\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3068\u306f\uff0c\u6955\u5186\u8ecc\u9053\u3092\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3059\u308b\u969b\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3042\u308b\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2 $u$ \u3068\u6642\u9593 $t$ \u3092\u95a2\u4fc2\u3065\u3051\u308b\u5f0f<\/strong><\/span>\u3067\u3042\u308a\uff0c\u6955\u5186\u8ecc\u9053\u306e\u96e2\u5fc3\u7387<\/strong><\/span>\u3092 $e$\uff0c\u5468\u671f<\/strong><\/span>\u3092 $T$ \u3068\u3059\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f0f\u306e\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n $$ u \u00a0 –\u00a0\u00a0 e \\sin u = \\frac{2\\pi}{T} t $$<\/p>\n <\/p>\n \u307e\u305a\uff0c\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c1\u6cd5\u5247\u304b\u3089\uff0c\u60d1\u661f\u306e\u8ecc\u9053\u306f\u6955\u5186\u8ecc\u9053\u3067\u3042\u308b\u3002\uff08\u5e73\u9762\u4e0a\u306b\u3042\u308b\u306e\u3067\u305d\u308c\u3092 $xy$ \u5e73\u9762\u3068\u3059\u308b\u3002\uff09<\/p>\n \u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $X, Y$ \u3067\u306f\uff0c\u9577\u534a\u5f84 $a$\uff0c\u96e2\u5fc3\u7387 $e$ \uff08\u3057\u305f\u304c\u3063\u3066\u77ed\u534a\u5f84 $b = a \\sqrt{1-e^2}$\uff09\u306e\u6955\u5186\u306e\u5f0f\u306f<\/p>\n $$\\frac{X^2}{a^2} + \\frac{Y^2}{a^2(1-e^2)} = 1$$<\/p>\n \u3067\u3042\u308b\u3002\u3053\u306e\u6955\u5186\u306f\uff0c\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2<\/strong><\/span> $u$ \u3092\u4f7f\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a<\/strong><\/span>\u3067\u304d\u308b\u3002<\/p>\n \\begin{eqnarray} \u6955\u5186\u306e\u4e2d\u5fc3\u304b\u3089 $X$ \u8ef8\u4e0a\u306b $a e$ \u3060\u3051\u96e2\u308c\u305f\u6955\u5186\u306e\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u3067\u66f8\u304f\u3068<\/p>\n \\begin{eqnarray} \u30b1\u30d7\u30e9\u30fc\u306e\u7b2c2\u6cd5\u5247<\/strong><\/span>\u3068\u306f\uff0c\u6955\u5186\u306e\u7126\u70b9\u306b\u4f4d\u7f6e\u3059\u308b\u592a\u967d\u3068\u6955\u5186\u4e0a\u3092\u904b\u52d5\u3059\u308b\u60d1\u661f\u3068\u3092\u7d50\u3076\u7dda\u5206\u304c\u5358\u4f4d\u6642\u9593\u306b\u63cf\u304f\u9762\u7a4d\uff0c\u3059\u306a\u308f\u3061\u9762\u7a4d\u901f\u5ea6\u304c\u4e00\u5b9a<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u3092\u8868\u3059\u3002\u6955\u5186\u306e\u9762\u7a4d $S = \\pi a^2 \\sqrt{1-e^2}$ \u306e\u6642\u9593\u5909\u5316\u7387 $\\displaystyle \\frac{dS}{dt}$ \u304c\u4e00\u5b9a\u3067\u3042\u308b\u304b\u3089<\/p>\n $$\\frac{dS}{dt} = \\mbox{const.} = \\frac{S}{T} = \\frac{\\pi a^2 \\sqrt{1-e^2}}{T} \\tag{1}$$<\/p>\n \u4e00\u65b9\u3067\uff0c\u9762\u7a4d\u901f\u5ea6 $\\displaystyle \\frac{dS}{dt}$ \u306f\uff0c\u529b\u5b66\u7684\u306a\u8996\u70b9\u304b\u3089\u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u89d2\u904b\u52d5\u91cf\u306e\u9762\u306b\u5782\u76f4\u306a\u6210\u5206\u306e\u534a\u5206\u3067\u3042\u308b\u304b\u3089<\/p>\n \\begin{eqnarray} \u3064\u307e\u308a\uff0c\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u306e\u6cd5\u5247\u3068\u306f\uff0c\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58\u5247\u306e\u3053\u3068\u3067\u3042\u3063\u305f\u3002<\/p>\n \u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3068<\/strong><\/span><\/span>\u306f\uff0c\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u306e\u6cd5\u5247\u306e\u7a4d\u5206\u306e\u3053\u3068<\/strong><\/span><\/span>\u3067\u3042\u308b\u3002<\/strong><\/span><\/span><\/p>\n $(1)$ \u304a\u3088\u3073 $(2)$ \u5f0f\u3088\u308a<\/p>\n \\begin{eqnarray} \u4e0a\u8a18\u3067\u7a4d\u5206\u5b9a\u6570\u306f $t = 0$ \u3067 $u = 0$ \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u3067\u30bc\u30ed\u3068\u3057\u3066\u3044\u308b\u3002<\/p>\n \u3053\u308c\u304c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3002<\/p>\n \u3061\u306a\u307f\u306b\uff0c<\/p>\n $$ r = a (1 – e \\cos u)$$<\/p>\n \u3067\u3042\u3063\u305f\u304b\u3089\uff0c<\/p>\n $$(1 – e\u00a0 \\cos u) \\frac{du}{dt} = \\frac{2 \\pi}{T}$$<\/p>\n \u3092\u4f7f\u3046\u3068\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u5c0e\u304b\u308c\u308b\u3002<\/p>\n $${\\color{blue}\\frac{1}{T} dt} = {\\color{red}\\frac{1}{2\\pi} \\frac{r}{a} du}$$<\/p>\n \u3053\u306e\u5f0f\u306f\u30b1\u30d7\u30e9\u30fc\u904b\u52d5\u306e\u6642\u9593\u7684\u306a\u5e73\u5747\u5024\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u306b\u4f7f\u3046\u3002\u305f\u3068\u3048\u3070<\/p>\n \\begin{eqnarray} \u6955\u5186\u306e\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u306f\u5a92\u4ecb\u5909\u6570 $u$ \uff08\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2\uff09\u306b\u3088\u3063\u3066\uff0c\u6642\u9593 $t$ \u3068\u95a2\u4fc2\u3065\u3051\u3089\u308c\u3066\u3044\u308b\u304c\uff0c$t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u306f\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u3002<\/p>\n \\begin{eqnarray} \u4e0a\u8a18\u306e3\u884c\u76ee\u306e\u5f0f\u304c\uff0c\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2 $u$ \u3068\u6642\u9593 $t$ \u3092\u95a2\u4fc2\u4ed8\u3051\u308b \u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f<\/strong> \u3067\u3042\u308b\u3002 \u306b\u5bfe\u3057\u3066\uff0c$u_i = u(t_i)$ \u3092\u6570\u5024\u7684\u306b\u6c42\u3081\uff0c\u7b49\u3057\u3044\u6642\u9593\u9593\u9694 $\\displaystyle \\Delta t = \\frac{T}{N}$ \u3054\u3068\u306e\u4f4d\u7f6e $x(u_i), y(u_i)$ \u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n $$g(u_i) \\equiv u_i – e \\sin u_i = \\frac{2\\pi}{T} t_i = \\frac{2\\pi}{N} \\times i$$\u3092 $u_i$ \u306b\u3064\u3044\u3066 \u6955\u5186<\/p>\n \\begin{eqnarray} \u3082\u91cd\u306d\u3066\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n\u6955\u5186\u8ecc\u9053\u306e\u5a92\u4ecb\u5909\u6570\u8868\u793a<\/h3>\n
\nX &=& a \\cos u\\\\
\nY &=& a\\sqrt{1 – e^2} \\sin u
\n\\end{eqnarray}<\/p>\n
\nx &=& X – a e = a (\\cos u – e) \\\\
\ny &=& Y = a\\sqrt{1 – e^2} \\sin u\\\\ \\ \\\\
\nr^2 &=& x^2 + y^2 \\\\
\n&=& a^2 (\\cos^2 u – 2 e \\cos u + e^2) + a^2 (1 – e^2) \\sin^2 u \\\\
\n&=& a^2 (1 – e \\cos u)^2 \\\\
\n\\therefore\\ \\ r &=& a (1 – e\\cos u)
\n\\end{eqnarray}<\/p>\n\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c2\u6cd5\u5247\uff08\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u306e\u6cd5\u5247\uff09<\/h3>\n
\n\\frac{dS}{dt} &=&\\frac{1}{2} \\left( \\boldsymbol{r}\\times \\dot{\\boldsymbol{r}}\\right)_z \\\\
\n&=& \\frac{1}{2}\\left( x \\dot{y} – \\dot{x} y\\right)\\\\
\n&=& \\frac{1}{2} a^2 \\sqrt{1 – e^2}\\left((\\cos u – e)\\cdot \\cos u\\, \\dot{u}
\n– (- \\sin u \\, \\dot{u}) \\cdot \\sin u\\right)\\\\
\n&=& \\frac{1}{2}a^2 \\sqrt{1 – e^2} (1 – e\u00a0 \\cos u) \\dot{u} \\tag{2}
\n\\end{eqnarray}<\/p>\n\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u5c0e\u51fa<\/h3>\n
\n\\frac{1}{2}a^2 \\sqrt{1 – e^2} (1 – e\u00a0 \\cos u) \\dot{u} &=& \\frac{\\pi a^2 \\sqrt{1-e^2}}{T}\\\\
\n(1 – e\u00a0 \\cos u) \\frac{du}{dt} &=& \\frac{2 \\pi}{T}\\\\
\n\\int (1 – e\u00a0 \\cos u)du &=& \\frac{2 \\pi}{T} \\int dt\\\\
\n\\therefore\\ \\ u – e \\sin u &=& \\frac{2 \\pi}{T} t
\n\\end{eqnarray}<\/p>\n\u30b1\u30d7\u30e9\u30fc\u904b\u52d5\u306e\u6642\u9593\u5e73\u5747<\/h3>\n
\n\\left\\langle\\frac{1}{r} \\right\\rangle &\\equiv& {\\color{blue}\\frac{1}{T}} \\int_0^T \\frac{1}{r} {\\color{blue}dt} \\\\
\n&=& {\\color{red}\\frac{1}{2\\pi}} \\int_0^{2\\pi} \\frac{1}{r} {\\color{red}\\frac{r}{a} du} \\\\
\n&=& \\frac{1}{a}
\n\\end{eqnarray}<\/p>\nMaxima-Jupyter \u3067\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f<\/h3>\n
\u30b1\u30d5\u309a\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092 Maxima \u3066\u3099\u6570\u5024\u7684\u306b\u89e3\u304f<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\nx &=& r \\cos\\varphi = a(\\cos u – e)\\\\
\ny &=& r \\sin\\varphi = a \\sqrt{1-e^2} \\sin u\\\\
\n\\frac{2\\pi}{T} t &=& u – e \\sin u
\n\\end{eqnarray}<\/p>\n
\n\u305d\u3053\u3067\uff0c\u5468\u671f $T$ \u3092 $N$ \u7b49\u5206\u3057\uff0c
\n$$t_i = \\frac{T}{N} \\times i, \\quad (i = 0, 1, \\dots, N)$$<\/p>\n\/* \u8ecc\u9053\u9577\u534a\u5f84 *\/<\/span>\r\na<\/span>:<\/span> 5$\r\n\r\n\/* \u96e2\u5fc3\u7387 *\/<\/span>\r\ne<\/span>:<\/span> 6\/<\/span>10$\r\n\r\n\/* \u5206\u5272\u6570 *\/<\/span>\r\nN<\/span>:<\/span> 36$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nfind_root()<\/code> \u95a2\u6570\u3067\u6570\u5024\u7684\u306b\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\ng<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> u<\/span> -<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nfor<\/span> i<\/span>:<\/span>0<\/span> thru<\/span> N<\/span> do<\/span> \r\n u<\/span>[<\/span>i<\/span>]<\/span>:<\/span> find_root<\/span>(<\/span>g<\/span>(<\/span>u<\/span>)<\/span> =<\/span> 2*<\/span>%pi<\/span>\/<\/span>N<\/span> *<\/span> i<\/span>, u<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nxu<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> a<\/span>*<\/span>(<\/span>cos<\/span>(<\/span>u<\/span>)<\/span> -<\/span> e<\/span>)<\/span>;\r\nyu<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> a<\/span>*<\/span>sqrt<\/span>(<\/span>1-<\/span>e<\/span>**<\/span>2<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nxy<\/span>:<\/span> makelist<\/span>([<\/span>xu<\/span>(<\/span>u<\/span>[<\/span>i<\/span>])<\/span>, yu<\/span>(<\/span>u<\/span>[<\/span>i<\/span>])]<\/span>, i<\/span>, 0<\/span>, N<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\nplot2d<\/span>([<\/span>discrete<\/span>, xy<\/span>]<\/span>, [<\/span>style<\/span>, [<\/span>points<\/span>, 1.<\/span>5]]<\/span>, \r\n [<\/span>same_xy<\/span>]<\/span>, [<\/span>x<\/span>, -<\/span>9<\/span>, 3]<\/span>, [<\/span>y<\/span>, -<\/span>6<\/span>, 6])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/fig-kepeq1.svg\")
<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\nr &=& \\frac{a (1-e^2)}{1 + e \\cos\\varphi}\\\\
\nx &=& r \\cos\\varphi\\\\
\ny &=& r \\sin\\varphi
\n\\end{eqnarray}<\/p>\nr<\/span>