of the orbit E at. and V r k r with k 0 a coupling constant If m is position r with velocity v Let t be the orthogonal reflection. replaced by the reduced mass l mM m M then the of the point s in the line L As time varies the position. coupling constant becomes k GmM with m and M the vector r moves along the orbit E and likewise s moves. masses of the two bodies and G the universal gravitational along the fall circle C It is good to investigate how the point. constant Using conservation of energy we show that the t moves. MARIS VAN HAANDEL graduated from GERT HECKMAN graduated from Leiden. Nijmegen University in 1993 with a thesis 2 University in 1980 with a thesis on Lie groups. on Riesz spaces He is a high school teacher He has been professor of geometry at Rad. in RSG Pantarin at Wageningen He has boud University since 1999 His wife teaches. been working together with Gert Heckman Greek and Latin in high school both their. for two years on a project on Newton and children are medical students Heckman is an. the Kepler laws with the objective of enthusiast for skating on the frozen canals in. writing a treatment suitable for high winter He hopes that global warming will. schools He lives in a small village near relent so that this hobby can continue. Nijmegen with his wife Yvette and their,one year old son Ruben IMAPP Radboud University. Nijmegen Netherlands, RSG Pantarijn Wageningen e mail G Heckman math ru nl. Netherlands,e mail marisvanhaandel wanadoo nl, 2009 The Author s This article is published with open access at Springerlink com 41. T HEOREM The point t equals K mH and therefore is T2 a3 4p2 m k. The mass m we have used so far is actually equal to the. reduced mass l mM m M with m the mass of the,P ROOF The line N spanned by n p 9 L is perpendic. planet and M the mass of the sun and this almost equals m. ular to L The point t is obtained from s by subtracting twice. if m M The coupling constant k is according to, the orthogonal projection of s r on the line N and therefore. Newton equal to GmM with G the universal gravitational. t s 2 s r n n n2 constant We therefore see that Kepler s harmonic third. law stating that T2 a3 is the same for all planets holds only. Now approximately for m M, s kr rH It might be a stimulating question for the students to. adapt the arguments of this section to the case of fixed. s r n H k r r p L H H k r L2 H energy H 0 Under this assumption the motion becomes. unbounded and traverses one branch of a hyperbola,n2 p2 L2 2m H k r L2. and therefore Feynman s Proof of Kepler s First Law. In this section we discuss a different geometric proof of. t kr rH n mH K mH Kepler s first law based on the hodograph H By definition. where K p 9 L kmr r is the Runge Lenz vector The H is the curve traced out by the velocity vector v in the. 0 is derived by a straightforward computa, final step K Kepler problem This proof goes back to Mo bius in 1843. and Hamilton in 1845 3 and has been forgotten and, tion using the compatibility relations and the Leibniz. rediscovered several times by Maxwell in 1877 2 and by. product rules for inner and outer products of vectors in R3. Feynman in 1964 in his Lost Lecture 7 among others. Let us assume as in the picture in the previous section. C OROLLARY The orbit E is an ellipse with foci 0 and t. that ivn n v with i the counterclockwise rotation around. and major axis equal to 2a k H, 0 over p 2 So the orbit E is assumed to be traversed. counterclockwise around the origin 0,P ROOF Indeed we have. jt rj jr 0j js rj jr 0j js 0j k H T HEOREM The hodograph H is a circle with center. c iK mL and radius k L, Hence E is an ellipse with foci 0 and t and major axis. P ROOF We shall indicate two proofs of this theorem The. The above proof has two advantages over the earlier first proof is analytic in nature and uses conservation of the. mentioned proofs of Kepler s first law The conserved vector Runge Lenz vector K by rewriting. t K mH is a priori well motivated in geometric terms K p L kmr r mvLn n kmr r. Moreover we use the gardener s definition of an ellipse The. gardener s definition so called because gardeners some as. times use this construction for making an oval flowerbed is vn n K mL kr rL. well known to Dutch freshmen In contrast the equation of or equivalently. an ellipse in polar coordinates is unknown to most fresh. men and so additional explanation would be needed for v iK mL ikr rL. that Yet another advantage of our proof is that the solution Hence the theorem follows from K 0. of the equation of motion is achieved by just finding enough There is a different geometric proof of the theorem. constants of motion of geometric origin whose integration discussed by Feynman which instead of using the. is performed trivially by the fundamental theorem of calcu conservation of the Runge Lenz vector K yields it as a. lus The proofs by Feynman and Newton in the next sections corollary The key point is to reparametrize the velocity. on the contrary rely at a crucial point on the existence and vector v from time t to angle h of the position vector r. uniqueness theorem for differential equations It turns out that the vector v h is traversing the hodograph H. We proceed to derive Kepler s third law along standard with constant speed k L Indeed we have from Newton s laws. lines 4 The ellipse E has numerical parameters the major. axis equals 2a the minor axis 2b and a2 b2 c2 dv dt kr dt. a b c 0 given by 2a k H 4c2 K2 m2H2 dh dh r dh, 2mHL2 m2k2 m2H2 The area of the region bounded by and Kepler s second law yields. r 2 dh 2 Ldt 2m,Combining these identities yields, with T the period of the orbit Indeed L 2m is the area of. the sector swept out by the position vector r per unit time dv. A straightforward calculation yields dh,42 THE MATHEMATICAL INTELLIGENCER. so indeed v h travels along H with constant speed k L Since. r reih a direct integration yields,v h c ikr rL c 0. and the hodograph becomes a circle with center c and radius. k L Comparison with the last formula in the first proof gives. 0 comes out as a corollary, All in all the circular nature of the hodograph H is more. or less equivalent to the conservation of the Runge Lenz. T HEOREM Let E be a smooth closed curve bounding a. convex region containing two points c and d Let r t tra. verse the curve E counterclockwise in time t such that the. areal speed with respect to the point c is constant Likewise. let r s traverse the curve E counterclockwise in time s such. that the areal speed with respect to the point d is equal to the. same constant, Let L be the tangent line to E at the point r and let e be. the intersection point of the line M which is parallel to L. Now turn the hodograph H clockwise around 0 by p 2. through the point c and the line through the points r and d. and translate by ic K mL This gives a circle D with. Then the ratio of the two accelerations is given by. center 0 and radius k L Since,kr rL K mL vn n iv d2r d2r. j j j 2 j jr ej3 jr cj jr dj2, the orbit E intersects the line through 0 and kr rL in a point. with tangent line L perpendicular to the line through k r rL. P ROOF Using the chain rule we get, and K mL For example the ellipse F with foci 0 and dr dr dt. K mL and major axis equal to k L has this property but ds dt ds. any scalar multiple kF with k 0 has the property as well. Because curves with the above property are uniquely d 2 r d 2 r dt 2 dr d 2 t. charcterized after an initial point on the curve is chosen we ds2 dt 2 ds dt ds. conclude that E kF for some k 0 This proves Kepler s. first law A comparison with the picture in the previous Because d2r dt2 is proportional to c r and likewise. section shows that E kF with k L H Indeed E has d2r ds2 is proportional to d r we see that. foci 0 and kK mL K mH t and its major axis is 2 2, equal to kk L k H 2a d2r d2r dt d 2 r dr d 2 t dt d2r. j 2j j 2j j 2 2 j j 2j, It is not clear to us whether Feynman was aware that he ds dt ds dt dt ds ds dt. was relying on the existence and uniqueness theorem for dt. jr ej jr cj, differential equations On page 164 of 7 the authors quote ds. Feynman Therefore the solution to the problem is an. Since the curve E is traversed with equal areal speed. ellipse or the other way around really is what I proved. relative to the two points c and d we get, that the ellipse is a possible solution to the problem And it. is this solution So the orbits are ellipses dr dr,j j jr ej j j jr dj. Apparently Feynman had trouble following Newton s dt ds. proof of Kepler s first law On page 111 of 7 the authors. and therefore also, write In Feynman s lecture this is the point at which he. finds himself unable to follow Newton s line of argument dt. jr ej jr dj, any further and so sets out to invent one of his own ds. In turn this implies that,Newton s Proof of Kepler s First Law 2. d2r d2r dt, In this section we discuss a modern version of the original j 2j j 2j jr ej jr cj. proof by Newton of Kepler s first law as given in 13 The ds dt ds. proof starts with a nice general result jr ej3 jr cj jr dj2. 2009 The Author s This article is published with open access at Springerlink com 43. which proves the theorem The year 1687 marks the birth of both modern mathe. matical analysis and modern theoretical physics As such. We shall apply this theorem in case E is an ellipse with the derivation of the Kepler laws from Newton s law of. center c and focus d Assume that r t traverses the ellipse E motion and law of universal gravitation is a rewarding. in harmonic motion say subject to teach to freshmen students In fact this was. the motivation for our work we plan to teach this, d2r material to high school students in their final year Of. dt 2 course the high school students first need to become. so the period for time t is assumed to be 2p acquainted with the basics of vector geometry and vector. calculus But after this familiarity is achieved nothing. else hinders the understanding of our proof of Kepler s. law of ellipses,For freshmen physics or mathematics students in. the university who are already familiar with vector. calculus our proof given here is fairly short and geo. metrically well motivated In our opinion of all proofs. this proof qualifies best to be discussed in an introduc. tory course,OPEN ACCESS, This article is distributed under the terms of the Creative. Commons Attribution Noncommercial License which, permits any noncommercial use distribution and repro. duction in any medium provided the original author s and. source are credited, Let b be the other focus of E and let f be the inter. section point of the line N passing through b and parallel. to L with the line through the points d and r Then we REFERENCES. find 1 V I Arnold Huygens and Barrow Newton and Hooke Birkha user. Boston 1990,jd ej je fj jf rj jb rj, 2 D Chakerian Central Force Laws Hodographs and Polar. which in turn implies that je rj is equal to the half major Reciprocals Mathematics Magazine 74 1 February 2001 3 18. axis a of the ellipse E We conclude from the formula in the 3 D Derbes Reinventing the wheel Hodographic solutions to the. previous theorem that the motion in time s along an ellipse Kepler problems Am J Phys 69 4 April 2001 481 489. with constant areal speed with respect to a focus is only 4 H Goldstein Classical Mechanics Addison Wesley 1980 2nd. possible in an attractive inverse square force field The edition. converse statement that an inverse square force field for 5 H Goldstein Prehistory of the Runge Lenz vector Am J. negative energy H indeed yields ellipses as orbits follows Phys 43 8 August 1975 737 738. from existence and uniqueness theorems for solutions of 6 H Goldstein More on the prehistory of the Laplace or Runge. Newton s equation F ma and the previously mentioned Lenz vector Am J Phys 44 11 November 1976 1123 1124. reasoning This is Newton s line of argument for proving. 7 D L Goodstein and J R Goodstein Feynman s Lost Lecture The. Kepler s first law,motion of planets around the sun Norton 1996. 8 V Guillemin and S Sternberg Variations on a Theme of Kepler. Conclusion Colloquium Publications AMS volume 42 1990. There exist other proofs of Kepler s law of ellipses from a. 9 J L Lagrange The orie des variations se culaires des e le ments des. higher viewpoint One such proof by Arnold uses complex. plane tes Oeuvres Gauthier Villars Paris Tome 5 1781 pp 125 207. analysis and is somewhat reminiscent of Newton s previ. in particular pp 131 132, ously described proof by comparing harmonic motion. 10 J Milnor On the Geometry of the Kepler Problem Amer Math. with motion under an 1 r2 force field 1 Apparently. Kasner had discovered the same method already back in Monthly 90 6 1983 353 365. 1909 12 Another proof by Moser is also elegant and 11 J Moser Regularization of the Kepler problem and the averaging. uses the language of symplectic geometry and canonical method on a manifold Comm Pure Appl Math 23 1970 609 636. transformations 8 10 11 However our goal here has 12 T Needham Visual Complex Analysis Oxford University Press. been to present a proof that is as basic as possible and at 1997. the same time is well motivated in terms of Euclidean 13 I Newton Principia Mathematica New translation by I B Cohen. geometry and A Whitman University of California Press Berkeley 1999. It is difficult to exaggerate the importance of the role 14 D Speiser The Kepler problem from Newton to Johann Bernoulli. of the Principia Mathematica in the history of science Archive for History of Exact Sciences 50 2 August 1996 103 116.
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