Производная Лагранжа

Производная Лагранжа, также известная как субстанциональная производная или материальная производная, — это производная, взятая в зависимости от системы координат, движущейся со скоростью u и часто используемая в гидроаэромеханике и классической механике. Она определена как от скалярной функции $\phi ({\vec {r}},t)$ координат и времени, так и от векторной ${\vec {v}}({\vec {r}},t)$ :

{\frac {D\phi }{Dt}}={\frac {\partial \phi }{\partial t}}+(\mathbf {u} \cdot \nabla )\phi

{\frac {D\mathbf {v} }{Dt}}={\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {v}

где $\nabla$ — это оператор набла, а ${\frac {\partial }{\partial t}}$ обозначает частную производную по t. Второе слагаемое есть конвективная производная данной функции.

Верно следующее тождество, когда берётся производная Лагранжа от интеграла:

{\frac {D}{Dt}}\int \limits _{V(t)}f(\mathbf {x} )\,dV=\int \limits _{V(t)}\left({\frac {\partial f}{\partial t}}+\nabla \cdot (f\mathbf {u} )\right)\,dV=\int \limits _{V(t)}\left({\frac {Df}{Dt}}+f(\nabla \cdot \mathbf {u} )\right)\,dV

Доказательство

Доказательство через правило дифференцирования сложных функций для частных производных. В тензорной нотации (с соглашением суммирования Эйнштейна) можно записать:

\left[{\frac {d\mathbf {B} }{dt}}\right]_{j}={\frac {d}{dt}}{\hat {B_{j}}}(t,x_{i}(t))={\frac {\partial B_{j}}{\partial t}}+{\frac {\partial B_{j}}{\partial x_{i}}}{\frac {\partial x_{i}}{\partial t}}={\frac {\partial B_{j}}{\partial t}}+{\frac {\partial x_{i}}{\partial t}}{\frac {\partial }{\partial x_{i}}}B_{j}={\frac {\partial B_{j}}{\partial t}}+\left[(\mathbf {u} \cdot \nabla )\mathbf {B} \right]_{j}

См. также

Конвективная производная

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