全充@ ウィキ
全充@ ウィキ

Lagrange の方法の一般的証明：一般化座標


一般化された座標 q_{{}^1} および q_{{}^2} が x および y の関数として

q_{{}^1}=q_{{}^1}(x\ ,\ y) ，q_{{}^1}=q_{{}^2}(x\ ,\ y) で与えられているとする．

\frac{\mbox{d}x}{\mbox{d}t}=\frac{\partial x}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial x}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t} ，

\frac{\mbox{d}^{{}_2}x}{\mbox{d}t^{{}_2}}=\frac{\partial x}{\partial q_{{}^1}}\frac{\mbox{d}^{{}_2}q_{{}^1}}{\mbox{d}t^{{}_2}}+\frac{\partial x}{\partial q_{{}^2}}\frac{\mbox{d}^{{}_2}q_{{}^2}}{\mbox{d}t^{{}_2}}

　　　　　+\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\(\frac{\partial^{{}_2}x}{\partial {q_{{}^1}}^{{}_2}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial^{{}_2}x}{\partial q_{{}^1}\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)

　　　　　+\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\(\frac{\partial^{{}_2}x}{\partial q_{{}^1}\partial q_{{}^2}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial^{{}_2}x}{\partial {q_{{}^2}}^{{}_2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)

\frac{\mbox{d}y}{\mbox{d}t}=\frac{\partial y}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial y}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t} ，

\frac{\mbox{d}^{{}_2}y}{\mbox{d}t^{{}_2}}=\frac{\partial y}{\partial q_{{}^1}}\frac{\mbox{d}^{{}_2}q_{{}^1}}{\mbox{d}t^{{}_2}}+\frac{\partial y}{\partial q_{{}^2}}\frac{\mbox{d}^{{}_2}q_{{}^2}}{\mbox{d}t^{{}_2}}

　　　　　+\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\(\frac{\partial^{{}_2}y}{\partial {q_{{}^1}}^{{}_2}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial^{{}_2}y}{\partial q_{{}^1}\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)

　　　　　+\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\(\frac{\partial^{{}_2}y}{\partial q_{{}^1}\partial q_{{}^2}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial^{{}_2}y}{\partial {q_{{}^2}}^{{}_2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)


m\frac{\mbox{d}^{{}^2}x}{\mbox{d}t^{{}^2}}=-\frac{\partial V}{\partial x} ，　　　m\frac{\mbox{d}^{{}^2}y}{\mbox{d}t^{{}^2}}=-\frac{\partial V}{\partial y} ．

に対し，x の式に \partial x/\partial q_{{}^1} y の式に \partial y/\partial q_{{}^1} をかけて辺々加える．

m\{\[\(\frac{\partial x}{\partial q_{{}^1}}\)^{{}_2}+\(\frac{\partial y}{\partial q_{{}^1}}\)^{{}_2}\]\frac{\mbox{d}^{{}_2}q_{{}^1}}{\mbox{d}t^{{}_2}}+\(\frac{\partial x}{\partial q_{{}^1}}\frac{\partial x}{\partial q_{{}^2}}+\frac{\partial y}{\partial q_{{}^1}}\frac{\partial y}{\partial q_{{}^2}}\)\frac{\mbox{d}^{{}_2}q_{{}^2}}{\mbox{d}t^{{}_2}}

　　　　　+\(\frac{{\partial x}}{\partial q_{{}^1}}\frac{\partial^{{}_2}x}{\partial {q_{{}^1}}^{{}_2}}+\frac{{\partial y}}{\partial q_{{}^1}}\frac{\partial^{{}_2}y}{\partial {q_{{}^1}}^{{}_2}}\)\(\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\)^{{}_2}

　　　　　+2\(\frac{\partial x}{\partial q_{{}^1}}\frac{\partial^{{}_2}x}{\partial q_{{}^1}\partial q_{{}^2}}+\frac{\partial y}{\partial q_{{}^1}}\frac{\partial^{{}_2}y}{\partial q_{{}^1}\partial q_{{}^2}}\)\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}

　　　　　+\(\frac{{\partial x}}{\partial q_{{}^1}}\frac{\partial^{{}_2}x}{\partial {q_{{}^2}}^{{}_2}}+\frac{{\partial y}}{\partial q_{{}^1}}\frac{\partial^{{}_2}y}{\partial {q_{{}^2}}^{{}_2}}\)\(\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)^{{}_2}\Bigg}

　　　　=-\(\frac{\partial V}{\partial x}\frac{\partial x}{\partial q_{{}^1}}+\frac{\partial V}{\partial y}\frac{\partial y}{\partial q_{{}^1}}\)=-\frac{\partial V}{\partial q_{{}^1}} ．

この左辺は，T を運動エネルギーとして

\frac{\mbox{d}}{\mbox{d}t}\(\frac{\partial T}{\partial \dot{q}_{{}_1}}\)-\frac{\partial T}{\partial q_{{}^1}}

に等しいことを示す．

T=\frac{m}{2}\Bigg[\bigg(\frac{\partial x}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial x}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\bigg)^{{}_2}+\bigg(\frac{\partial y}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial y}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\bigg)^{{}_2}\Bigg] ．

と書き換え，つぎに， \dot{q}_{{}^1}=\mbox{d}q_{{}^1}/\mbox{d}t に注意して，T の表式を微分すると

\frac{\partial T}{\partial \dot{q}_{{}_1}}=m\[\(\frac{\partial x}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial x}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)\frac{\partial x}{\partial q_{{}^1}}+\(\frac{\partial y}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial y}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)\frac{\partial y}{\partial q_{{}^1}}\] ，

\frac{\mbox{d}}{\mbox{d}t}\(\frac{\partial T}{\partial \dot{q}_{{}_1}}\)=m\[\(\frac{\partial x}{\partial q_{{}^1}}\frac{\mbox{d}^{{}_2}q_{{}^1}}{\mbox{d}t^{{}_2}}+\frac{\partial x}{\partial q_{{}^2}}\frac{\mbox{d}^{{}_2}q_{{}^2}}{\mbox{d}t^{{}_2}}\)\frac{\partial x}{\partial q_{{}^1}}+\(\frac{\partial y}{\partial q_{{}^1}}\frac{\mbox{d}^{{}_2}q_{{}^1}}{\mbox{d}t^{{}_2}}+\frac{\partial y}{\partial q_{{}^2}}\frac{\mbox{d}^{{}_2}q_{{}^2}}{\mbox{d}t^{{}_2}}\)\frac{\partial y}{\partial q_{{}^1}}\]

　　　　　　　　　　+m\Bigg{\(\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\)^{{}_2}\[\frac{\partial}{\partial q_{{}^1}}\(\frac{\partial x}{\partial q_{{}^1}}\)^{{}_2}+\frac{\partial}{\partial q_{{}^1}}\(\frac{\partial y}{\partial q_{{}^1}}\)^{{}_2}\]

　　　　　　　　　　　　+\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\[\frac{\partial}{\partial q_{{}^2}}\(\frac{\partial x}{\partial q_{{}^1}}\)^{{}_2}+\frac{\partial}{\partial q_{{}^2}}\(\frac{\partial y}{\partial q_{{}^1}}\)^{{}_2}+\frac{\partial}{\partial q_{{}^1}}\(\frac{\partial x}{\partial q_{{}^1}}\frac{\partial x}{\partial q_{{}^2}}\)+\frac{\partial}{\partial q_{{}^1}}\(\frac{\partial y}{\partial q_{{}^1}}\frac{\partial y}{\partial q_{{}^2}}\)\]

　　　　　　　　　　　　+\(\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)^{{}_2}\[\frac{\partial}{\partial q_{{}^2}}\(\frac{\partial x}{\partial q_{{}^1}}\frac{\partial x}{\partial q_{{}^2}}\)+\frac{\partial}{\partial q_{{}^1}}\(\frac{\partial y}{\partial q_{{}^1}}\frac{\partial y}{\partial q_{{}^2}}\)\]\Bigg}


\frac{\mbox{d}}{\mbox{d}t}\[\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\(\frac{\partial x}{\partial q_{{}^1}}\)^{{}_2}\]=\frac{\mbox{d}^{{}_2}q_{{}^1}}{\mbox{d}t^{{}_2}}\(\frac{\partial x}{\partial q_{{}^1}}\)^{{}_2}+\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}\frac{\partial}{\partial q_{{}^1}}\(\frac{\partial x}{\partial q_{{}^1}}\)^{{}_2}


および

\frac{\partial T}{\partial q_{{}^1}}=m\Bigg[\(\frac{\partial x}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial x}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)\(\frac{\partial^{{}_2} x}{\partial {q_{{}^1}}^{{}_2}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial^{{}_2} x}{\partial q_{{}^1}\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)

　　　　　+\(\frac{\partial y}{\partial q_{{}^1}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial y}{\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)\(\frac{\partial^{{}_2} y}{\partial {q_{{}^1}}^{{}_2}}\frac{\mbox{d}q_{{}^1}}{\mbox{d}t}+\frac{\partial^{{}_2} y}{\partial q_{{}^1}\partial q_{{}^2}}\frac{\mbox{d}q_{{}^2}}{\mbox{d}t}\)\Bigg] ．

これら二つの表式から

タグ：

Lagrangeの方程式 一般化座標
「Lagrange の方法の一般的証明：一般化座標」をウィキ内検索
最終更新：2012年04月14日 22:16