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«Çû¤Î¤¢¤ë¾ì¹ç¡¢¤³¤ÎÎϳطϤμ«Í³ÅÙ $n$ ¤Ï $N$ ¤è¤ê¤â¾®¤µ¤ÊÃͤȤʤ롣 ¤½¤Î¾ì¹ç¡¢Â¿¤¯¤Î¥±¡¼¥¹¤Ç $x_i$ ¤È¤Ï°Û¤Ê¤ëÊ̤ÎÊÑ¿ô¤òÍøÍѤ¹¤ëÊý¤¬¤è¤¤¤³¤È¤¬Â¿¤¤¡£ Î㤨¤Ð¡¢±ß±¿Æ°¤ò¤·¤Æ¤¤¤ë¼ÁÅÀ¤Ê¤é¤Ð $\theta$ °ì¤Ä¤ò»È¤¨¤Ð¤è¤¤¤·¡¢ ¸ß¤¤¤ËÅùµ÷Î¥¤ËÀܳ¤µ¤ì¤¿Æó¼ÁÅÀ¤Ê¤é¤Ð¡¢½Å¿´¤ÎºÂɸ¤È¡¢ Æó¼ÁÅÀ¤ò·ë¤ÖľÀþ¤ÎÊý¸þ¤ò¼¨¤¹¶ËºÂɸŪÆó¤Ä¤Î³ÑÅÙ¤ò»È¤¦¤Î¤¬ÊØÍø¤Ç¤¢¤ë¡£ ¤³¤¦¤·¤¿ºÂɸ¤ò°ìÈ̲½ºÂɸ¡Êgeneralized coordinate¡Ë¤È¸Æ¤Ó¡¢ $q_j: (j=1,\cdots,n)$ ¤È¤¹¤ë¡£

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$$\sum_i^Nm_i\ddot x_i\delta x_i$$

$$= \sum_i^Nm_i\ddot x_i \left(\sum_j^n\D{\delta x_i}{q_j}\delta q_j\right)$$

$$= \sum_i^N\sum_j^n\left[\frac d{dt} \left(m_i\dot x_i\D{x_i}{q_j}\right) -m_i\dot x_i\frac d{dt} \left(\D{x_i}{q_j}\right)\right]\delta q_j$$

$$= \sum_j^n\sum_i^N\left[\frac d{dt} \left(m_iv_i\D{x_i}{q_j}\right)... ...\dot q_k +\frac{\partial^2x_i}{\partial q_j\partial t} \right)\right]\delta q_j$$

$$= \sum_j^n\sum_i^N\left[\frac d{dt} \left(m_iv_i\D{v_i}{\dot q_j}\right) -m_iv_i\D{v_i}{q_j}\right]\delta q_j$$

$$= \sum_j^n\left[ \frac d{dt}\left(\D T{\dot q_j}\right) -\D{T}{q_j}\right]\delta q_j$$ (A.9)

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$$T=\sum_i^N\frac12m_iv_i^2$$ (A.10)

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$$\D{v_i}{\dot q_j} = \D{\dot x_i}{\dot q_j} =\D{x_i}{q_j}$$

$$\D{v_i}{q_j} = \D{}{q_j}\left(\frac{dx_i}{dt}\right) =\D{}{q_j}\left(\sum_k^n\D{dx_i}{q_k}\dot q_k +\D{dx_i}t\right)$$ (A.11)

¤Î´Ø·¸¤òÍøÍѤ¹¤ë¡£ ³Æ±¦ÊÕ¤¬ 3¹ÔÌܤˡ¢³Æº¸ÊÕ¤¬ 4¹ÔÌܤ˸½¤ï¤ì¤Æ¤¤¤ë¡£ ¤³¤ì¤é¤Î¼°¤ÎͶƳ¤«¤é¤ï¤«¤ë¤è¤¦¤Ë¡¢$T$ ¤Ï¸«³Ý¤±¤Ï $v_i$ ¤Î´Ø¿ô¤Ç¤¢¤ë¤¬¡¢³Æ $v_i$ ¤Ï $q_j$ ¤È $\dot q_j$ ¤Ç½ñ¤­´¹¤¨¤é¤ì¤Æ¤¤¤ë¤³¤È¤òÁ°Äó¤È¤·¤Æ¤¤¤ë¡£ ¤·¤¿¤¬¤Ã¤Æ $T(q,\dot q)$ ¤Ç¤¢¤ë¡£

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$$\sum_i^NF_i\delta x_i =\sum_i^NF_i \left(\sum_j^n\D{x_i}{q_j}\del... ...sum_j^n\left(\sum_i^NF_i \D{x_i}{q_j}\right)\delta q_j =\sum_j^nF^g_j\delta q_j$$ (A.12)

ºÇ¸å¤Î³ç¸Ì¤ò $F^g_j$ ¤È¤·¤¿¤¬¡¢¤³¤ì¤Ï°ìÈ̲½ÎÏ¡Êgeneralized force¡Ë¤È¸Æ¤Ð¤ì¤Æ¤¤¤ë¡£

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$$\sum_j^n\left[ \frac d{dt}\left(\D T{\dot q_j}\right) -\D{T}{q_j}-F^g_j\right]\delta q_j=0$$ (A.13)

$\delta q_j$ ¤ÏÆÈΩ¤ËÊÑÆ°¤Ç¤­¤ë¤Î¤Ç¡¢¼¡¤Î¼°¤¬ÆÀ¤é¤ì¤ë¡£

$$\frac d{dt}\left(\D T{\dot q_j}\right) -\D T{q_j}=F^g_j \hspace{1cm} (j=1,\cdots,n)$$ (A.14)

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ÆäËÊݸÎϾì¤Î¾ì¹ç¤Ï¡¢ $F_i=-\mathop{\mathop{\emph ¢¦}\nolimits }\nolimits _{x_i} U$ ¤È¤·¤Æ¡¢ ¼¡¤Î¤è¤¦¤ËÊÑ·Á¤Ç¤­¤ë¡£

$$\sum_i^NF_i{\delta x_i} =-\sum_i^N\sum_j^n\mathop{\mathop{\emph ¢... ...limits }\nolimits _{x_i} U \D{x_i}{q_j}\delta q_j =-\sum_j^n\D U{q_j}\delta q_j$$ (A.15)

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$$L(q_1,\cdots,q_n,\dot q_1,\cdots,\dot q_n) =T(q_1,\cdots,q_n,\dot q_1,\cdots,\dot q_n)-U(q_1,\cdots,q_n)$$ (A.16)

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$$\frac d{dt}\left(\D L{\dot q_j}\right) -\D L{q_j}=0 \hspace{1cm} (j=1,\cdots,n)$$ (A.17)

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$$L(q,\dot q)=T(q,\dot q)-U(q)$$ (A.18)

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