fix(HJ): A.02 — Remove Stackel/Levi-Civita, add gradient derivation, physical constants

concepts/advanced/separation.tex

@@ -2,6 +2,8 @@
 
 This subsection develops the method of separation of variables for the Hamilton--Jacobi equation, showing how the choice of coordinate system determines whether the PDE reduces to a set of ordinary quadratures.
 
+\nt{Recap: The Hamilton--Jacobi equation from A.01 is a single first-order nonlinear PDE. Here we learn how to solve it by separation of variables.}
+
 \dfn{Separation of variables for the HJ equation}{
 Suppose the Hamiltonian has no explicit time dependence and the Hamilton--Jacobi equation is $\mcH(q_1,\dots,q_n,\pdv{\mcS}{q_1},\dots,\pdv{\mcS}{q_n}) = E$. The time variable is separated by setting
 \[
@@ -17,6 +19,8 @@ This is the time-independent Hamilton--Jacobi equation. It contains $n$ partial
 
 \nt{The condition $\pdv{\mcH}{t} = 0$ is necessary for the simple time separation $\mcS = W - Et$. When the Hamiltonian depends explicitly on time, a different separation ansatz or a time-dependent canonical transformation is required. In the time-independent case, energy is a constant of motion and serves as the first separation constant.}
 
+\nt{The separation constant $E$ is the total energy of the system, arising because $t$ is a cyclic coordinate in the extended phase space whenever the Hamiltonian is time-independent.}
+
 A particularly simple situation arises when one or more coordinates are cyclic. A generalized coordinate $q_i$ is cyclic, or ignorable, when it is absent from the Hamiltonian, which means $\pdv{\mcH}{q_i} = 0$. For such a coordinate, Hamilton's equation gives $\dot{p}_i = 0$, so the conjugate momentum is conserved. Within the Hamilton--Jacobi framework this translates directly: since $p_i = \pdv{\mcS}{q_i}$ and $\pdv{\mcH}{q_i} = 0$, the derivative
 \[
 \pdv{\mcS}{q_i} = \alpha_i
@@ -36,10 +40,19 @@ W(q_1,\dots,q_n) = W_1(q_1) + W_2(q_2) + \cdots + W_n(q_n).
 \]
 Each function $W_i(q_i)$ satisfies an ordinary differential equation involving one separation constant, and the complete integral is obtained by evaluating $n$ quadratures.}
 
-\nt{The additive separation theorem gives a sufficient condition for separability. A more general theory was developed by Levi-Civita and later refined by Stackel. The Levi-Civita separability conditions state that the Hamilton--Jacobi equation is separable in a given orthogonal coordinate system if and only if the Hamiltonian can be written as a sum, each term depending on only one coordinate and its conjugate momentum. Equivalently, the coefficient matrix of the quadratic kinetic-energy form, when written in these coordinates, must be a Stackel matrix. A Stackel matrix $S_{ij}$ is an $n\times n$ matrix whose $(i,j)$ entry depends only on the single coordinate $q_i$, and whose determinant is nonzero almost everywhere. The inverse of the Stackel matrix then relates the separation constants to the Hamiltonian components, producing the $n$ separated ODEs.}
-
 Several standard orthogonal coordinate systems admit separable Hamilton--Jacobi equations for important classes of potentials. The gradient-squared operator takes different forms in each system, and the metric coefficients determine whether a given potential allows additive separation.
 
+\ex{Gradient in spherical coordinates}{
+In orthogonal curvilinear coordinates $(q_1,q_2,q_3)$, the line element is $\dd s^2 = h_1^2\,\dd q_1^2 + h_2^2\,\dd q_2^2 + h_3^2\,\dd q_3^2$, where $h_i$ are the scale factors. The magnitude-squared of a gradient follows from the metric as
+\[
+|\nabla W|^2 = \frac{1}{h_1^2}\left(\pdv{W}{q_1}\right)^2 + \frac{1}{h_2^2}\left(\pdv{W}{q_2}\right)^2 + \frac{1}{h_3^2}\left(\pdv{W}{q_3}\right)^2.
+\]
+For spherical coordinates $(r,\theta,\phi)$, the line element is $\dd s^2 = \dd r^2 + r^2\,\dd\theta^2 + r^2\sin^2\theta\,\dd\phi^2$, so the scale factors are $h_r = 1$, $h_\theta = r$, $h_\phi = r\sin\theta$. Substituting:
+\[
+|\nabla W|^2 = \left(\pdv{W}{r}\right)^2 + \frac{1}{r^2}\left(\pdv{W}{\theta}\right)^2 + \frac{1}{r^2\sin^2\theta}\left(\pdv{W}{\phi}\right)^2.
+\]
+This explicit form enters the time-independent Hamilton--Jacobi equation as $\,|\nabla W|^2/(2m) + V = E$ and the geometric factors $1/r^2$ and $1/(r^2\sin^2\theta)$ control which potentials permit additive separation.}
+
 \mprop{Coordinate systems and separability for the HJ equation}{
 The table below summarizes the gradient operator squared $|\nabla W|^2$ in commonly used orthogonal coordinate systems, and the classes of potentials that permit additive separation of the Hamilton--Jacobi equation with $\mcH = |\nabla W|^2/(2m) + V = E$:
 
@@ -81,6 +94,10 @@ The last two terms depend only on $\theta$ while the first and rightmost terms d
 \]
 Each equation integrates by quadrature, and $W_\phi(\phi) = \alpha_\phi\,\phi$. These three quadratures constitute the complete integral for the free particle in spherical coordinates.}
 
+\nt{Physical meaning of $\alpha_\phi$: In spherical coordinates the azimuthal angle $\phi$ is always cyclic for rotationally symmetric Hamiltonians. The separation constant $\alpha_\phi$ is the $z$-component of angular momentum, denoted $L_z$. It is a constant of motion because the system is invariant under rotations about the $z$-axis.}
+
+\nt{Physical meaning of $\alpha^2$: The constant $\alpha^2$ that appears when separating the angular variables $\theta$ and $\phi$ identifies with $L^2$, the squared total angular momentum. It measures the magnitude of rotational motion and produces the centrifugal barrier $L^2/(2mr^2)$ in the radial equation.}
+
 \qs{Separation for a particle in a uniform gravitational field}{
 A particle of mass $m$ moves in the $xy$-plane under a uniform gravitational field $g$ acting in the negative $y$-direction. The Hamiltonian is
 \[