\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306b\u3064\u3044\u3066\u304a\u3055\u3089\u3044\u3057\u3066\u304a\u304f\u3002<\/p>\n
\u3053\u306e\u30b5\u30a4\u30c8\u306e\u4ed6\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u306f\uff0c<\/p>\n
\u306a\u3069\u3068\u533a\u5225\u3057\u3066\u8868\u8a18\u3057\u3066\u3044\u308b\u304c\uff0c\u3053\u306e\u7bc0\u306b\u3064\u3044\u3066\u306f\uff0c4\u5143\u30d9\u30af\u30c8\u30eb\u306f\u73fe\u308c\u305a\uff0c3\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u306e\u307f\u306a\u306e\u3067\uff0c<\/p>\n
\u4e07\u6709\u5f15\u529b\u306e1\u4f53\u554f\u984c\u3068\u306f\uff0c\u4e07\u6709\u5f15\u529b\u3092\u53ca\u307c\u3059\u91cd\u529b\u6e90\u3067\u3042\u308b\u8cea\u91cf $M$ \u306e\u5929\u4f53\u304c\u539f\u70b9\u306b\u9759\u6b62\u3057\u3066\u304a\u308a\uff0c\u305d\u306e\u307e\u308f\u308a\u3092\u8cea\u91cf $m$ \u306e\u30c6\u30b9\u30c8\u5929\u4f53<\/strong><\/span>\uff08\u8cea\u91cf\u304c\u5341\u5206\u5c0f\u3055\u3044\u305f\u3081\uff0c\u91cd\u529b\u6e90\u306e\u5929\u4f53\u306b\u529b\u3092\u53ca\u307c\u3057\u3066\u52d5\u304b\u3059\u3053\u3068\u306f\u306a\u3044\u3068\u4eee\u5b9a\u3067\u304d\u308b\u5929\u4f53\u306e\u3053\u3068\uff09\u304c\u904b\u52d5\u3057\u3066\u3044\u308b\uff0c\u3068\u3059\u308b\u72b6\u6cc1\u8a2d\u5b9a\u306e\u554f\u984c\u306e\u3053\u3068\u3002<\/p>\n \u3053\u306e\u5834\u5408\uff0c\u539f\u70b9\u304b\u3089\u30c6\u30b9\u30c8\u5929\u4f53\u307e\u3067\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3092 $\\boldsymbol{r}$ \u3068\u3059\u308b\u3068\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n $$m \\frac{d^2 \\boldsymbol{r}}{dt^2} = – \\frac{G M m}{r^3} \\boldsymbol{r}$$<\/p>\n \u3042\u308b\u3044\u306f\u4e21\u8fba\u306e $m$ \u3092\u30ad\u30e3\u30f3\u30bb\u30eb\u3055\u305b\u3066<\/p>\n $$\\frac{d^2 \\boldsymbol{r}}{dt^2} = – \\frac{G M}{r^3} \\boldsymbol{r}$$<\/p>\n \u306e2\u3064\u306e\u5929\u4f53\u304c\u4e92\u3044\u306b\u4e07\u6709\u5f15\u529b\u3092\u53ca\u307c\u3057\u5408\u3063\u3066\u904b\u52d5\u3059\u308b\u3002\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\uff0c<\/p>\n $$m_1 \\frac{d^2\\boldsymbol{r}_1}{dt^2}\u00a0 = – \\frac{Gm_1 m_2 (\\boldsymbol{r}_1 – \\boldsymbol{r}_2)}{|\\boldsymbol{r}_1 – \\boldsymbol{r}_2|^3} \\tag{1}$$<\/p>\n $$m_2 \\frac{d^2\\boldsymbol{r}_2}{dt^2}\u00a0 = – \\frac{Gm_2 m_1 (\\boldsymbol{r}_2 – \\boldsymbol{r}_1)}{|\\boldsymbol{r}_2 – \\boldsymbol{r}_1|^3} \\tag{2}$$<\/p>\n \\(\\mbox{(1)}+\\mbox{(2)}\\) \u3088\u308a<\/p>\n $$\\frac{d^2}{dt^2} \\left( m_1 \\boldsymbol{r}_1 + m_2 \\boldsymbol{r}_2 \\right) = \\boldsymbol{0}$$<\/p>\n \u8cea\u91cf\u4e2d\u5fc3<\/strong><\/span>\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\boldsymbol{R}\\) \u3092<\/p>\n $$\\boldsymbol{R} \\equiv \\frac{m_1 \\boldsymbol{r}_1 + m_2 \\boldsymbol{r}_2}{M}, \\(\\displaystyle \\frac{1}{m_2}\\times\\mbox{(2)}-\\frac{1}{m_1}\\times\\mbox{(1)}\\) \u3088\u308a\uff0c\u76f8\u5bfe\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb<\/strong><\/span> \\(\\boldsymbol{r} \\equiv \\boldsymbol{r}_2 – \\boldsymbol{r}_1\\) \u306b\u5bfe\u3057\u3066<\/p>\n $$\\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 = – \\frac{GM \\boldsymbol{r}}{r^3} 2\u3064\u306e\u5929\u4f53\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{r}_1, \\boldsymbol{r}_2\\) \u306b\u95a2\u3059\u308b\u9023\u7acb\u5fae\u5206\u65b9\u7a0b\u5f0f\u3060\u3063\u305f2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u304c \\(\\boldsymbol{r}\\) \u306b\u5bfe\u3059\u308b1\u7d44\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306b\u306a\u308b\u3053\u3068\u3092\uff0c\u696d\u754c\u7528\u8a9e\u3067<\/p>\n 2\u4f53\u554f\u984c\u306f1\u4f53\u554f\u984c\u306b\u5e30\u7740\u3059\u308b<\/strong><\/span><\/p>\n \u3068\u3044\u3046\u3002<\/p>\n \u307e\u305a\uff0c\\(\\mbox{(3)}\\) \u5f0f\u306b \\(\\boldsymbol{r}\\) \u3092\u5916\u7a4d\u3057\u3066\u3084\u308b\u3068<\/p>\n $$\\boldsymbol{r}\\times \\frac{d^2\\boldsymbol{r}}{dt^2} = -\\frac{GM}{r^3} \\boldsymbol{r}\\times \\boldsymbol{r} = \\boldsymbol{0}$$<\/p>\n $$\\therefore\\\u00a0 \\boldsymbol{r}\\times \\frac{d^2\\boldsymbol{r}}{dt^2} = \u5b9a\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{\\ell}\\) \u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u89d2\u904b\u52d5\u91cf\u306b\u76f8\u5f53\u3059\u308b\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308a\uff0c\u4e00\u5b9a\u3067\u3042\u308b\u305d\u306e\u5411\u304d\u3092 \\(z\\) \u8ef8\u306b\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002 \\(\\ell\\) \u3092\u6975\u5ea7\u6a19\u3067\u3042\u3089\u308f\u3059\u3068<\/p>\n \\begin{eqnarray} \u3082\u3046\u4e00\u3064\u306e\u4fdd\u5b58\u91cf\u306f\uff0c\\(\\mbox{(3)}\\) \u5f0f\u306b \\(\\displaystyle\\frac{d\\boldsymbol{r}}{dt}\\) \u3092\u5185\u7a4d\u3057\u3066\u3084\u308b\u3068\u5f97\u3089\u308c\u308b\u3002<\/p>\n \\begin{eqnarray} \\(\\varepsilon\\) \u306f\u5358\u4f4d\u8cea\u91cf\u3042\u305f\u308a\u306e\u529b\u5b66\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\uff09\u306b\u76f8\u5f53\u3059\u308b\u5b9a\u6570\u3067\u3042\u308a\uff0c\u6975\u5ea7\u6a19\u3067\u3042\u3089\u308f\u3059\u3068<\/p>\n \\begin{eqnarray} \u904b\u52d5\u304c\u6709\u754c\u306a\u675f\u7e1b\u8ecc\u9053\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n $$ 0 < r_{\\rm min} \\leq r \\leq r_{\\rm max}$$<\/p>\n \u3068\u306a\u308b\u3002$r = r_{\\rm min}$ \u304a\u3088\u3073 $r = r_{\\rm max}$ \u3067\u306f $r$ \u304c\u6975\u5024\u3092\u3068\u308b\u3053\u3068\u304b\u3089 $\\displaystyle \\frac{dr}{dt} = 0$ \u3068\u306a\u308b\u304b\u3089 (B) \u5f0f\u304b\u3089<\/p>\n \\begin{eqnarray} \u3053\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092 $\\varepsilon, \\ell^2$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n \\begin{eqnarray} \u3053\u308c\u3089\u3092<\/p>\n $$r_{\\rm max} \\equiv a (1 + e), \\quad r_{\\rm min} \\equiv a (1 -e)$$<\/p>\n \u3059\u306a\u308f\u3061<\/p>\n \\begin{eqnarray} \u3067\u5b9a\u7fa9\u3055\u308c\u308b $a, \\, e$ \u3092\u4f7f\u3063\u3066\u66f8\u304d\u8868\u3059\u3068<\/p>\n \\begin{eqnarray} \u3068\u306a\u308b\u3002\u73fe\u6bb5\u968e\u3067\u306f $a$ \u306f $r_{\\rm max} $ \u3068 $r_{\\rm min} $ \u306e\u5e73\u5747\u5024\uff0c$e$ \u306f\u305d\u308c\u3089\u306e\u5dee\u304b\u3089\u3064\u304f\u3089\u308c\u308b\u7121\u6b21\u5143\u91cf\u3068\u3044\u3046\u610f\u5473\u5408\u3044\u3057\u304b\u306a\u3044\u304c\uff0c\u5f8c\u3005\u306b\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u8ecc\u9053\u304c\u6955\u5186\u3067\u3042\u308b\u3053\u3068\u304c\u89e3\u3051\u305f\u3042\u304b\u3064\u304d\u306b\u306f\uff0c$a$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\uff0c$e$ \u306f\u96e2\u5fc3\u7387\u3068\u547c\u3070\u308c\u308b\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n $r$ \u3092 $\\phi$ \u3092\u901a\u3057\u3066 $t$ \u306e\u95a2\u6570\u3067\u3042\u308b\u3068\u3059\u308b\u3068<\/p>\n $$ \\frac{dr}{dt} = \\frac{dr}{d\\phi}\\,\\frac{d\\phi}{dt} = \\frac{\\ell}{r^2} \\frac{dr}{d\\phi}$$<\/p>\n \u3067\u3042\u308b\u304b\u3089 (B) \u5f0f\u306f<\/p>\n \\begin{eqnarray} \\(r\\) \u304c\u3053\u3068\u3054\u3068\u304f\u5206\u6bcd\u306b\u73fe\u308c\u3066\u3044\u308b\u72b6\u6cc1\u3092\u9451\u307f\uff0c\\(\\displaystyle s \\equiv \\frac{1}{r}\\)\u00a0 \u3068\u3044\u3046\u5909\u6570\u3067\u66f8\u304d\u76f4\u3059\u3068<\/p>\n \\begin{eqnarray} \u3053\u3053\u3067 \u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u306b\u304a\u3051\u308b\u7c92\u5b50\u306e\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/a>\u3068\u306e\u985e\u4f3c\u6027\u306b\u522e\u76ee\u305b\u3088\u3002<\/p>\n \u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f\u306f\uff0c\u3042\u3089\u305f\u3081\u3066\u5909\u6570\u3092 \u3053\u306e\u5f0f\u306f\u5909\u6570\u5206\u96e2\u5f62\u306b\u3067\u304d\u3066<\/p>\n $$\\pm \\frac{d(b u_0)}{\\sqrt{1 – (b u_0)^2}} = d\\phi$$<\/p>\n \u300c\u88dc\u8db3<\/a>\u300d\u306b\u66f8\u3044\u305f\u3088\u3046\u306b\uff0c\u521d\u671f\u6761\u4ef6\u3068\u3057\u3066 \\(\\phi = 0\\) \u306e\u3068\u304d\u306b \\(u_0\\) \u3057\u305f\u304c\u3063\u3066 \\(r\\) \u304c\u6975\u5024\u3092\u3068\u308b\u3068\u3059\u308c\u3070\uff0c\u305f\u3060\u3061\u306b\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6c42\u307e\u308b\u3002 $$\\therefore \\ r = \\frac{a(1-e^2)}{1 + e\\cos\\phi}$$<\/p>\n \u3068\u306a\u308a\uff0c\u3053\u306e\u8ecc\u9053\u306f\uff0c\u9577\u534a\u5f84<\/strong><\/span> \\(a\\)\uff0c\u96e2\u5fc3\u7387<\/strong><\/span> \\(e\\) \u306e\u6955\u5186<\/strong><\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u4e00\u76ee\u77ad\u7136\u3068\u306a\u308b\u3002<\/p>\n \\(a\\)\uff0c\\(e\\) \u3092\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09\u3092\u4f7f\u3063\u3066\u66f8\u304d\u76f4\u3057\u3066\u3084\u308b\u3068\uff0c\u675f\u7e1b\u8ecc\u9053\u306e\u5834\u5408\u306b \\( \\varepsilon = -|\\varepsilon|\\) \u3068\u306a\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068 \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306f\uff0c1\u4f53\u554f\u984c\u306b\u5e30\u7740\u3057\u3066\uff08\u675f\u7e1b\u8ecc\u9053\u306e\u5834\u5408\u306b\u306f\uff09\u6955\u5186\u8ecc\u9053\u304c\u89e3\u3068\u306a\u308b\u3053\u3068\u304c\u89e3\u6790\u7684\u306b\u793a\u3055\u308c\u305f\u308f\u3051\u3060\u304c\uff0c\u6211\u3005\u304c\u7d20\u6734\u306b\u601d\u3046\u3068\u3053\u308d\u306e\u300c\u89e3\u300d\u3068\u306f\u5c11\u3057\u8da3\u304c\u9055\u3046\u3002<\/p>\n \u305f\u3068\u3048\u3070\uff0c\u4e00\u69d8\u91cd\u529b\u5834\u4e2d\u306e\u659c\u65b9\u6295\u5c04\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b \\(x\\) \u5ea7\u6a19\u3068 \\(y\\) \u5ea7\u6a19\u304c\u6642\u9593 \\(t\\) \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u3042\u304b\u3089\u3055\u307e\u306b\u89e3\u3051\u305f\u3002<\/p>\n \\begin{eqnarray} \u3057\u304b\u3057\uff0c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u89e3\u3067\u3042\u308b\u6955\u5186\u8ecc\u9053\u306f\uff0c\\(r\\) \u304c \\(\\phi\\) \u306e\u95a2\u6570\u3068\u3057\u3066<\/p>\n $$r = \\frac{a(1-e^2)}{1 + e\\cos\\phi}$$<\/p>\n \u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3060\u3051\u3067\uff0c\u6642\u9593 \\(t\\) \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u3042\u304b\u3089\u3055\u307e\u306b\u89e3\u3051\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u3042\u308b\u6642\u523b \\(t\\) \u306e\u3068\u304d\u306e\u5929\u4f53\u306e\u4f4d\u7f6e \\( (x(t), y(t)) \\)\u00a0 \u3042\u308b\u3044\u306f \\( (r(t), \\phi(t)) \\) \u3092\u77e5\u308a\u305f\u3044\u3068\u3044\u3046\u5834\u5408\u306b\u306f\u5225\u306e\u5de5\u592b\u304c\u5fc5\u8981\u306b\u306a\u308b\u3002\u3053\u306e\u3042\u305f\u308a\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e Memo \u3092\u53c2\u7167\u3002<\/p>\n2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n
\n
\n\\quad M \\equiv m_1 + m_2$$
\n\u3068\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u4e0a\u5f0f\u304b\u3089
\n$$\\frac{d^2\\boldsymbol{R}}{dt^2} = \\boldsymbol{0}$$
\n\u3053\u308c\u304b\u3089\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\uff0c\u8cea\u91cf\u4e2d\u5fc3\u306f\u9759\u6b62\u3057\u3066\u3044\u308b\u3068\u3057\u3066\u8a71\u3092\u9032\u3081\u308b\u3002<\/p>\n
\n\\tag{3}$$<\/p>\n\u4fdd\u5b58\u91cf\uff08\u904b\u52d5\u306e\u5b9a\u6570\uff09<\/h3>\n
\u89d2\u904b\u52d5\u91cf\u4fdd\u5b58<\/h4>\n
\n\\frac{d}{dt}\\left(\\boldsymbol{r}\\times\u00a0 \\frac{d\\boldsymbol{r}}{dt} \\right) = \\boldsymbol{0}$$
\n$$\\therefore\\\u00a0 \\boldsymbol{r}\\times\u00a0 \\frac{d\\boldsymbol{r}}{dt} = \\mbox{const.} \\equiv \\boldsymbol{\\ell}$$<\/p>\n
\n$$\\boldsymbol{\\ell} = (0, 0, \\ell)$$
\n\u3055\u3089\u306b\uff0c
\n$$\\boldsymbol{r}\\cdot\\boldsymbol{\\ell} =
\n\\boldsymbol{r}\\cdot\\left(\\boldsymbol{r}\\times\u00a0 \\frac{d\\boldsymbol{r}}{dt} \\right) =
\n\\frac{d\\boldsymbol{r}}{dt}\\cdot(\\boldsymbol{r}\\times\\boldsymbol{r}) = 0,
\n\\quad\\therefore\\ \\boldsymbol{r} \\perp \\boldsymbol{\\ell}$$
\n\u3068\u306a\u308a\uff0c\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f \\(\\boldsymbol{r}\\) \u3092\\(xy\\) \u5e73\u9762\uff08\u8d64\u9053\u9762\uff09\u4e0a\u306b\u3068\u308b\u3053\u3068\u304c\u3067\u304d\u3066
\n$$\\boldsymbol{r} = (x, y, 0) = (r\\cos\\phi, r\\sin\\phi, 0)$$<\/p>\n
\n\\ell = |\\boldsymbol{\\ell}| =x\\frac{dy}{dt}\u00a0 -y\\frac{dx}{dt} = r^2 \\frac{d\\phi}{dt},
\n\\quad \\therefore\\ \\frac{d\\phi}{dt} = \\frac{\\ell}{r^2} \\tag{A}
\n\\end{eqnarray}<\/p>\n\u30a8\u30cd\u30eb\u30ae\u30fc\u4fdd\u5b58<\/h4>\n
\n\\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d^2\\boldsymbol{r}}{dt^2}\u00a0 &=&
\n– \\frac{GM}{r^3} \\boldsymbol{r}\\cdot\\frac{d\\boldsymbol{r}}{dt}\\\\
\n\\frac{d}{dt}\\left(\\frac{1}{2}\u00a0 \\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d\\boldsymbol{r}}{dt}\\right)&=&
\n– \\frac{GM}{r^3} \\frac{d}{dt}\\left(\\frac{1}{2}\u00a0 \\boldsymbol{r}\\cdot\\boldsymbol{r}\\right)\\\\
\n&=&- \\frac{GM}{r^3} \\frac{d}{dt}\\left(\\frac{1}{2}\u00a0 r^2\\right) \\\\
\n&=& \\frac{d}{dt} \\left(\\frac{GM}{r}\\right)\\\\
\n\\end{eqnarray}
\n$$\\therefore\\ \\ \\frac{d}{dt} \\left(\\frac{1}{2}\u00a0 \\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d\\boldsymbol{r}}{dt}-\\frac{GM}{r} \\right) = 0$$
\n$$\\therefore\\ \\\u00a0 \\frac{1}{2}\u00a0 \\frac{d\\boldsymbol{r}}{dt}\\cdot\\frac{d\\boldsymbol{r}}{dt}-\\frac{GM}{r}\u00a0 =
\n\\mbox{const.} \\equiv \\varepsilon$$<\/p>\n
\n\\varepsilon &=& \\frac{1}{2} \\left\\{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2 \\right\\} – \\frac{GM}{r} \\\\
\n&=& \\frac{1}{2}\\left\\{\\left(\\frac{dr}{dt}\\right)^2 +r^2\\left(\\frac{d\\phi}{dt}\\right)^2 \\right\\} – \\frac{GM}{r}\\\\
\n&=& \\frac{1}{2}\\left\\{\\left(\\frac{dr}{dt}\\right)^2 +\\frac{\\ell^2}{r^2} \\right\\} – \\frac{GM}{r} \\\\
\n\\therefore\\ \\ \\left(\\frac{dr}{dt}\\right)^2 &=& 2 \\varepsilon + \\frac{2 G M}{r} – \\frac{\\ell^2}{r^2} \\tag{B}
\n\\end{eqnarray}<\/p>\n\u904b\u52d5\u304c\u6709\u754c\u306a\u5834\u5408<\/h3>\n
\n0 &=& 2 \\varepsilon + \\frac{2 G M}{r_{\\rm min}} – \\frac{\\ell^2}{r_{\\rm min}^2} \\\\
\n0 &=& 2 \\varepsilon + \\frac{2 G M}{r_{\\rm max}} – \\frac{\\ell^2}{r_{\\rm max}^2}
\n\\end{eqnarray}<\/p>\n
\n\\varepsilon &=& -\\frac{G M}{r_{\\rm max} + r_{\\rm min}} \\\\
\n\\ell^2 &=& \\frac{2 G M \\,r_{\\rm max}\\, r_{\\rm min}}{r_{\\rm max} + r_{\\rm min}}
\n\\end{eqnarray}<\/p>\n
\na &\\equiv& \\frac{r_{\\rm max} + r_{\\rm min}}{2} \\\\
\ne &\\equiv& \\frac{r_{\\rm max} -r_{\\rm min}}{r_{\\rm max} + r_{\\rm min}}
\n\\end{eqnarray}<\/p>\n
\n\\varepsilon &=& -\\frac{G M}{2 a} \\\\
\n\\ell^2 &=&\u00a0 G M a ( 1 -e^2)
\n\\end{eqnarray}<\/p>\n\u8ecc\u9053\u3092\u6c7a\u3081\u308b\u5f0f<\/h3>\n
\n\\left(\\frac{\\ell}{r^2} \\frac{dr}{d\\phi}\\right)^2 &=& 2 \\varepsilon + \\frac{2 G M}{r} – \\frac{\\ell^2}{r^2} \\\\
\n\\therefore\\ \\ \\left(\\frac{1}{r^2} \\frac{dr}{d\\phi}\\right)^2 &=& \\frac{2 \\varepsilon}{\\ell^2} + \\frac{2 G M}{\\ell^2 }\\frac{1}{r}\u00a0 -\\frac{1}{r^2} \\\\
\n&=& -\\frac{1}{a^2 (1 -e^2)} + \\frac{2}{a (1 -e^2)} \\frac{1}{r}\u00a0 -\\frac{1}{r^2}
\n\\end{eqnarray}<\/p>\n
\n\\left(\\frac{ds}{d\\phi}\\right)^2 &=& -\\frac{1}{a^2 (1 -e^2)} + \\frac{2}{a (1 -e^2)} s\u00a0 -s^2 \\\\
\n&=& -\\frac{1}{a^2 (1 -e^2)}\u00a0 -\\left(s -\\frac{1}{a (1 -e^2)} \\right)^2 + \\frac{1}{a^2 (1 -e^2)^2} \\\\
\n&=& \\frac{1}{b^2}\u00a0 -\\left(s -\\frac{1}{a (1 -e^2)} \\right)^2
\n\\end{eqnarray}<\/p>\n
\n$$\\frac{1}{b} \\equiv \\frac{e}{a (1 -e^2)} $$<\/p>\n\u89e3\uff1a\u6955\u5186\u8ecc\u9053<\/h3>\n
\n$$u_0 \\equiv s -\\frac{1}{a (1 -e^2)}$$
\n\u3068\u7f6e\u304d\u76f4\u3059\u3068\uff0c
\n$$\\left(\\frac{du_0}{d\\phi}\\right)^2 = \\frac{1}{b^2} – u_0^2$$<\/p>\n
\n\\begin{eqnarray}
\nu_0 &=& s -\\frac{1}{a (1 -e^2)} = \\frac{\\cos\\phi}{b} = \\frac{e \\cos\\phi}{a (1 -e^2)} \\\\
\n\\therefore\\ \\ s &=& \\frac{1}{r} =\\frac{1 + e \\cos\\phi}{a (1 -e^2)}
\n\\end{eqnarray}<\/p>\n
\n$$a = \\frac{GM}{2|\\varepsilon|}, \\quad e = \\sqrt{1 – \\frac{2 |\\varepsilon| \\ell^2}{(GM)^2}}$$<\/p>\n\u6955\u5186\u8ecc\u9053\u3067\u3042\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u305f\u3082\u306e\u306e…<\/h4>\n
\nx(t) &=& x_0 + v_{0x} t \\\\
\ny(t) &=& y_0 + v_{0y} t – \\frac{1}{2} g t^2
\n\\end{eqnarray}<\/p>\n\n