$\Phi, \Lambda$ are both scalars dependent upon, and $\mathbf u$ is a vector independent of coordinates. I'm trying to express $\Lambda$ in terms from $\mathbf U \cdot \nabla\Lambda = \Phi$ and to start with, since I'm just familiar with the basics, it looks pretty hopless. So:
- Get help from people more expert in this area
- Trawl through the list of standard vector identitites involving $\nabla$
- Search for an online mathematics program to solve it.
- Make it simpler to start with by solving in one dimension, and progress from there
what strategy would/did you use in tackling this problem? Solving it would be a bonus ;)
Edit: I've added the context of the problem since some people think this will help:
I'm trying to work out the conserved canonical momentum for a static electric field, using Noether's theorem on the relativistic electromagnetic Lagrangian $L = \frac{m_0c^2}{\gamma} + \frac{e}{c}\mathbf{u \cdot A} - e\Phi(x,y,z)$ $L$ needs to be independent of coordinates which can be done by transforming \mathbf A\rightarrow \mathbf A'= \mathbf A + \nabla\Lambda The conserved canonical momentum $ P = \frac\partial{\partial \mathbf u} ( \frac{-m_0c^2}{\gamma} + \frac{e}{c}\mathbf{u \cdot (A+\nabla\Lambda}) - e\Phi)$ With no magnetic field $A=0$ $P = -m_0\mathbf u\gamma +\frac e c \frac\partial {\partial\mathbf u}\mathbf u \cdot\nabla\Lambda$ becomes $P = -m_0\mathbf u\gamma +\frac e c (\nabla\Lambda + \mathbf u \cdot\frac\partial {\partial\mathbf u}\mathbf\nabla\Lambda)$ To get any further, I need to know the form $\Lambda$ must take, which comes from making $L$ independent of the coordinates before, and so $\nabla( \frac{e}{c}\mathbf u \nabla\Lambda - e\Phi) = 0$