Extend energy arguments to coupled fields

Try

Two fields can exchange information while their combinations travel independently. Consider Uₜ+A Uₓ=0 with U=(U₁,U₂) and A having rows (0,2) and (2,0). The equations are U₁ₜ+2U₂ₓ=0 and U₂ₜ+2U₁ₓ=0.

Add and subtract them. The sums are not merely algebraic conveniences: they identify characteristic variables, the combinations that propagate at individual speeds.

Predict

Predict the speeds of w₊=U₁+U₂ and w₋=U₁−U₂. Then reconstruct U₁ and U₂ from those variables. Check the factor 1/2 in that reconstruction before applying initial data.

Experiment

The equations become w₊ₜ+2w₊ₓ=0 and w₋ₜ−2w₋ₓ=0.

Variable	Speed	Evolution
w₊	2	w₊(x−2t,0)
w₋	−2	w₋(x+2t,0)

Then U₁=(w₊+w₋)/2 and U₂=(w₊−w₋)/2. Start with U₂=0 and a pulse in U₁. Explain why it splits into two traveling pieces and why U₂ becomes nonzero even though it started at zero.

Understand

For a constant real symmetric matrix A, multiply the system by U transpose. Symmetry makes U transpose A Uₓ the derivative of U transpose A U divided by 2. Thus ∂ₜ(|U|²/2)+∂ₓ(U transpose A U/2)=0. Total energy is conserved for periodic data, sufficient decay or boundary conditions eliminating the flux. Without those conditions, energy can enter or leave through the boundary.

The spectral theorem for real symmetric matrices supplies an orthonormal eigenbasis, reducing the constant-coefficient system to independent transports at real eigenvalues. If A depends smoothly on x, the energy calculation gains the interior term U transpose AₓU/2. A bound on Aₓ yields an inequality and a Grönwall estimate rather than exact conservation. General variable-coefficient existence theory is a further theorem, not established by diagonalizing one constant matrix.

Vacuum Maxwell equations provide a higher-dimensional example: Eₜ=c² curl B and Bₜ=−curl E, with no charges or currents and constant material parameters after normalization. Their energy density is (|E|²+c²|B|²)/2. Its time derivative is c²(E·curl B−B·curl E)=−c² div(E×B). Hence the energy flux is c²E×B.

Taking divergence of the evolution equations gives time derivatives of div E and div B equal to zero for smooth fields, because divergence of a curl vanishes. The constraints must hold initially to remain the desired zero constraints. Evolution does not automatically repair an initially nonzero divergence error. Boundary conditions must also be compatible; the vacuum calculation is not a complete theory for arbitrary media or conductors.

Apply

Diagonalize a different two-by-two symmetric transport matrix and calculate its maximum signal speed. Then state which boundary flux would need control in an energy estimate. For Maxwell, separate the evolution equations from the initial divergence constraints and explain what a numerical divergence diagnostic actually measures.

Continue

The final lesson derives a wave-type field equation from stationary action and checks its conserved energy. This connects variational reasoning with the boundary and energy mechanisms used throughout the course.