{"id":7419,"date":"2024-01-26T16:42:57","date_gmt":"2024-01-26T07:42:57","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=7419"},"modified":"2025-06-07T10:23:39","modified_gmt":"2025-06-07T01:23:39","slug":"%e9%ab%98%e3%81%95-h-%e3%81%8b%e3%82%89%e3%81%ae%e6%96%9c%e6%96%b9%e6%8a%95%e5%b0%84%e3%81%ae%e6%9c%80%e5%a4%a7%e6%b0%b4%e5%b9%b3%e5%88%b0%e9%81%94%e8%b7%9d%e9%9b%a2%e3%82%92%e9%99%b0%e9%96%a2","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/7419\/","title":{"rendered":"\u9ad8\u3055 h \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b"},"content":{"rendered":"

\u300c\u9ad8\u3055 h \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u5230\u9054\u8ddd\u96e2\u3092\u6c42\u3081\u308b\u6e96\u5099<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002<\/p>\n

\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3068\u306a\u308b\u6253\u3061\u51fa\u3057\u89d2\u5ea6 $\\theta$ \u3092\uff0c\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3044\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066 SymPy \u3084 Maxima \u3092\u4f7f\u3063\u3066\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n

\n
<\/div>\n
\n
\n

SymPy \u3067\u9ad8\u3055 $h$ \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
from<\/span> sympy<\/span> import<\/span> *<\/span>\r\n# 1\u6587\u5b57\u5909\u6570\u306e Symbol \u306e\u5ba3\u8a00\u304c\u7701\u7565\u3067\u304d\u308b<\/span>\r\nfrom<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\n\r\ninit_printing<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u659c\u65b9\u6295\u5c04\u306e\u89e3<\/h4>\n
\\begin{eqnarray}
\nX(T, \\theta) &=& \\cos\\theta\\cdot T \\\\
\nY(T, \\theta) &=& H+ \\sin\\theta\\cdot T – \\frac{1}{2} T^2
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
var<\/span>(<\/span>'H'<\/span>)<\/span>\r\ndef<\/span> X<\/span>(<\/span>T<\/span>,<\/span> theta<\/span>):<\/span>\r\n    return<\/span> cos<\/span>(<\/span>theta<\/span>)<\/span>*<\/span>T<\/span>\r\n\r\ndef<\/span> Y<\/span>(<\/span>T<\/span>,<\/span> theta<\/span>):<\/span>\r\n    return<\/span> H<\/span> +<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>*<\/span>T<\/span> -<\/span> T<\/span>**<\/span>2<\/span>\/<\/span>2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
X<\/span>(<\/span>T<\/span>,<\/span> theta<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
$\\displaystyle T \\cos{\\left(\\theta \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
Y<\/span>(<\/span>T<\/span>,<\/span> theta<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
$\\displaystyle H – \\frac{T^{2}}{2} + T \\sin{\\left(\\theta \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6ede\u7a7a\u6642\u9593 $T_1$<\/h4>\n
\u6ede\u7a7a\u6642\u9593 $T_1\\ \\ (>0)$ \u306f\u4ee5\u4e0b\u306e\u5f0f\u3092\u6e80\u305f\u3059\u3002<\/p>\n
$$Y(T_1, \\theta) = 0 \\quad\\Rightarrow\\quad T_1 = T_1(\\theta)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
var<\/span>(<\/span>'T1'<\/span>)<\/span>\r\nsolve<\/span>(<\/span>Y<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>),<\/span> T1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
$\\displaystyle \\left[ – \\sqrt{2 H + \\sin^{2}{\\left(\\theta \\right)}} + \\sin{\\left(\\theta \\right)}, \\ \\sqrt{2 H + \\sin^{2}{\\left(\\theta \\right)}} + \\sin{\\left(\\theta \\right)}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3046<\/h4>\n
\u7c21\u5358\u306a\u5834\u5408\u306f\uff0c$T_1$ \u306f $\\theta$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\uff0c\u4e00\u822c\u306b\u306f\u305d\u3046\u306f\u3044\u304b\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u305d\u3053\u3067\uff0c$T_1$ \u306f $\\theta$ \u306e\u9670\u95a2\u6570\u3067\u3042\u308b\u3068\u3057\u3066\u8a71\u3092\u9032\u3081\u308b\u3002<\/p>\n
$Y(T_1(\\theta), \\theta) = 0$ \u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n
$$\\frac{d Y(T_1(\\theta), \\theta)}{d\\theta} =
\n\\frac{\\partial Y}{\\partial T_1} \\frac{d T_1}{d\\theta} +
\n\\frac{\\partial Y}{\\partial \\theta}
\n= 0$$<\/p>\n
\u3088\u308a\uff0c$\\displaystyle \\frac{d T_1}{d\\theta}$ \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
T1<\/span> =<\/span> Function<\/span>(<\/span>'T1'<\/span>)(<\/span>theta<\/span>)<\/span>\r\n\r\ndY<\/span> =<\/span> diff<\/span>(<\/span>Y<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>),<\/span> theta<\/span>)<\/span>\r\nsol1<\/span> =<\/span> solve<\/span>(<\/span>dY<\/span>,<\/span> diff<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>))<\/span>\r\nsol1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
$\\displaystyle \\left[ \\frac{T_{1}{\\left(\\theta \\right)} \\cos{\\left(\\theta \\right)}}{T_{1}{\\left(\\theta \\right)} – \\sin{\\left(\\theta \\right)}}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[7]:<\/div>\n
\n
\n
\n
dT1<\/span> =<\/span> sol1<\/span>[<\/span>0<\/span>]<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>),<\/span> dT1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[7]:<\/div>\n
$\\displaystyle \\frac{d}{d \\theta} T_{1}{\\left(\\theta \\right)} = \\frac{T_{1}{\\left(\\theta \\right)} \\cos{\\left(\\theta \\right)}}{T_{1}{\\left(\\theta \\right)} – \\sin{\\left(\\theta \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $L$<\/h4>\n
\u6ede\u7a7a\u6642\u9593 $T_1(\\theta)$ \u306e\u9593\u306b\u6c34\u5e73\u65b9\u5411\u306b\u9032\u3080\u8ddd\u96e2\u306f<\/p>\n
$$L = X(T_1(\\theta), \\theta)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[8]:<\/div>\n
\n
\n
\n
def<\/span> L<\/span>(<\/span>theta<\/span>):<\/span>\r\n    return<\/span> X<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$L$ \u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u3092\u6c42\u3081\u308b<\/h4>\n
$\\displaystyle \\frac{dL}{d\\theta} = 0$ \u3068\u306a\u308b\u89d2\u5ea6 $\\theta \\equiv \\theta_{\\rm max}$ \u3092\u6c42\u3081\u308b\u3002\u9670\u95a2\u6570\u5b9a\u7406\u306e\u7d50\u679c\u3092 .subs()<\/code> \u3067\u4ee3\u5165\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[9]:<\/div>\n
\n
\n
\n
dL<\/span> =<\/span> diff<\/span>(<\/span>L<\/span>(<\/span>theta<\/span>),<\/span> theta<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>diff<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>),<\/span> dT1<\/span>)<\/span>\r\ndL<\/span> =<\/span> simplify<\/span>(<\/span>dL<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>L<\/span>(<\/span>theta<\/span>),<\/span> theta<\/span>),<\/span> dL<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[9]:<\/div>\n
$\\displaystyle \\frac{d}{d \\theta} T_{1}{\\left(\\theta \\right)} \\cos{\\left(\\theta \\right)} = \\frac{\\left(- T_{1}{\\left(\\theta \\right)} \\sin{\\left(\\theta \\right)} + 1\\right) T_{1}{\\left(\\theta \\right)}}{T_{1}{\\left(\\theta \\right)} – \\sin{\\left(\\theta \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$L$ \u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u306f\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f<\/p>\n
\\begin{eqnarray}
\nY(T_1(\\theta), \\theta) &=& 0 \\\\
\n\\frac{d}{d\\theta} L(\\theta) &=& 0
\n\\end{eqnarray}<\/p>\n
\u3092\uff0c$\\sin\\theta$ \u3068 $T_1(\\theta)$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[10]:<\/div>\n
\n
\n
\n
sol2<\/span> =<\/span> solve<\/span>([<\/span>Y<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>),<\/span> dL<\/span>],<\/span> \r\n             [<\/span>sin<\/span>(<\/span>theta<\/span>),<\/span> T1<\/span>])<\/span>\r\nsol2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[10]:<\/div>\n
$\\displaystyle \\left[ \\left( – \\frac{\\sqrt{2}}{2 \\sqrt{H + 1}}, \\ – \\sqrt{2 H + 2}\\right), \\ \\left( \\frac{\\sqrt{2}}{2 \\sqrt{H + 1}}, \\ \\sqrt{2 H + 2}\\right)\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\sin\\theta > 0, \\ T_1(\\theta) > 0$ \u3067\u3042\u308b\u304b\u3089\uff0c\u89e3\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[11]:<\/div>\n
\n
\n
\n
sinth<\/span> =<\/span> sol2<\/span>[<\/span>1<\/span>][<\/span>0<\/span>]<\/span>\r\nsinth<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[11]:<\/div>\n
$\\displaystyle \\frac{\\sqrt{2}}{2 \\sqrt{H + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$$ \\sin\\theta_{\\rm max} = \\frac{1}{\\sqrt{2 (1+H)}}$$\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u304b\u3089\uff0c$\\tan\\theta_{\\rm max}$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[12]:<\/div>\n
\n
\n
\n
simplify<\/span>(<\/span>sinth<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>sinth<\/span>**<\/span>2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[12]:<\/div>\n
$\\displaystyle \\frac{1}{\\sqrt{\\frac{2 H + 1}{H + 1}} \\sqrt{H + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[13]:<\/div>\n
\n
\n
\n
# \u4e8c\u4e57\u3057\u3066\u5e73\u65b9\u6839<\/span>\r\nsqrt<\/span>(<\/span>_<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[13]:<\/div>\n
$\\displaystyle \\sqrt{\\frac{1}{2 H + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u306a\u304b\u306a\u304b\u7c21\u5358\u5316\u3057\u3066\u304f\u308c\u307e\u305b\u3093\u304c\uff0c\u8981\u3059\u308b\u306b<\/p>\n
$$\\tan\\theta_{\\rm max} = \\frac{1}{\\sqrt{1+2 H}}$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2<\/h4>\n
$$L_{\\rm max} = X(T_1(\\theta_{\\rm max}), \\theta_{\\rm max})$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[14]:<\/div>\n
\n
\n
\n
T1max<\/span> =<\/span> sol2<\/span>[<\/span>1<\/span>][<\/span>1<\/span>]<\/span>\r\nthmax<\/span> =<\/span> asin<\/span>(<\/span>sol2<\/span>[<\/span>1<\/span>][<\/span>0<\/span>])<\/span>\r\n\r\nLmax<\/span> =<\/span> X<\/span>(<\/span>T1max<\/span>,<\/span> thmax<\/span>)<\/span>\r\nsimplify<\/span>(<\/span>Lmax<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[14]:<\/div>\n
$\\displaystyle \\sqrt{\\frac{2 H + 1}{H + 1}} \\sqrt{H + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[15]:<\/div>\n
\n
\n
\n
# \u4e8c\u4e57\u3057\u3066\u5e73\u65b9\u6839<\/span>\r\nsqrt<\/span>(<\/span>_<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[15]:<\/div>\n
$\\displaystyle \\sqrt{2 H + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u3082\u306a\u304b\u306a\u304b\u7c21\u5358\u5316\u3057\u3066\u304f\u308c\u307e\u305b\u3093\u304c\uff0c\u8981\u3059\u308b\u306b<\/p>\n
$$L_{\\rm max} = \\sqrt{1 + 2H}$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5225\u89e3<\/h4>\n
\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3044\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u307e\u3059\u304c\uff0c$\\sin\\theta$ \u3092\u6d88\u53bb\u3059\u308b\u65b9\u91dd\u3067\u3084\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{dL}{d\\theta} = 0$ \u3092 $\\theta$ \u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[16]:<\/div>\n
\n
\n
\n
Eq<\/span>(<\/span>dL<\/span>,<\/span> 0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[16]:<\/div>\n
$\\displaystyle \\frac{\\left(- T_{1}{\\left(\\theta \\right)} \\sin{\\left(\\theta \\right)} + 1\\right) T_{1}{\\left(\\theta \\right)}}{T_{1}{\\left(\\theta \\right)} – \\sin{\\left(\\theta \\right)}} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[17]:<\/div>\n
\n
\n
\n
solsin<\/span> =<\/span> solve<\/span>(<\/span>dL<\/span>,<\/span> sin<\/span>(<\/span>theta<\/span>))<\/span>\r\nsolsin<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[17]:<\/div>\n
$\\displaystyle \\left[ \\frac{1}{T_{1}{\\left(\\theta \\right)}}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e $\\sin\\theta$ \u3092 $Y(T_1, \\theta) = 0$ \u306b\u4ee3\u5165\u3057\u30661\u5909\u6570 $T_1$ \u306b\u95a2\u3059\u308b\u65b9\u7a0b\u5f0f\u306b\u3057\uff0c$T_1$ \u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[18]:<\/div>\n
\n
\n
\n
Eq<\/span>(<\/span>Y<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>sin<\/span>(<\/span>theta<\/span>),<\/span> solsin<\/span>[<\/span>0<\/span>]),<\/span> 0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[18]:<\/div>\n
$\\displaystyle H – \\frac{T_{1}^{2}{\\left(\\theta \\right)}}{2} + 1 = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[19]:<\/div>\n
\n
\n
\n
eqT1<\/span> =<\/span> Y<\/span>(<\/span>T1<\/span>,<\/span> theta<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>sin<\/span>(<\/span>theta<\/span>),<\/span> solsin<\/span>[<\/span>0<\/span>])<\/span>\r\nsols<\/span> =<\/span> solve<\/span>(<\/span>eqT1<\/span>,<\/span> T1<\/span>)<\/span>\r\nsols<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[19]:<\/div>\n
$\\displaystyle \\left[ – \\sqrt{2 H + 2}, \\ \\sqrt{2 H + 2}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$T_1 > 0$ \u3067\u3059\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[20]:<\/div>\n
\n
\n
\n
T1max<\/span> =<\/span> sols<\/span>[<\/span>1<\/span>]<\/span>\r\nT1max<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[20]:<\/div>\n
$\\displaystyle \\sqrt{2 H + 2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u3068\u304d\u306e $\\theta$ \u306e\u5024\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[21]:<\/div>\n
\n
\n
\n
thmax<\/span> =<\/span> asin<\/span>(<\/span>1<\/span>\/<\/span>T1max<\/span>)<\/span>\r\nthmax<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[21]:<\/div>\n
$\\displaystyle \\operatorname{asin}{\\left(\\frac{1}{\\sqrt{2 H + 2}} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\sin \\theta_{\\rm max} = $ …<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[22]:<\/div>\n
\n
\n
\n
sin<\/span>(<\/span>thmax<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[22]:<\/div>\n
$\\displaystyle \\frac{1}{\\sqrt{2 H + 2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\tan \\theta_{\\rm max} = $ …<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[23]:<\/div>\n
\n
\n
\n
tan<\/span>(<\/span>thmax<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[23]:<\/div>\n
$\\displaystyle \\frac{1}{\\sqrt{1 – \\frac{1}{2 H + 2}} \\sqrt{2 H + 2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[24]:<\/div>\n
\n
\n
\n
sqrt<\/span>(<\/span>factor<\/span>(<\/span>_<\/span>)<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[24]:<\/div>\n
$\\displaystyle \\sqrt{\\frac{1}{2 H + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u5927\u5230\u9054\u8ddd\u96e2 $L_{\\rm max}$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[25]:<\/div>\n
\n
\n
\n
X<\/span>(<\/span>T1max<\/span>,<\/span> thmax<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[25]:<\/div>\n
$\\displaystyle \\sqrt{1 – \\frac{1}{2 H + 2}} \\sqrt{2 H + 2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[26]:<\/div>\n
\n
\n
\n
sqrt<\/span>(<\/span>simplify<\/span>(<\/span>_<\/span>)<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[26]:<\/div>\n
$\\displaystyle \\sqrt{2 H + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Maxima \u3067\u9ad8\u3055 $h$ \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u659c\u65b9\u6295\u5c04\u306e\u89e3<\/h4>\n
\\begin{eqnarray}
\nX(T, \\theta) &=& \\cos\\theta\\cdot T \\\\
\nY(T, \\theta) &=& H+ \\sin\\theta\\cdot T – \\frac{1}{2} T^2
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
X<\/span>(<\/span>T<\/span>, theta<\/span>)<\/span>:=<\/span> cos<\/span>(<\/span>theta<\/span>)<\/span>*<\/span>T<\/span>;\r\nY<\/span>(<\/span>T<\/span>, theta<\/span>)<\/span>:=<\/span> H<\/span> +<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>*<\/span>T<\/span> -<\/span> T<\/span>**<\/span>2\/<\/span>2<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}X\\left(T , \\vartheta\\right):=\\cos \\vartheta\\,T\\]<\/div>\n<\/div>\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}Y\\left(T , \\vartheta\\right):=H+\\sin \\vartheta\\,T+\\frac{-T^2}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6ede\u7a7a\u6642\u9593 $T_1$<\/h4>\n
\u6ede\u7a7a\u6642\u9593 $T_1\\ \\ (>0)$ \u306f\u4ee5\u4e0b\u306e\u5f0f\u3092\u6e80\u305f\u3059\u3002<\/p>\n
$$Y(T_1, \\theta) = 0 \\quad\\Rightarrow\\quad T_1 = T_1(\\theta)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
solve<\/span>(<\/span>Y<\/span>(<\/span>T1<\/span>, theta<\/span>)<\/span> =<\/span> 0<\/span>, T1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}\\left[ T_{1}=\\sin \\vartheta-\\sqrt{\\sin ^2\\vartheta+2\\,H} , T_{1}=\\sqrt{\\sin ^2\\vartheta+2\\,H}+\\sin \\vartheta \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3046<\/h4>\n
\u7c21\u5358\u306a\u5834\u5408\u306f\uff0c$T_1$ \u306f $\\theta$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\uff0c\u4e00\u822c\u306b\u306f\u305d\u3046\u306f\u3044\u304b\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u305d\u3053\u3067\uff0c$T_1$ \u306f $\\theta$ \u306e\u9670\u95a2\u6570\u3067\u3042\u308b\u3068\u3057\u3066\u8a71\u3092\u9032\u3081\u308b\u3002<\/p>\n
$Y(T_1(\\theta), \\theta) = 0$ \u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n
$$\\frac{d Y(T_1(\\theta), \\theta)}{d\\theta} =
\n\\frac{\\partial Y}{\\partial T_1} \\frac{d T_1}{d\\theta} +
\n\\frac{\\partial Y}{\\partial \\theta}
\n= 0$$<\/p>\n
\u3088\u308a\uff0c$\\displaystyle \\frac{d T_1}{d\\theta}$ \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
depends<\/span>(<\/span>T1<\/span>, theta<\/span>)<\/span>;\r\n\r\ndY<\/span>:<\/span> diff<\/span>(<\/span>Y<\/span>(<\/span>T1<\/span>, theta<\/span>)<\/span>=<\/span>0<\/span>, theta<\/span>)<\/span>;\r\nsol1<\/span>:<\/span> solve<\/span>(<\/span>dY<\/span>, diff<\/span>(<\/span>T1<\/span>, theta<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}\\left[ T_{1}\\left(\\vartheta\\right) \\right] \\]<\/div>\n<\/div>\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\frac{d}{d\\,\\vartheta}\\,T_{1}\\,\\sin \\vartheta+T_{1}\\,\\cos \\vartheta-T_{1}\\,\\left(\\frac{d}{d\\,\\vartheta}\\,T_{1}\\right)=0\\]<\/div>\n<\/div>\n
\n
Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\left[ \\frac{d}{d\\,\\vartheta}\\,T_{1}=-\\frac{T_{1}\\,\\cos \\vartheta}{\\sin \\vartheta-T_{1}} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
dT1<\/span>:<\/span> rhs<\/span>(<\/span>sol1<\/span>[<\/span>1])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}-\\frac{T_{1}\\,\\cos \\vartheta}{\\sin \\vartheta-T_{1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $L$<\/h4>\n
\u6ede\u7a7a\u6642\u9593 $T_1(\\theta)$ \u306e\u9593\u306b\u6c34\u5e73\u65b9\u5411\u306b\u9032\u3080\u8ddd\u96e2\u306f<\/p>\n
$$L = X(T_1(\\theta), \\theta)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
L<\/span>(<\/span>theta<\/span>)<\/span>:=<\/span> X<\/span>(<\/span>T1<\/span>, theta<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}L\\left(\\vartheta\\right):=X\\left(T_{1} , \\vartheta\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$L$ \u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u3092\u6c42\u3081\u308b<\/h4>\n
$\\displaystyle \\frac{dL}{d\\theta} = 0$ \u3068\u306a\u308b\u89d2\u5ea6 $\\theta \\equiv \\theta_{\\rm max}$ \u3092\u6c42\u3081\u308b\u3002\u9670\u95a2\u6570\u5b9a\u7406\u306e\u7d50\u679c\u3092 .ev()<\/code> \u3067\u4ee3\u5165\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
dL<\/span>:<\/span> diff<\/span>(<\/span>L<\/span>(<\/span>theta<\/span>)<\/span>, theta<\/span>)<\/span>$\r\ndL<\/span>:<\/span> trigsimp<\/span>(<\/span>ev<\/span>(<\/span>dL<\/span>, sol1<\/span>[<\/span>1]))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}\\frac{T_{1}^2\\,\\sin \\vartheta-T_{1}}{\\sin \\vartheta-T_{1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$L$ \u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u306f\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f<\/p>\n
\\begin{eqnarray}
\nY(T_1(\\theta), \\theta) &=& 0 \\\\
\n\\frac{d}{d\\theta} L(\\theta) &=& 0
\n\\end{eqnarray}<\/p>\n
\u3092\uff0c$\\sin\\theta$ \u3068 $T_1(\\theta)$ \u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[7]:<\/div>\n
\n
\n
\n
sol2<\/span>:<\/span> solve<\/span>([<\/span>Y<\/span>(<\/span>T1<\/span>, theta<\/span>)<\/span> =<\/span> 0<\/span>, dL<\/span> =<\/span> 0]<\/span>, \r\n            [<\/span>sin<\/span>(<\/span>theta<\/span>)<\/span>, T1<\/span>])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}\\left[ \\left[ \\sin \\vartheta=-\\frac{\\sqrt{2}\\,\\sqrt{H+1}}{2\\,H+2} , T_{1}=-\\sqrt{2}\\,\\sqrt{H+1} \\right] , \\left[ \\sin \\vartheta=\\frac{\\sqrt{2}\\,\\sqrt{H+1}}{2\\,H+2} , T_{1}=\\sqrt{2}\\,\\sqrt{H+1} \\right] \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\sin\\theta > 0, \\ T_1(\\theta) > 0$ \u3067\u3042\u308b\u304b\u3089\uff0c\u89e3\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[8]:<\/div>\n
\n
\n
\n
sinth<\/span>:<\/span> rhs<\/span>(<\/span>sol2<\/span>[<\/span>2][<\/span>1])<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{12}$}\\frac{1}{\\sqrt{2}\\,\\sqrt{H+1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$$ \\sin\\theta_{\\rm max} = \\frac{1}{\\sqrt{2 (1+H)}}$$\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u304b\u3089\uff0c$\\tan\\theta_{\\rm max}$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[9]:<\/div>\n
\n
\n
\n
sinth<\/span>\/<\/span>sqrt<\/span>(<\/span>1-<\/span>sinth<\/span>**<\/span>2<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}\\frac{1}{\\sqrt{H+1}\\,\\sqrt{\\frac{2\\,H+1}{H+1}}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[10]:<\/div>\n
\n
\n
\n
\/* \u4e8c\u4e57\u3057\u3066\u5e73\u65b9\u6839 *\/<\/span>\r\nsqrt<\/span>(<\/span>%<\/span>**<\/span>2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}\\frac{1}{\\sqrt{2\\,H+1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u306a\u304b\u306a\u304b\u7c21\u5358\u5316\u3057\u3066\u304f\u308c\u307e\u305b\u3093\u304c\uff0c\u8981\u3059\u308b\u306b<\/p>\n
$$\\tan\\theta_{\\rm max} = \\frac{1}{\\sqrt{1+2 H}}$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2<\/h4>\n
$$L_{\\rm max} = X(T_1(\\theta_{\\rm max}), \\theta_{\\rm max})$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[11]:<\/div>\n
\n
\n
\n
T1max<\/span>:<\/span> rhs<\/span>(<\/span>sol2<\/span>[<\/span>2][<\/span>2])<\/span>$\r\nthmax<\/span>:<\/span> asin<\/span>(<\/span>1\/<\/span>sqrt<\/span>(<\/span>2*<\/span>(<\/span>1+<\/span>H<\/span>)))<\/span>$\r\n\r\nX<\/span>(<\/span>T1max<\/span>, thmax<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}\\sqrt{H+1}\\,\\sqrt{\\frac{2\\,H+1}{H+1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[12]:<\/div>\n
\n
\n
\n
\/* \u4e8c\u4e57\u3057\u3066\u5e73\u65b9\u6839 *\/<\/span>\r\nsqrt<\/span>(<\/span>%<\/span>**<\/span>2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{18}$}\\sqrt{2\\,H+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u308c\u3082\u306a\u304b\u306a\u304b\u7c21\u5358\u5316\u3057\u3066\u304f\u308c\u307e\u305b\u3093\u304c\uff0c\u8981\u3059\u308b\u306b<\/p>\n
$$L_{\\rm max} = \\sqrt{1 + 2H}$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5225\u89e3<\/h4>\n
\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3044\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u307e\u3059\u304c\uff0c$\\sin\\theta$ \u3092\u6d88\u53bb\u3059\u308b\u65b9\u91dd\u3067\u3084\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{dL}{d\\theta} = 0$ \u3092 $\\theta$ \u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[13]:<\/div>\n
\n
\n
\n
dL<\/span> =<\/span> 0<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[13]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}\\frac{T_{1}^2\\,\\sin \\vartheta-T_{1}}{\\sin \\vartheta-T_{1}}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[14]:<\/div>\n
\n
\n
\n
solsin<\/span>:<\/span> solve<\/span>(<\/span>dL<\/span>=<\/span>0<\/span>, sin<\/span>(<\/span>theta<\/span>))<\/span>;\r\nsoltheta<\/span>:<\/span> solve<\/span>(<\/span>solsin<\/span>[<\/span>1]<\/span>, theta<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
\n
solve: using arc-trig functions to get a solution.\r\nSome solutions will be lost.\r\n<\/pre>\n<\/div>\n<\/div>\n
\n
Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{20}$}\\left[ \\sin \\vartheta=\\frac{1}{T_{1}} \\right] \\]<\/div>\n<\/div>\n
\n
Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{21}$}\\left[ \\vartheta=\\arcsin \\left(\\frac{1}{T_{1}}\\right) \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e $\\theta$ \u3092 $Y(T_1, \\theta) = 0$ \u306b\u4ee3\u5165\u3057\u30661\u5909\u6570 $T_1$ \u306b\u95a2\u3059\u308b\u65b9\u7a0b\u5f0f\u306b\u3057\uff0c$T_1$ \u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[15]:<\/div>\n
\n
\n
\n
eqT1<\/span>:<\/span> ev<\/span>(<\/span>Y<\/span>(<\/span>T1<\/span>, theta<\/span>)<\/span> =<\/span> 0<\/span>, soltheta<\/span>[<\/span>1])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[15]:<\/div>\n
\\[\\tag{${\\it \\%o}_{22}$}-\\frac{T_{1}^2}{2}+H+1=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[16]:<\/div>\n
\n
\n
\n
sols<\/span>:<\/span> solve<\/span>(<\/span>eqT1<\/span>, T1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{23}$}\\left[ T_{1}=-\\sqrt{2}\\,\\sqrt{H+1} , T_{1}=\\sqrt{2}\\,\\sqrt{H+1} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$T_1 > 0$ \u3067\u3059\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[17]:<\/div>\n
\n
\n
\n
T1max<\/span>:<\/span> rhs<\/span>(<\/span>sols<\/span>[<\/span>2])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{24}$}\\sqrt{2}\\,\\sqrt{H+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u3068\u304d\u306e $\\theta$ \u306e\u5024\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[18]:<\/div>\n
\n
\n
\n
thmax<\/span>:<\/span> ev<\/span>(<\/span>rhs<\/span>(<\/span>soltheta<\/span>[<\/span>1])<\/span>, sols<\/span>[<\/span>2])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[18]:<\/div>\n
\\[\\tag{${\\it \\%o}_{25}$}\\arcsin \\left(\\frac{1}{\\sqrt{2}\\,\\sqrt{H+1}}\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\sin \\theta_{\\rm max} = $ …<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[19]:<\/div>\n
\n
\n
\n
sin<\/span>(<\/span>thmax<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[19]:<\/div>\n
\\[\\tag{${\\it \\%o}_{26}$}\\frac{1}{\\sqrt{2}\\,\\sqrt{H+1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\tan \\theta_{\\rm max} = $ …<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[20]:<\/div>\n
\n
\n
\n
tan<\/span>(<\/span>thmax<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[20]:<\/div>\n
\\[\\tag{${\\it \\%o}_{27}$}\\frac{1}{\\sqrt{H+1}\\,\\sqrt{\\frac{2\\,H+1}{H+1}}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[21]:<\/div>\n
\n
\n
\n
sqrt<\/span>(<\/span>%<\/span>**<\/span>2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{28}$}\\frac{1}{\\sqrt{2\\,H+1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u5927\u5230\u9054\u8ddd\u96e2 $L_{\\rm max}$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[22]:<\/div>\n
\n
\n
\n
X<\/span>(<\/span>T1max<\/span>, thmax<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{29}$}\\sqrt{H+1}\\,\\sqrt{\\frac{2\\,H+1}{H+1}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[23]:<\/div>\n
\n
\n
\n
sqrt<\/span>(<\/span>%<\/span>**<\/span>2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[23]:<\/div>\n
\\[\\tag{${\\it \\%o}_{30}$}\\sqrt{2\\,H+1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u300c\u9ad8\u3055 h \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u5230\u9054\u8ddd\u96e2\u3092\u6c42\u3081\u308b\u6e96\u5099\u300d\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002<\/p>\n
\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3068\u306a\u308b\u6253\u3061\u51fa\u3057\u89d2\u5ea6 $\\theta$ \u3092\uff0c\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3044\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066 SymPy \u3084 Maxima \u3092\u4f7f\u3063\u3066\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[14,12,25,21],"tags":[],"class_list":["post-7419","post","type-post","status-publish","format-standard","hentry","category-maxima","category-sympy","category-25","category-21","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/7419","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=7419"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/7419\/revisions"}],"predecessor-version":[{"id":8778,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/7419\/revisions\/8778"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=7419"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=7419"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=7419"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}