Buc1234

• Generalized hydrodynamics
• Spectral moment

Electromagnetism

$\nabla ^{2}\mathbf {E} ={\frac {n^{2}(x)}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}$
EM wave equation
 ↓
$\left(\partial _{t}S\right)^{2}-c^{2}|\nabla S|^{2}=0$
Eikonal equation
 ↓
$k\partial _{z}I+\nabla \cdot (I\nabla \Phi )=0$
Transport of intensity equation
 ↓
${\dot {l}}=\{H,l\}$
Light transport equation

Oceanology

${\frac {\partial ^{2}\eta }{\partial x^{2}}}=g{\frac {\partial }{\partial x}}\left(h{\frac {\partial \eta }{\partial x}}\right)$
Wave equation
 ↓
$\partial _{t}S+\omega (x,\nabla S)=0$
Ray equation
 ↓
$\partial _{t}{\mathcal {A}}+\nabla \cdot ({\mathcal {A}}\,v_{g})=0$
Wave action equation

Quantum mechanics

$i\hbar {\frac {\partial \psi }{\partial t}}={\hat {H}}\psi$
Schrödinger equation
 ↓
$\partial _{t}S+H(x,\nabla S,t)=0$
Hamilton–Jacobi equation
 ↓
$\partial _{t}\rho +\nabla \cdot (\rho \,v)=0$
Continuity equation
 ↓
$\partial _{t}\rho =\{H,\rho \}$
Liouville equation

General Relativity

${\hat {\mathcal {H}}}\Psi [g_{ij}]=0$
Wheeler-DeWitt equation
 ↓
${\mathcal {H}}\left(g_{ij},{\frac {\delta S}{\delta g_{ij}}}\right)=0$
Hamilton-Jacobi-Einstein equation

In his milestone 1926 article, Quantisierung als Eigenwertproblem, Schrödinger introduced (what is called today) the Schrödinger equation for the hydrogen atom by taking the Hamilton-Jacobi equation of an electron in a Coulomb potential:

${\frac {\partial S({\vec {x}},t)}{\partial t}}+{\frac {1}{2m}}{\frac {\partial S({\vec {x}},t)}{\partial {\vec {x}}}}\cdot {\frac {\partial S({\vec {x}},t)}{\partial {\vec {x}}}}+{\frac {e^{2}}{|{\vec {x}}|}}=0$

and replacing derivatives with $-i\hbar \partial$:

$-i\hbar {\frac {\partial \psi ({\vec {x}},t)}{\partial t}}-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi ({\vec {x}},t)+{\frac {e^{2}}{|{\vec {x}}|}}\,\psi ({\vec {x}},t)=0.$

In a shortly subsequent article, Schrödinger offers a rationale for this procedure: the eikonal approximation of a wave equation, which defines the geometrical optic approximation where wave packets follow definite trajectories, can be obtained by the opposite procedure. If we interpret classical mechanics as the eikonal approximation to a wave mechanics, we can guess the wave equation by this procedure. Given the immense success of Schrödinger’s leap, trying the same strategy for gravity is obviously tempting. This is what led DeWitt and Wheeler to their equation:

$\left[\left(q_{ab}q_{cd}-{\frac {1}{2}}q_{ac}q_{bd}\right){\frac {\delta }{\delta q_{ac}}}{\frac {\delta }{\delta q_{bd}}}+\det(q)\,R[q]\right]\Psi [q]=0.$

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