解析力学
d d t  ( ∂ L ∂ q ̇ i ) = d d t  ( ∂ L ∂ x ̇ ∂ x ̇ ∂ q ̇ i + ∂ L ∂ y ̇ ∂ y ̇ ∂ q ̇ i ) = d d t  ( ∂ L ∂ x ̇ ∂ x ∂ q i + ∂ L ∂ y ̇ ∂ y ∂ q i ) = d d t  ( ∂ L ∂ x ̇ ) ∂ x ∂ q i + d d t  ( ∂ L ∂ y ̇ ) ∂ y ∂ q i + ∂ L ∂ x ̇ ∂ x ̇ ∂ q i + ∂ L ∂ y ̇ ∂ y ̇ ∂ q i ∂ L ∂ q i = ∂ L ∂ x ∂ x ∂ q i + ∂ L ∂ y ∂ y ∂ q i + ∂ L ∂ x ̇ ∂ x ̇ ∂ q i + ∂ L ∂ x ̇ ∂ x ̇ ∂ q i {\displaystyle {\begin{aligned}\mathop {\frac {d}{dt}} \left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)&=\mathop {\frac {d}{dt}} \left({\frac {\partial L}{\partial {\dot {x}}}}{\frac {\partial {\dot {x}}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {y}}}}{\frac {\partial {\dot {y}}}{\partial {\dot {q}}_{i}}}\right)\\&=\mathop {\frac {d}{dt}} \left({\frac {\partial L}{\partial {\dot {x}}}}{\frac {\partial x}{\partial q_{i}}}+{\frac {\partial L}{\partial {\dot {y}}}}{\frac {\partial y}{\partial q_{i}}}\right)\\&=\mathop {\frac {d}{dt}} \left({\frac {\partial L}{\partial {\dot {x}}}}\right){\frac {\partial x}{\partial q_{i}}}+\mathop {\frac {d}{dt}} \left({\frac {\partial L}{\partial {\dot {y}}}}\right){\frac {\partial y}{\partial q_{i}}}+{\frac {\partial L}{\partial {\dot {x}}}}{\frac {\partial {\dot {x}}}{\partial q_{i}}}+{\frac {\partial L}{\partial {\dot {y}}}}{\frac {\partial {\dot {y}}}{\partial q_{i}}}\\{\frac {\partial L}{\partial q_{i}}}&={\frac {\partial L}{\partial x}}{\frac {\partial x}{\partial q_{i}}}+{\frac {\partial L}{\partial y}}{\frac {\partial y}{\partial q_{i}}}+{\frac {\partial L}{\partial {\dot {x}}}}{\frac {\partial {\dot {x}}}{\partial q_{i}}}+{\frac {\partial L}{\partial {\dot {x}}}}{\frac {\partial {\dot {x}}}{\partial q_{i}}}\end{aligned}}}