fix(HJ): Rewrite chapter 3 intro — motivation, optics analogy, structure

chapters/advanced.tex

@@ -1,8 +1,115 @@
 \chapter{Advanced Analytical Mechanics}
 
-The Hamilton-Jacobi (HJ) formulation is the final reformulation of classical mechanics, expressing the entire dynamics of a system as a single first-order partial differential equation for a scalar function $S$, called the \textbf{principal function}. Solving the HJ equation by separation of variables often yields complete solutions more directly than the Lagrange or Hamilton equations -- especially for systems with symmetries and cyclic coordinates. The HJ framework also provides the classical foundation for the WKB approximation and connects to the Schrodinger equation in the $\hbar \to 0$ limit.
+Newtonian mechanics, Lagrangian mechanics, and Hamiltonian mechanics each reformulate
+the same physics in progressively more abstract language. Newton\normalsize{}'s laws write
+second-order differential equations for particle positions. The Lagrangian principle
+of least action recasts this as a variational problem, automatically accommodating
+constraints and generalized coordinates. Hamilton\normalsize{}'s canonical equations split
+the second-order problem into $2n$ coupled first-order equations on phase space,
+revealing the underlying symplectic structure of mechanics. Each of these formulations
+is fundamentally \textbf{trajectory-based}: you solve for a particle\normalsize{}'s specific
+path through space and time.
 
-This chapter is organized in three parts. Section 3.1 develops the HJ equation from Hamiltonian mechanics and introduces separation of variables, action-angle variables, and electromagnetic minimal coupling. Section 3.2 applies the HJ formalism to classical mechanics problems: the free particle, projectile motion, the simple harmonic oscillator, the Kepler (two-body) problem, and the rigid rotator on a sphere. Section 3.3 treats problems from electromagnetism, including charged particles in uniform $\vec{E}$-fields, cyclotron motion, and $\vec{E}\times\vec{B}$ drift, showing that the HJ approach recovers all standard results with a unified method.
+The Hamilton--Jacobi framework changes the question entirely. Instead of solving for
+a single trajectory, it asks: \textit{what is the global structure of all possible
+trajectories?} The answer is encoded in a single scalar function $S(q_1,\dots,q_n,t)$,
+called Hamilton\normalsize{}'s principal function, whose spatial gradient equals the
+canonical momentum:
+\[
+p_i = \frac{\partial S}{\partial q_i}.
+\]
+This relation elevates momentum from a dynamical variable to a property of a
+\textbf{field} defined over configuration space. Solving mechanics becomes a matter
+of finding the field $S$ that satisfies the Hamilton--Jacobi equation, a single
+first-order nonlinear partial differential equation. The Hamilton--Jacobi formulation
+is the most abstract of the four classical frameworks, but that abstraction is precisely
+what makes it useful: by shifting from trajectories to fields, it exposes structure
+that is invisible at the level of individual paths.
+
+The field perspective of the Hamilton--Jacobi formalism is not an accident. It
+reflects a deep analogy with geometric optics, first noticed by Maupertuis and
+made explicit by Hamilton himself. In geometrical optics, light propagates as
+rays, each orthogonal to surfaces of constant phase called \textbf{wavefronts}.
+The wavefronts are level sets of a scalar function called the \textbf{eikonal},
+and the local direction of each ray is determined by the gradient of the eikonal.
+Hamilton recognized that mechanics has the exact same structure. Particle trajectories
+play the role of light rays, surfaces of constant action $S$ play the role of
+optical wavefronts, and the gradient relation $p_i = \partial S/\partial q_i$
+mirrors the optical relation between wavefront normals and ray directions. In
+this view, the Hamilton--Jacobi equation is the mechanical analog of the
+\textbf{eikonal equation} of optics:
+\[
+\left|\nabla S\right|^2 = 2m\bigl(E - V(\vec{r})\bigr).
+\]
+Just as light rays bend when the refractive index changes, particle trajectories
+curve when the potential energy varies in space. The analogy runs even deeper:
+in both cases, the dynamics of rays is completely determined by the level-geometry
+of a single scalar field.
+
+This analogy is not merely poetic. It makes concrete the three most powerful
+features of the Hamilton--Jacobi approach. \textbf{First}, separation of variables
+for the Hamilton--Jacobi PDE reveals conserved quantities that are often obscured
+in the Newtonian or even Hamiltonian formulation. When the equation separates in
+a particular coordinate system, each additive separation constant corresponds to
+a constant of motion, and the choice of coordinates that enables separation is
+itself a signature of the system\normalsize{}'s hidden symmetry. Spherical coordinates
+separate for central potentials; parabolic coordinates separate for the Kepler
+problem and expose the Runge--Lenz vector\normalsize{}'s associated conservation law.
+\textbf{Second}, for periodic or bound systems, the \textbf{action-angle variables}
+$(J,w)$ provide a direct route to the system\normalsize{}'s frequencies without solving
+any differential equation. The action variable
+\[
+J = \frac{1}{2\pi} \oint p \, \mathrm{d}q
+\]
+is the area enclosed by the orbit in phase space, divided by $2\pi$. The frequency
+follows immediately as a partial derivative of the Hamiltonian with respect to
+the action:
+\[
+\omega = \frac{\partial H}{\partial J}.
+\]
+The angle variable $w$ advances uniformly in time, acting as a clock that tracks
+the system\normalsize{}'s progress through one period. Systems with commensurate
+frequencies close their trajectories, while incommensurate frequencies fill out
+invariant tori in phase space --- the geometric origin of resonant and chaotic
+behavior. \textbf{Third}, the Hamilton--Jacobi equation is the classical limit of
+quantum mechanics. In the Wentzel--Kramers--Brillouin (WKB) approximation, the
+quantum wave function is written as
+\[
+\psi(\vec{r},t) = A(\vec{r},t)\, \mathrm{e}^{iS(\vec{r},t)/\hbar}.
+\]
+Substituting this ansatz into the Schrödinger equation and collecting the leading
+order in $\hbar \to 0$ reproduces the Hamilton--Jacobi equation exactly. The older
+Bohr--Sommerfeld quantization rule,
+\[
+J = n h \qquad (n = 0,1,2,\dots),
+\]
+was the first successful attempt to quantize classical action, emerging naturally
+from the action-angle formalism nearly a decade before the modern theory of
+quantum mechanics. The Hamilton--Jacobi framework is the conceptual bridge that
+connects the orbit picture of classical physics to the wave picture of quantum
+physics.
+
+Everything in this chapter rests on the mechanics and electromagnetism you already
+know. The mechanics problems --- free particle, projectile motion, the simple harmonic
+oscillator, the Kepler two-body problem, and the rigid rotator --- draw on your
+work with kinematics, energy conservation, momentum, rotation, and central forces
+from the earlier mechanics units. The electromagnetism problems --- charged particles
+in uniform electric fields, cyclotron motion in magnetic fields, and
+$\vec{E} \times \vec{B}$ drift --- build on your treatment of Lorentz forces,
+equipotentials, and magnetic particle motion. The Hamilton--Jacobi formalism unifies
+all of these results under one method. For problems you have already solved by
+elementary means, it provides a deeper structural understanding. For problems that
+resist elementary approaches, it supplies a systematic technique grounded in the
+same variational principles you used to derive Lagrange\normalsize{}'s equations.
+
+This chapter is organized in three parts. Section 3.1 develops the HJ equation from
+Hamiltonian mechanics and introduces separation of variables, action-angle variables,
+and electromagnetic minimal coupling. Section 3.2 applies the HJ formalism to classical
+mechanics problems: the free particle, projectile motion, the simple harmonic oscillator,
+the Kepler (two-body) problem, and the rigid rotator on a sphere. Section 3.3 treats
+problems from electromagnetism, including charged particles in uniform $\vec{E}$-fields,
+cyclotron motion, and $\vec{E}\times\vec{B}$ drift, showing that the HJ approach recovers
+all standard results with a unified method.
 
 \section{Hamilton-Jacobi Fundamentals}