\chapter{Advanced Analytical Mechanics}

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.

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}

\input{concepts/advanced/hj-equation}

\input{concepts/advanced/separation}

\input{concepts/advanced/action-angle}

\input{concepts/advanced/hj-em-coupling}

\section{Mechanics Problems via HJ}

\input{concepts/advanced/free-particle-hj}

\input{concepts/advanced/projectile-hj}

\input{concepts/advanced/sho-hj}

\input{concepts/advanced/kepler-hj}

\input{concepts/advanced/rigid-rotator-hj}

\section{Electromagnetism Problems via HJ}

\input{concepts/advanced/uniform-e-field-hj}

\input{concepts/advanced/cyclotron-hj}

\input{concepts/advanced/crossed-fields-hj}

\input{concepts/advanced/kepler-coulomb-hj}
