(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 171014, 5022]*) (*NotebookOutlinePosition[ 171911, 5053]*) (* CellTagsIndexPosition[ 171867, 5049]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["\n"], "Text", Evaluatable->False], Cell["Classical Kepler Problem", "Section", Evaluatable->False], Cell[TextData["\n"], "Text", Evaluatable->False], Cell["Clear[\"Global`*\"]", "Input", PageWidth->Infinity], Cell["<< Graphics`Graphics`", "Input", PageWidth->Infinity], Cell["\<\ Potential is V(r) = - G m1 m2 / r [ 1 - G (m1+m2)/(2c^2 r) + O(1/c^4) ]\ \>", "Text", Evaluatable->False], Cell["v[u_] := -k/r + b/r^2 /. r -> 1/u", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["D[v[u],u]", "Input", PageWidth->Infinity], Cell[BoxData[ \(\(-k\) + 2\ b\ u\)], "Output"] }, Open ]], Cell["u''[phi] + u[phi] == - (m/l^2) D[v[u[phi]],u[phi]] ;", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["Apart[#] & /@ %", "Input", PageWidth->Infinity], Cell[BoxData[ RowBox[{ RowBox[{\(u[phi]\), "+", RowBox[{ SuperscriptBox["u", "\[DoublePrime]", MultilineFunction->None], "[", "phi", "]"}]}], "==", \(\(k\ m\)\/l\^2 - \(2\ b\ m\ u[phi]\)\/l\^2\)}]], "Output"] }, Open ]], Cell["eq = % ;", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["?/@", "Input", PageWidth->Infinity], Cell[BoxData[ \("Map[f, expr] or f /@ expr applies f to each element on the first level \ in expr. 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.17645 .25002 L .18847 .22151 L .20287 .19601 L .23663 .15279 L .27003 .12201 L .30959 .09453 L .34906 .07387 L .38583 .05933 L .42877 .04712 L .45263 .04234 L .4743 .03915 L .51271 .03608 L .55131 .03625 L .58557 .03914 L .61982 .04469 L .64956 .05178 L .67469 .05954 L .72553 .08089 L .76903 .10655 L .79731 .12816 L .82232 .15179 L .85616 .19465 L .871 .22066 L .88162 .245 L .88864 .26667 L .89164 .27883 L .89381 .29002 L .89555 .30217 L .89661 .31369 L .89708 .32402 L .89707 .33529 L .89645 .3475 L .89583 .35433 L .89509 .36053 L .89317 .37262 L .89076 .38404 L .88438 .40629 L .87638 .4269 L .86499 .44971 L .84966 .47399 L .83005 .49878 L .80375 .52524 L .77299 .54971 L .73901 .571 L Mistroke .6907 .59362 L .65887 .60456 L .64244 .60914 L .62627 .61297 L .59526 .61858 L .57781 .62076 L .55826 .62239 L .53809 .62319 L .5161 .62305 L .47498 .61995 L .43382 .61302 L .41008 .60717 L .38801 .60043 L .34744 .58441 L .30552 .562 L .27002 .53699 L .23967 .5094 L .21077 .47473 L .18633 .43306 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.53421 L .23108 .50017 L .21515 .48072 L .20004 .45848 L .17987 .41814 L .16781 .37808 L .16275 .32882 L .1672 .28393 L .17933 .24222 L .19686 .20587 L .22217 .16929 L .25468 .13508 L .2864 .10966 L .3182 .08951 L .35445 .07147 L .3941 .05659 L .43713 .04529 L .47515 .03904 L .51716 .03593 L Mistroke .55449 .03641 L .58691 .03931 L .61484 .04371 L .64335 .05011 L .6711 .05832 L .72166 .07897 L .76067 .10098 L .79564 .12674 L .82551 .1552 L .8479 .18257 L .8666 .21222 L .87966 .23995 L .8845 .25308 L .88828 .26537 L .89127 .2772 L .89346 .28802 L .89529 .30001 L .89599 .30629 L .89656 .31292 L .8971 .32535 L .89702 .33693 L .89628 .34965 L .89556 .35677 L .89471 .36326 L .89274 .37483 L .89036 .3857 L .88288 .41057 L .87263 .43503 L .85794 .46158 L .8404 .48638 L .81967 .50998 L .76762 .55342 L .72864 .57654 L .7038 .58825 L .67795 .59834 L .65148 .6067 L .62025 .61424 L .58944 .61938 L .55907 .62233 L .52034 .62316 L .48199 .62074 L .45914 .61775 L .4379 .61388 L .41298 .60796 L .38915 .60081 L .34159 .58167 L .29888 .5578 L .25536 .52453 L .23547 .50498 L .21564 .48138 L Mistroke .1902 .44095 L .17349 .40004 L .1639 .35278 L .16278 .32585 L .16443 .30142 L .17541 .25302 L .18588 .22689 L .19802 .20387 L .22587 .16482 L .26169 .1289 L .29584 .10319 L .33493 .08059 L .373 .06393 L .40797 .05243 L .44833 .04309 L .49086 .03742 L .52909 .03575 L .56253 .03691 L .59866 .04094 L .61774 .04427 L .63448 .0479 L .66333 .05582 L .6911 .06555 L .73327 .0849 L .77162 .10835 L .80492 .13482 L .83311 .16376 L .85386 .19115 L .86858 .21592 L .87917 .23874 L .8878 .26366 L .89103 .27612 L .89372 .28947 L .89553 .30205 L .8966 .31353 L .8971 .32484 L .89707 .33535 L .8965 .34688 L .89526 .35919 L .89347 .37095 L .89122 .38206 L .88549 .40292 L .87595 .42786 L .86225 .45449 L .84721 .4774 L .82657 .50266 L .79869 .52968 L .76094 .55786 L .71634 .5826 L .6919 .59315 L Mistroke .66672 .60212 L .63963 .60985 L .61276 .61569 L .58063 .62045 L .54328 .62306 L .50709 .62269 L .46529 .61867 L .44348 .615 L .41976 .60972 L .37646 .59636 L .33593 .5789 L .2934 .55418 L .25607 .52517 L .22545 .49367 L .20202 .46168 L .18191 .42315 L .16824 .38014 L .16292 .33841 L .16627 .28889 L .17273 .26145 L .18153 .23674 L .20408 .19409 L .23592 .15353 L .27326 .11943 L .31215 .09299 L .3489 .07393 L .38941 .05811 L .43383 .04598 L .45855 .04135 L .48111 .03835 L .50434 .03646 L .52487 .03577 L .54776 .03609 L .56859 .03738 L .60342 .04169 L .63321 .0476 L .66327 .0558 L .69228 .06601 L .73588 .08629 L .77377 .10986 L .82483 .15446 L .84582 .17973 L .86142 .20313 L .8748 .22861 L .88073 .24267 L .88531 .25555 L .89162 .27874 L .89395 .29089 L .89572 .3037 L .89668 .31481 L Mistroke .8971 .3252 L .89703 .33658 L .89635 .34881 L .8952 .35966 L .89337 .3715 L .89107 .38271 L .88843 .39312 L .88203 .4129 L .87308 .43411 L .86027 .4578 L .84306 .48296 L .80067 .52796 L .7691 .55241 L .73339 .57405 L .68605 .59539 L .65535 .6056 L .62462 .61333 L .59544 .61855 L .56219 .62212 L .53196 .62325 L .49684 .62206 L .45729 .61745 L .41315 .608 L .37073 .59421 L .33217 .577 L .29486 .55516 L .2623 .53062 L .23009 .49906 L .20129 .46051 L .18053 .4198 L .16797 .37887 L .16302 .34089 L .16466 .29954 L .17555 .2526 L .19332 .21214 L .2219 .16961 L .25539 .13442 L .29037 .10687 L .32504 .08573 L .36449 .06725 L .40683 .05274 L .44859 .04304 L .47152 .03948 L .49207 .03731 L .52856 .03575 L .56477 .03707 L .58304 .03883 L .60178 .04142 L .6335 .04767 L .66786 .05725 L Mistroke .69604 .06752 L .74416 .09091 L .78032 .11463 L .81239 .14179 L .8391 .171 L .85617 .19466 L .87063 .21991 L .88249 .24735 L .89023 .27278 L .89323 .28676 L .89455 .29464 L .89552 .30195 L .89624 .30896 L .89671 .31531 L .89704 .32263 L .89714 .32951 L Mfstroke .1 .1 1 r .89714 .32951 m .89688 .34091 L .8962 .35129 L .89486 .36331 L .89306 .37472 L .88848 .39518 L .88139 .41772 L .87073 .44307 L .85566 .47067 L .81628 .52225 L .78907 .54878 L .7524 .57759 L .70911 .60438 L .66052 .62755 L .62625 .64035 L .59081 .65091 L .54767 .66046 L .52154 .66458 L .49659 .66741 L .44776 .66984 L .41862 .66936 L .39088 .66754 L .33789 .66024 L .28019 .64609 L .25054 .63599 L .2186 .62262 L .1628 .59163 L .11374 .55306 L .07558 .51039 L .04495 .45801 L .03327 .42593 L .02646 .39483 L .02381 .3616 L .02547 .33103 L .03174 .29783 L .04313 .2638 L .07014 .21333 L .10924 .16518 L .15336 .12631 L .19744 .09693 L .25186 .06957 L .28088 .05815 L .31147 .04815 L .36644 .03492 L .41487 .02786 L .44234 .02566 L .46743 .02479 L .51178 .02586 L .53496 .02778 L .55545 .03027 L Mistroke .57682 .03369 L .59834 .038 L .63429 .04731 L .66467 .05739 L .71345 .0785 L .76038 .10599 L .79455 .13207 L .82422 .16081 L .84846 .19089 L .86409 .21558 L .87574 .23873 L .88391 .25941 L .89056 .28185 L .89315 .29348 L .89535 .30636 L .8962 .31286 L .89692 .31988 L .89774 .33266 L .89791 .34483 L .89745 .35776 L .89653 .36893 L .89494 .38129 L .89267 .39407 L .8899 .40631 L .8829 .42977 L .87242 .45599 L .85699 .48562 L .83927 .51275 L .81529 .54262 L .75476 .59854 L .70424 .63236 L .64363 .66277 L .61282 .67484 L .57686 .68641 L .53349 .69706 L .48966 .70435 L .43967 .70858 L .38207 .70802 L .32517 .70147 L .27203 .68936 L .214 .66823 L .18243 .6524 L .15447 .63508 L .10406 .59267 L .0805 .56487 L .06262 .53737 L .04104 .4868 L .03131 .4297 L .03402 .37999 L .04822 .32626 L Mistroke .0751 .27189 L .11484 .21933 L .15979 .1766 L .20484 .14383 L .24861 .11868 L .29565 .09735 L .3464 .07985 L .39175 .06849 L .43577 .06103 L .47475 .05726 L .50758 .05613 L .54074 .05692 L .57625 .05999 L .60594 .06441 L .63722 .07104 L .66412 .07849 L .70605 .09378 L .74427 .11238 L .77898 .13419 L .80898 .15821 L .8492 .20233 L .86398 .22451 L .87676 .24866 L .88615 .27178 L .89262 .29335 L .89545 .30595 L .89751 .31784 L .89844 .32476 L .89913 .3311 L .89972 .33849 L .90009 .34553 L .90026 .35247 L .90025 .35989 L .89968 .37344 L .89905 .3811 L .89825 .38837 L .89564 .40526 L .89265 .41939 L .88853 .43485 L .87838 .46389 L .86083 .50071 L .83849 .53635 L .81007 .57224 L .7693 .61288 L .71866 .65212 L .65931 .68726 L .62659 .70272 L .58786 .71794 L .54769 .73052 L .50811 .73991 L Mistroke .45814 .74768 L .40021 .75101 L .34302 .74806 L .28935 .73906 L .23487 .72262 L .18641 .70012 L .13844 .66735 L .09576 .62315 L .06402 .56859 L .04799 .51108 L .04597 .45708 L .05668 .39811 L .06731 .3678 L .08187 .33686 L .11515 .28522 L .15879 .23625 L .2125 .19125 L .26116 .15979 L .31353 .13326 L .36172 .11438 L .40385 .10171 L .44522 .09252 L .48121 .08707 L .51807 .0839 L .55423 .08323 L .58309 .08447 L .60825 .08692 L .63381 .09077 L .65916 .09606 L .68378 .10271 L .70472 .10967 L .7461 .1275 L .77944 .14673 L .80891 .1686 L .83168 .18987 L .84968 .21058 L .86428 .23116 L .87572 .25095 L .88558 .2723 L .89279 .29247 L .89593 .30348 L .89881 .31574 L .90122 .32902 L .90289 .34198 L .90354 .34917 L .90397 .35605 L .90431 .36918 L .90423 .37672 L .90396 .38376 L .90346 .39163 L Mistroke .90267 .40009 L .90076 .41454 L .89824 .42847 L .89457 .4443 L .88949 .46191 L .87706 .49504 L .85754 .53394 L .83177 .57359 L .79989 .61263 L .7594 .65247 L .71116 .69041 L .64833 .72869 L .60924 .74744 L .56252 .76544 L .50985 .78045 L .45818 .78995 L .39497 .79454 L .33339 .79105 L .27652 .77982 L .2262 .76194 L .1756 .73362 L .13013 .69404 L .0954 .64528 L .0816 .61475 L .07282 .58539 L .06637 .52578 L .07277 .46826 L .08128 .43638 L .0938 .40338 L .1239 .34751 L .16459 .29379 L .21583 .24328 L .26378 .20652 L .31471 .17543 L .35779 .15431 L .40205 .1369 L .44709 .12328 L .48625 .11465 L .50881 .111 L .52919 .10856 L .5467 .1071 L .56489 .10622 L .59358 .10621 L .60892 .10691 L .62267 .10798 L .64865 .11118 L .6623 .11351 L .67423 .11593 L .69417 .12085 L .71425 .12697 L Mistroke .74682 .13974 L .77668 .15513 L .80421 .17331 L .82503 .1904 L .84402 .20941 L .85886 .2274 L .87285 .24807 L .88357 .26764 L .89212 .28708 L .89652 .29924 L .90066 .31291 L .90358 .32466 L .90622 .33806 L .90734 .34522 L .90835 .35309 L .90918 .36159 L .90974 .36975 L .91006 .37761 L .91016 .38606 L .90972 .40169 L .90852 .41699 L .90629 .43406 L .90319 .4509 L .89945 .46686 L .88585 .50892 L .86582 .55272 L .83896 .59731 L .80462 .64211 L .75396 .69376 L .68084 .74937 L .63589 .77555 L .58027 .80111 L .55257 .81126 L .52164 .82068 L .46272 .83312 L .43157 .83677 L .39729 .83835 L .33468 .83411 L .27524 .82012 L .22437 .79804 L .18117 .76856 L .14307 .7286 L .11646 .68411 L .10095 .63925 L .09364 .58327 L .0985 .52108 L .11396 .46354 L .136 .41312 L .16472 .36516 L .19565 .32452 L Mistroke .23565 .28203 L .27383 .2487 L .35183 .19617 L .3932 .17499 L .43075 .1592 L .46652 .147 L .50279 .13737 L .53469 .13114 L .56209 .12748 L .5853 .12563 L .60917 .12497 L .63241 .12557 L .65266 .12716 L .67471 .13008 L .69409 .13375 L .7269 .14257 L .75774 .1543 L .78687 .1691 L .82739 .19775 L .84454 .21382 L .86127 .2328 L .87379 .24997 L .88566 .26968 L .89537 .28963 L .90287 .30898 L .90997 .33336 L .91301 .34763 L .91529 .36174 L .91682 .37538 L .91771 .3885 L .91804 .40354 L .91759 .42025 L .91621 .43792 L .91498 .44826 L .91355 .45797 L .90984 .4776 L .90506 .49724 L .88904 .54491 L .87599 .57433 L .86074 .60323 L .81893 .66557 L .76228 .72814 L .69368 .78432 L .64678 .81393 L .59833 .83837 L .5406 .86013 L .48259 .87432 L .44801 .8791 L .41604 .88098 L .37904 .87991 L Mistroke .34438 .87542 L .31164 .86768 L .27835 .85567 L .22424 .82434 L .19833 .80175 L .17725 .7775 L .15808 .74788 L .14238 .71304 L .12831 .65314 L .12682 .59631 L .13717 .53085 L .15685 .4717 L .18175 .42072 L .20881 .3778 L .27822 .29667 L .31621 .26295 L .35152 .23638 L .39242 .21038 L .42823 .19133 L .4957 .16393 L .52943 .15416 L .54504 .15051 L .5592 .14768 L .57254 .14543 L .58643 .14352 L .60944 .14136 L .62187 .14073 L .63448 .14048 L .65587 .14098 L .67444 .14241 L .69074 .14448 L .70758 .14745 L .72449 .15136 L .75891 .16249 L .78738 .17549 L .8101 .18889 L .83215 .20516 L .84921 .22054 L .86576 .23857 L .8913 .27548 L .90411 .30116 L .91419 .32825 L .91786 .34095 L .92125 .35518 L .92429 .37167 L .9264 .38789 L .92725 .39726 L .92787 .40749 L .92822 .42672 L .92794 .43794 L Mistroke .92734 .44878 L .92629 .4611 L .92465 .4748 L .92278 .48705 L .92022 .50084 L .914 .52726 L .9057 .55471 L .89573 .58168 L .86975 .63648 L .8272 .70264 L .76278 .77545 L .72559 .80817 L .67994 .84144 L .62883 .87111 L .57763 .89371 L .5224 .9106 L .46795 .91967 L .40367 .92002 L .34613 .90927 L .28811 .88453 L .26194 .867 L .23615 .84404 L .21475 .81871 L .19774 .79203 L .175 .73653 L .16758 .70333 L .16401 .67171 L .16358 .63659 L .16685 .59924 L .17255 .5671 L .18137 .5332 L .20322 .47496 L .23001 .42317 L .25788 .38041 L .31687 .3106 L .3729 .26071 L .43189 .22033 L .4901 .19042 L .51719 .17954 L .54494 .17031 L .568 .16412 L .59108 .15925 L .61069 .15618 L .63112 .15405 L .64869 .15312 L .66434 .153 L .68102 .15366 L .69786 .15518 L .71292 .1573 L .72594 .15975 L Mistroke .74054 .16321 L .75551 .16761 L .7807 .17714 L .80474 .18912 L .82785 .20384 L .84614 .21824 L .86419 .23551 L .87954 .25331 L .89243 .27124 L .91435 .31175 L .92443 .3381 L .92899 .35328 L .9332 .3705 L .93624 .38646 L .93883 .40526 L .94038 .4235 L .94105 .44172 L .94078 .46237 L .9392 .48597 L .93628 .50978 L .93222 .53279 L .92548 .56159 L .91531 .59504 L .88746 .66162 L .86802 .69743 L .84148 .7384 L .80976 .77925 L .76851 .82303 L .72288 .86225 L .6751 .89501 L .62168 .92306 L .56679 .94327 L .50946 .95529 L .45677 .9578 L .39739 .94953 L .3453 .93025 L .30233 .90258 L .2688 .86957 L .23969 .82631 L .21835 .77461 L .20662 .71842 L .20434 .65655 L .21075 .59887 L .22261 .54837 L .24224 .49292 L .26867 .43826 L .32437 .35512 L .38027 .29461 L .43885 .24664 L .4904 .21444 L Mistroke .53379 .19363 L .57756 .17807 L .60044 .17207 L .62055 .168 L .63107 .16633 L .64073 .16508 L .6581 .16352 L .67588 .16286 L .6855 .16291 L .69413 .16321 L .70885 .16428 L .72394 .16614 L .73756 .16853 L .74951 .1712 L .77697 .17954 L .80098 .18968 L 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