Hamilton-Jacobi as model reduction, extension to Newtonian particle mechanics, and a wave mechanical curiosity

This paper reinterprets the Hamilton-Jacobi equation as a model reduction that eliminates velocity degrees of freedom, thereby extending its applicability to general Newtonian systems with non-conservative forces and deriving a dissipative Schrödinger equation through a geometric optics approximation.

The Gist

The Big Picture: From Chaos to a Simple Map

Imagine you are trying to predict the path of a swarm of bees. In standard physics (Newtonian mechanics), to know where a bee will be in the future, you need to know two things right now: where it is and how fast it's moving. It's like driving a car; you need to know your current location and your current speed to predict where you'll be in 10 minutes.

This paper proposes a clever shortcut. It asks: What if we could forget about the speed entirely and just look at a special "map" that tells us exactly where the bee is going based solely on its current location?

The author calls this a "Model Reduction." Instead of tracking two variables (position and speed) for every particle, he tries to eliminate the speed variable and describe the whole system using just one variable: a "potential map" (called $S$).

The Core Idea: The "Speed Map"

In classical physics, if you have a conservative system (like a planet orbiting a star with no friction), there is a famous equation called the Hamilton-Jacobi (H-J) equation. Think of this equation as a recipe for baking a cake. If you follow the recipe perfectly, you get a "map" (the function $S$).

• The Old Way: You calculate the path of a particle step-by-step, checking its speed at every moment.
• The New Way (This Paper): You solve the H-J equation to get the map. Once you have the map, the speed of the particle is automatically determined just by looking at the slope of the map at that spot.
  • Analogy: Imagine a giant, hilly landscape (the map). If you place a marble anywhere on it, the marble doesn't need a separate instruction on how fast to roll; the steepness of the hill tells it how fast to go. The paper shows how to create this "landscape" even when things get messy.

The Twist: What if there is Friction?

Here is the paper's main breakthrough. The old H-J equation only worked for "perfect" worlds where energy is conserved (no friction, no air resistance). If you have a car braking or a ball sinking in mud, the old math breaks down.

Acharya says: "Let's fix the map for messy worlds too."

He extends the H-J equation to include dissipative forces (friction, drag, damping).

• The Metaphor: Imagine the "hill" in our landscape isn't just a static shape. If there is friction, the hill itself is slightly "sticky" or "spongy." The author derives a new version of the map equation that accounts for this stickiness.
• The Result: You can now create a "Speed Map" for systems that lose energy. This is a big deal because it allows us to simplify complex, messy real-world problems (like fluid dynamics or materials with internal friction) into a single, elegant equation.

The "Wave" Surprise: From Particles to Quantum

The paper takes a fascinating turn in Section 2.1. It asks: What happens if we treat this "Speed Map" not as a physical landscape, but as the phase of a wave?

In physics, there is a famous connection between particles and waves (Quantum Mechanics). The author uses a trick called the Geometric Optics Approximation (think of light rays vs. light waves).

1. He takes his new "friction-inclusive" map.
2. He translates it into a wave equation.
3. The Surprise: He derives a Dissipative Schrödinger Equation.

Why is this cool?
The Schrödinger equation is the heart of quantum mechanics. Usually, it describes how quantum waves evolve without losing energy. But here, the author shows that if you start with a particle experiencing friction (dissipation), and you translate that into a wave, you get a new kind of Schrödinger equation that includes a "loss" term.

• Analogy: It's like taking a sound wave traveling through a wall (which gets quieter due to friction) and writing down the mathematical rule for that specific type of fading sound. It connects the messy world of friction directly to the elegant world of quantum waves.

The "Many Sheets" Problem

The paper also addresses a tricky logical issue.

• The Problem: In the real world, you can start a particle at the same spot with different speeds. But the "Map" ($S$) usually only gives you one speed for a given spot.
• The Solution: The author explains that to cover all possibilities, you don't just need one map; you need a stack of maps (like a deck of cards).
  • Card 1 represents all particles starting with speed A.
  • Card 2 represents all particles starting with speed B.
  • By having this "deck" of solutions, you can describe any possible starting condition. This is a crucial insight for solving the equations completely.

Summary: Why Does This Matter?

1. Simplification: It offers a way to describe complex particle systems by removing the need to track speed explicitly, replacing it with a single "potential" function.
2. Realism: It extends this simplification to real-world scenarios involving friction and energy loss, which the old math couldn't handle easily.
3. Connection: It builds a bridge between classical mechanics (friction, particles) and wave mechanics (quantum equations), suggesting that even "messy" dissipative systems have a hidden wave-like structure.

In a nutshell: The author has found a way to draw a single, simplified "traffic map" for a city where cars are slowing down due to traffic jams (friction), and then showed that this map looks suspiciously like the rules governing quantum waves. It's a new, elementary, and powerful way to look at how things move.

Technical Summary

1. Problem Statement

The classical Hamilton-Jacobi (H-J) equation is traditionally derived within the framework of Lagrangian or Hamiltonian mechanics, specifically for conservative systems. It serves as a tool for integrating equations of motion or as a bridge to wave mechanics (Schrödinger equation).

However, two significant limitations exist in the standard formulation:

1. Scope: It is generally restricted to conservative forces (gradients of a potential). It does not naturally accommodate non-conservative or dissipative forces (e.g., friction, drag) which are ubiquitous in real-world Newtonian particle systems.
2. Derivation: Standard derivations rely on the calculus of variations (action functionals) or canonical transformations.

The paper addresses the problem of deriving the H-J equation as a model reduction of Newtonian particle mechanics that eliminates velocity degrees of freedom. The goal is to extend this formulation to general Newtonian systems (including dissipative forces) and explore its connection to a "dissipative" wave equation.

2. Methodology

The author employs a direct, elementary approach based on invariant manifolds and ansatz reduction, avoiding the calculus of variations in the primary derivation.

• Model Reduction via Ansatz:
  The author considers a system of $N$ particles governed by Newton's second law:
  $$\dot{v}_i = f_i(x, v, t), \quad \dot{x}_i = v_i$$
  Instead of solving for trajectories $(x(t), v(t))$ directly, the author posits a reduced set of trajectories where the velocity is a function of position and time only:
  $$v_i(t) = G_i(x(t), t)$$
  This represents a time-dependent invariant manifold of the dynamical system.

• Derivation of the Governing PDE:
  By substituting the ansatz into Newton's equations, a necessary condition is derived for the functions $G_i$ to satisfy the dynamics:
  $$\frac{\partial G_i}{\partial x_j} G_j + \frac{\partial G_i}{\partial t} - f_i(x, G, t) = 0$$
  This is a first-order PDE system for the vector field $G$.

• Gradient Ansatz (Scalar Potential):
  To connect to the classical H-J equation, the author restricts $G_i$ to be the gradient of a scalar function $S$ (Action):
  $$G_i = \frac{1}{m} \frac{\partial S}{\partial x_i}$$
  Substituting this into the governing PDE yields a generalized H-J equation.

• Force Decomposition:
  The force $f_i$ is decomposed into conservative parts (gradients of potentials $V$ and $\tilde{F}$) and non-conservative parts ($D_i$). This allows the derivation of a generalized equation that explicitly includes dissipative terms.

• Wave Mechanical Connection:
  Using a geometric optics approximation (WKB-like expansion) where $S$ is the phase of a complex wave function $\Psi = A e^{iS/\hbar}$, the author derives a corresponding wave equation.

3. Key Contributions and Results

A. Generalized Hamilton-Jacobi Equation for Dissipative Systems

The paper derives a generalized H-J equation for systems with linear damping and non-conservative forces.

• For a system with a potential $V$ and linear damping force $D = -\nu v$, the generalized equation is:
  $$\frac{\partial S}{\partial t} + \frac{1}{2m}|\nabla S|^2 + V + \nu S = 0$$
  (up to an additive function of time).
• Significance: This provides a physical justification for adding a "viscosity" term ($\nu S$) to the classical H-J equation. It explains why such terms appear in regularization contexts (e.g., to prevent shock formation in $\nabla S$).

B. Extension to General Non-Conservative Forces

For forces that cannot be written as gradients (e.g., "curl forces" or complex dissipative systems), the scalar ansatz is insufficient.

• The author generalizes the ansatz to include a divergence-free vector field $R$:
  $$G_i = \frac{1}{m} \frac{\partial S}{\partial x_i} + R_i, \quad \text{with } \nabla \cdot R = 0$$
• This leads to a coupled system of equations (Eq. 19) for the scalar field $S$ and the vector field $R$. This system represents a Hamilton-Jacobi-like system for general Newtonian forces, effectively reducing the second-order Newtonian system to a first-order system on an extended state space.

C. Derivation of a Dissipative Schrödinger Equation

By applying the geometric optics approximation to the generalized H-J equation with damping ($\nu > 0$), the author derives a nonlinear wave equation:
$$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\Delta \Psi + V \Psi + \nu \hbar \arctan\left(\frac{\text{Im}(\Psi)}{\text{Re}(\Psi)}\right) \Psi$$

• Result: The presence of classical dissipation ($\nu$) in the particle mechanics leads to a nonlinear term in the associated Schrödinger equation. This contrasts with the standard linear Schrödinger equation derived from conservative mechanics.

D. Re-derivation of Classical H-J via Action Variation

In Section 3, the author provides a self-contained derivation of the classical H-J equation using the action functional and extremal paths.

• Key Insight: The paper emphasizes that a single solution $S(x,t)$ to the H-J equation cannot describe all possible classical trajectories starting from arbitrary initial conditions $(x_0, v_0)$.
• Multiple Sheets: To cover the full phase space, one requires a family of solutions (multiple "sheets" $S^{(I)}$), each corresponding to a specific set of initial velocity constraints. This highlights the multi-valued nature of the action function in global mechanics.

4. Significance and Outlook

• New Perspective on Model Reduction: The paper reframes the H-J equation not just as a mathematical trick for integration, but as a rigorous model reduction technique that eliminates velocity variables. This viewpoint is novel and elementary, requiring no variational calculus for the primary derivation.
• Bridging Mechanics and Wave Theory: It establishes a direct link between classical dissipative mechanics and nonlinear wave mechanics. The result suggests that "dissipative quantum mechanics" (in a classical limit sense) naturally involves nonlinear wave equations.
• Continuum Mechanics Implications: The author suggests that this formalism could be extended to continuum mechanics. By treating the dual variables of dissipative continuum systems (via duality principles), one might define a "many-body" H-J problem for fields, potentially leading to new variational formulations for dissipative continua.
• Mathematical Challenges: The paper acknowledges that the resulting systems (Eq. 19) are formidable nonlinear PDEs. It points to emerging research in "relaxed" variational dual solutions and Mean Field Game theory as potential tools for solving these complex systems.

Conclusion

Amit Acharya's work successfully generalizes the Hamilton-Jacobi framework beyond conservative systems. By treating the H-J equation as a model reduction of Newtonian dynamics, the author derives consistent equations for dissipative systems and uncovers a natural connection to nonlinear Schrödinger-type equations. This approach offers a fresh, elementary perspective on the relationship between particle mechanics, wave mechanics, and the role of dissipation.