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I would like to know how to define initial conditions if there are constraints regarding the initial data, for example in EMT, when dealing with the time evolution of EM field, we cannot arbitrarily give an initial $E$ and $B$, they should satisfy the scalar conditions, the constraints reduce the degrees of freedom, then how can we define the initial conditions. I need to study these concepts, if possible in a simpler situation, these are the topics that i should cover

  1. How to define initial conditions for a differential equation

  2. How to deal with initial conditions when there are constraints relating the initial data.

Qmechanic
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Thejas
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2 Answers2

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You are correct that there will be constraints on the initial data (such as $\nabla\cdot\vec{E}=\rho_c$ and $\nabla\cdot\vec{B}=0$, and perhaps the choice of gauge) which must be satisfied at the initial time and during the subsequent evolution, at least approximately. This means you are not free to specify a "random" field configuration for the initial conditions, since such a random choice may not satisfy these constraints in general.

So the question is: how do you pick initial data which does satisfy the constraints? One approach is to first compute the charge distribution of your matter fields, and then solve for the electromagnetic field configuration which corresponds to that charge distribution (and satisfies the constraints).

For example, suppose you are trying to satisfy $\nabla\cdot\vec{E}=\rho_c$. Assume the electric field can be written as $$\vec{E}=\vec{E}^T+\nabla U$$ where $\vec{E}^T$ is the transverse part of the electric field and $U$ is a scalar potential (see: Helmholtz decomposition). Assuming $\nabla\cdot\vec{E}^T=0$, the constraint $\nabla\cdot\vec{E}=\rho_c$ becomes $$\nabla^2 U=\rho_c.$$ The problem is now reduced to specifying $\vec{E}^T$ such that it is transverse and solving for $U$. But because $\vec{E}^T$ represents freely-specifiable initial radiation, we can set it to zero. Then, the problem just becomes one of solving the equation $\nabla^2 U=\rho_c$ for the scalar function $U$ and setting the electric field via $\vec{E}=\nabla U$. Of course, $\nabla^2 U=\rho_c$ is just an elliptic differential equation which can be solved using your method of choice (e.g. numerically using relaxation).

For further reading on this topic, you may be interested in

James W. York, Jr. and Tsvi Piran. The initial value problem and beyond. In Richard A. Matzner and Lawrence C. Shepley, editors, Spacetime and Geometry: The Alfred Schild Lectures, pages 147–176. University of Texas Press, Austin (Texas), 1982.

Superbee
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One option in this case is to take the Wheeler-Feynman view that there should be no initial unsourced fields -- in other words, that all fields should be sourced by charged matter -- but that requires expanding your analysis to include the field sources, and it also requires more than just the free-field equations. More commonly, people just assume the Sommerfeld Radiation Condition that there are no incoming free fields at all, other than those with known sources.

More generally, note that most physicists treat the complete initial state of a modelled system as an externally controllable input to the "model" in question, so we are setting those values anyway when we solve a problem. Since we set them by hand, if they are supposed to obey some constraint, we just have to make sure that we set them according to that constraint.

If we don't treat the initial values as a external input, your question becomes a lot more interesting! For example, if you're solving an EM problem using action extremization, then some of the input to the problem is a future constraint, and a portion of the initial state is an output, solved by extremizing the action. In that case, for classical EM, I'd highly recommend changing the equations around to be in terms of the four-vector potential $A_\mu$ (or, equivalently, the scalar potential $V$ and the 3-vector potential $\vec{A}$), where the constraints you're worried about (on $\vec{E}$ and $\vec{B}$) become automatic. You still have a choice of gauge in this view, but you can use the freedom of choosing a gauge to not have to worry about constraining the initial data in any particular way.

Ken Wharton
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