Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 13C / Solution

WriteVariation of $\varphi$ gives $\nabla \cdot \mathbf{u}=0$. Variation of $\alpha$ givesand variation of $\beta$ givesUsing incompressibility, the middle equation becomesThese are the three required Euler-Lagrange equations.

Let $D_t={\partial }_t+\mathbf{u}\cdot \nabla$. Since $D_t\alpha =D_t\beta =0$, differentiating $u_i={\partial }_i\varphi +\beta {\partial }_i\alpha$ givesNowbecause $\mathbf{u}\cdot \nabla \alpha =-{\alpha }_t$. HencewhereThis is the Clebsch-potential variational derivation of incompressible Euler flow.