Hello everybody,
it would be very nice if someone could help me here, please.
I am investigating the laplace equation.
\begin{equation}
\begin{split}
Laplace \phi &= 0 \text{ in Q}\\\
\phi &= 0 \text{ at } \Gamma_1\\\
\gamma \frac{\partial \phi}{\partial n} &= 0 \text{ at } \Gamma_2 \text{ and } \Gamma_3\\\
\gamma \frac{\partial \phi}{\partial n} &= - \frac{1}{|\Gamma_4|}*\dot{Q_2}(t)*\int \phi dx \text{ at } \Gamma_4.
\end{split}
\end{equation}
after a few calculations with the electrical potential $$\phi$$ as the test function, this results in the weak formulation
\begin{equation}
\int_{Q} \gamma |\nabla \phi|^2dx - \frac{1}{|\Gamma_4|}\dot{Q}_2(t)\int_{\Gamma_4}\phi(x)dx = 0 \text{ for all } v \in V.
\end{equation}
I HAVE to use the Newmark-beta-method. But this method needs the second order.
Also if i am using the classical approach function
\phi \approx \tilde{\phi} = \sum_{I=1}^{N} c_i(x) \phi_i(x)\\
and for the gradient
\nabla \phi \approx \tilde{\nabla\phi} = \sum_{I=1}^{N} c_i(x) \nabla \phi_i(x)
Then i get
\gamma[\sum_{I=1}^{N} c_i(x) \int \nabla \phi_i \nabla \phi_j dx] = \frac{1}{|\Gamma_4|} I_2(t) \int_{\Gamma_4} \phi_j (x).
So one can see that this still doesn’t fit and i think it is the wrong way.
Ok. Then i tried the Lagrange Formalism from mechanics applied to electrodynamics with the lagragian
L(x(t), \dot{x}(t), t) = \underbrace{\frac{m}{2}|\dot{x}|^2}_{\text{kinetic energy}} - \underbrace{e\phi(x,t) + e\dot{x} * A(x,t)}_{\text{potential energy}}.
Then doing the Lagrange Formalism i get
m|\ddot{x}| = eE(x,t) + \underbrace{e\dot{x} \frac{\partial A(x,t)}{\partial x} - \frac{1}{e} \frac{\partial}{\partial x} A(x,t) \dot{x}}_{e(\dot{x} \times \underbrace{\nabla \times A(x,t))}_{B(x,t)}}.
Please excuse the lack of vector arrows. I hope it’s clear anyway.
So great. This would be nice, but i really don’t know how to handle A and if this approach is ok or not.
Thank you and kindly greetings