My question is: can the same differential problem (PDE, Action minimization...) be treated as a Boundary Value Problem or as an Initial Value Problem, depending on the nature of the constraints I require? Or are BVP and IVP different in how they're formulated?
The question comes from a problem I've recently encountered, which I describe for context. Me and another teammate tried to solve the same problem using two different approaches: we wanted to find the $x$ that minimises an action $S(x)=\int dy dz \mathcal{L}(y,z,x,x_y,x_z)$ on a domain $\Omega$ which is a $(0,0)-(1,1)$ square in $(z,y)$, and the constraints that we have are:
- $z=0$ Neumann $x_y=0$ $\rightarrow A$
- $y=1$ Dirichlet $x=L$ $\rightarrow B$
- $z=1$ Dirichlet $x=L$ $\rightarrow C$
My approach was to simply minimize the action, solving the BVP: since there were no Dirichlet conditions on $z=0$, I set arbitrary conditions $D$ and then went to minimize with respect to the said conditions.
My colleague obtained the Equations of Motion (Euler Lagrange Equations) and treated the problem as an IVP, asking "I want the solution to evolve in $z$ to reach the Dirichlet condition $C$, what should my starting point be?".
Now, our trouble comes from the fact that the problem posed in the second way would be overconstrained if the arbitrary Dirichlet condition $D$ wasn't the "right one", and thus couldn't be solved, but the first approach finds a solution for every $D$ (even though the "right one" is only the one that minimizes the action.
Is the mistake in interpreting the same problem as a BVP and an IVP? Or does the mistake reside elsewhere?