1
$\begingroup$

I originally posted this question on Physics SE, but I think it is more like a math question since I need rigorous justification.

Could anyone please provide any insight to the below question:

Let us work in a box $(t,\overrightarrow{x}) \in [0,1] \times [0,1]^3$. For any function on this box, we impose some Dirichlet boundary condition on the temporal direction and periodic boundary conditions on the spatial directions.

Let $i,j,k,l \in \{1,2,3\}$, which we use as spatial indices.

Now, think of some action functional \begin{equation} S[f_l,T_{ij}]=\int_0^1 dt \int_{[0,1]^3}d^3\overrightarrow{x}\mathcal{L}(f_l(\overrightarrow{x}, t),T_{ij}(\overrightarrow{x}, t),\frac{\partial f_l}{\partial t}, \partial_k f_l) \end{equation} subject to the constraints \begin{equation} T_{ij}-\frac{\partial_i f_j + \partial_j f_i}{2}=0. \end{equation}

(All field components are set to be real-valued and smooth.)

I am using the field-theoretic language now, so the Lagrange multipliers introduced to enforce these constraints must form some (symmetric) tensor field $\Psi_{ij}(\overrightarrow{x}, t)$, so that the modfied action is

\begin{equation} S_{mod}[f_l,T_{ij},\Psi_{ij}]=\int_0^1 dt \int_{[0,1]^3}d^3\overrightarrow{x}\Bigl[\mathcal{L}(f_l,T_{ij},\frac{\partial f_l}{\partial t}, \partial_k f_l)-\Psi_{ij} \cdot \Bigl(T_{ij}-\frac{\partial_i f_j + \partial_j f_i}{2}\Bigr)\Bigr]. \end{equation}

where the summation convention is used and the constraints are realized as the Euler-Lagrange equations of $S_{mod}$ with respect to $\Psi_{ij}$'s.

Now, my question is:

"Can we assume that $\Psi$ does NOT depend on time explicitly, that is, $\frac{\partial \Psi_{ij}}{\partial t}=0$?"

I think that since the constraint does NOT contain any time derivative, it is valid on each time slice. Thus, there is no impediment for assuming $\frac{\partial \Psi_{ij}}{\partial t}=0$.

However, I cannot give a more rigorous justification for this assumption...

Improve this question
$\endgroup$
4
  • $\begingroup$ It seems to me that your question is more about the physical than the mathematical validity of the assumption, and so not really appropriate for MO. You can always add an assumption to a mathematical model; the non-mathematical question is whether it's physically meaningful. Or am I missing the point, and you are asking whether there is something about the assumptions that forces $\frac{\partial\Psi_{i j}}{\partial t}$ to equal $0$? $\endgroup$
    – LSpice
    Commented Aug 30, 2022 at 16:07
  • $\begingroup$ Why don’t you impose the constraint directly by eliminating $T_{ij}$ in the action? $\endgroup$
    – Nawaf Bou-Rabee
    Commented Aug 30, 2022 at 16:22
  • 3
    $\begingroup$ @LSpice the OP seems to be asking about mathematical properties of the critical point of the constrained action; at least that’s how I read the question. In particular, if the critical point has the property that the Lagrange multiplier component is time independent. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Aug 30, 2022 at 16:25
  • 1
    $\begingroup$ If the constraint has to be enforced independently on every time slice $t$, the corresponding Lagrange multipliers $\Psi (t)$ must all be independent of one another. So, you cannot assume any relationship between them, in particular, you can't assume that $\Psi $ is constant in time. Maybe it would help if you reduced the clutter and just examined your question on a maximally simple mechanical example. $\endgroup$
    – Michael Engelhardt
    Commented Aug 31, 2022 at 3:25

0

Reset to default

You must log in to answer this question.