Hamilton--Jacobi theory for non-conservative field theories in the $k$-contact framework

This paper establishes a comprehensive Hamilton–Jacobi theory for non-conservative classical field theories within the $k$-contact framework by introducing evolution $k$-contact $k$-vector fields, developing both $z$-independent and $z$-dependent approaches, and validating the formalism through diverse applications ranging from dissipative wave equations to relativistic thermodynamics.

The Gist

Imagine you are trying to predict how a complex system changes over time. In the world of physics, there are two main types of systems: conservative ones (like a perfect pendulum in a vacuum that swings forever) and non-conservative ones (like a real-world pendulum that slows down because of air resistance and friction).

This paper is about building a new mathematical "map" to understand the second type: systems that lose energy, or dissipative systems, but on a much larger scale than just a single pendulum. Instead of looking at a single point in time, they are looking at fields—things that exist everywhere in space and time, like sound waves, electric signals, or heat spreading through a metal plate.

Here is a breakdown of what the authors did, using simple analogies:

1. The Problem: The "Friction" of the Universe

Most classic physics math (Hamiltonian mechanics) was built for perfect, frictionless worlds. When you add friction (dissipation), the old math breaks down or becomes very messy.

• The Analogy: Imagine trying to navigate a city using a map that only shows streets but ignores traffic jams and road closures. You can get to your destination, but the route you calculate won't match reality.
• The Paper's Goal: The authors created a new "map" (a mathematical framework called k-contact geometry) that naturally includes the "traffic jams" (dissipation) so you can navigate non-conservative fields accurately.

2. The New Tool: "k-Contact" Geometry

The authors use a framework called k-contact geometry.

• The Analogy: Think of a standard map (symplectic geometry) as a flat piece of paper. It works great for simple things. But the real world is 3D and complex.
• The "k" Factor: The "k" in their theory represents multiple dimensions of time or space acting at once. Instead of just tracking how a system changes from "now" to "next second," this theory tracks how it changes across a whole grid of space and time simultaneously.
• The "Contact" Part: They added extra variables (called dissipative variables, or $z$) to the map. Think of these as "energy meters" attached to every point in the system. As the system evolves, these meters tick down, recording exactly how much energy is being lost to friction or heat.

3. Two Ways to Read the Map

The paper develops two different ways to use this new map to solve problems, which they call Hamilton-Jacobi theories.

Approach A: The "z-Independent" Way (The Static Blueprint)

• How it works: You look at the system's state without worrying about the specific "energy meter" readings at every single moment. You treat the energy loss as a background rule.
• The Analogy: Imagine you are designing a car engine. You know it will lose some fuel to heat, so you design the engine based on that general rule, without tracking the exact temperature of every bolt in real-time.
• The Result: This gives you a clean, simplified equation that tells you how the main parts of the system (like the position of a wave) move, ignoring the messy details of how the energy is lost, as long as the loss follows a simple rule.

Approach B: The "z-Dependent" Way (The Live Dashboard)

• How it works: You include the "energy meter" readings ($z$) directly into your map. You track the system and its energy loss simultaneously.
• The Analogy: This is like driving the car while watching the dashboard. You see the speed, the fuel level, and the engine temperature all changing together. You are solving for the path and the energy loss at the same time.
• The Result: This is more flexible. It allows for complex situations where the friction changes depending on how fast you are going or how hot the engine gets. It's a "live" simulation rather than a static blueprint.

4. The "Gauge" Mystery

One of the paper's key findings is that for these complex systems, there isn't just one mathematical description for a single physical situation.

• The Analogy: Imagine you are describing a route from New York to Boston. You could say "Go North," or "Go 50 miles, then turn East." Both get you there, but they describe the path differently. In this math, there are many different "routes" (mathematical fields) that describe the exact same physical reality.
• The Paper's Insight: The authors figured out how to handle this "choice." They showed that even though the math has this flexibility (which they call gauge freedom), the final physical prediction (where the wave ends up) remains the same.

5. Real-World Examples Tested

To prove their new map works, they applied it to four different real-world scenarios:

1. The Damped Telegrapher/Klein-Gordon Equation: Modeling how electrical signals fade as they travel down a wire (like an old-fashioned telegraph line).
2. The Dissipative Hunter-Saxton Equation: Modeling waves in liquid crystals (like the stuff in your LCD screen) that lose energy.
3. A Simple Dissipative Field: A basic test case to show how the math handles systems where you can't easily predict the future state just from the current one.
4. Relativistic Thermodynamics: Modeling how heat and entropy (disorder) flow in a system moving at high speeds, treating heat flow as a physical field just like electricity.

Summary

In short, this paper builds a new, robust mathematical toolkit for understanding real-world physics where energy is lost.

• It moves beyond "perfect" physics to handle friction and heat.
• It works for fields (things spread out in space), not just single particles.
• It offers two ways to solve problems: a simplified "blueprint" method and a detailed "live dashboard" method.
• It successfully models complex phenomena like fading electrical signals and heat flow, proving that this new "k-contact" geometry is a powerful way to describe the messy, energy-losing universe we actually live in.

Technical Summary

1. Problem Statement

Classical field theories are traditionally modeled using conservative frameworks (e.g., symplectic or multisymplectic geometry). However, many physical systems, particularly those involving dissipation (energy loss) or non-conservative forces, require a geometric framework that naturally incorporates these effects.

• The Gap: While contact geometry ($k=1$) successfully describes dissipative mechanical systems, extending this to field theories (systems with multiple independent variables, $k > 1$) has been challenging. Existing formulations often lack a unified geometric structure for non-conservative fields or fail to address the specific gauge freedoms and integrability issues inherent in higher-dimensional contact structures.
• The Goal: To develop a rigorous Hamilton–Jacobi (HJ) theory for non-conservative classical field theories within the co-oriented $k$-contact geometry framework. This involves defining appropriate dynamical equations, addressing the non-uniqueness of Hamiltonian vector fields in $k$-contact settings, and formulating HJ equations in both $z$-independent and $z$-dependent approaches.

2. Methodology and Geometric Framework

The authors utilize $k$-contact geometry, a generalization of contact geometry where the manifold $M$ is equipped with an $\mathbb{R}^k$-valued 1-form $\eta = \sum \eta_\alpha \otimes e_\alpha$ satisfying specific non-degeneracy conditions ($\ker \eta \oplus \ker d\eta = TM$).

Key Geometric Tools Introduced:

• Evolution $k$-Contact $k$-Vector Fields: The authors extend the concept of "evolution vector fields" from contact mechanics to field theory. Unlike standard Hamiltonian $k$-vector fields ($X_h$) which satisfy $\iota_{X_h}\eta = -h$, evolution fields ($E_h$) satisfy $\iota_{E_h}\eta = 0$. This distinction allows for two parallel formulations of the dynamics.
• Gauge Freedom ($\ker \chi$): A critical finding is that for $k > 1$, the map $\chi: L^k TM \to T^*M \times \mathbb{R}$ (relating vector fields to Hamiltonian functions) has a non-trivial kernel. Consequently, a single Hamiltonian function $h$ does not uniquely determine a Hamiltonian $k$-vector field. The authors define gauge $k$-vector fields as elements of $\ker \chi$ and formulate the theory in terms of equivalence classes of vector fields modulo this kernel.
• Hamilton–De Donder–Weyl (HdDW) Equations: The paper derives both standard and evolution versions of the HdDW equations. For Hamiltonians affine in the dissipative variables ($z_\alpha$), the authors prove that the projected dynamics on the configuration space $Q$ are identical for both standard and evolution formulations, despite the difference in the equations governing the dissipative variables.
• Integrability and Coisotropy: The paper distinguishes between Legendrian submanifolds (relevant for $z$-independent approaches) and maximally coisotropic submanifolds (relevant for $z$-dependent approaches).

3. Key Contributions

A. Two Distinct Hamilton–Jacobi Formulations

The paper develops two families of HJ theories based on the choice of the base manifold for the projection:

1. $z$-Independent Approach (Action-Independent):

   • Base Manifold: Configuration space $Q$.
   • Section: Holonomic sections $\gamma: Q \to J_{Q,k}$ (where $J_{Q,k} = L^k T^*Q \times \mathbb{R}^k$).
   • Condition: The image of $\gamma$ must be a Legendrian submanifold.
   • HJ Equation:
     • Standard: $h \circ \gamma = 0$.
     • Evolution: $d(h \circ \gamma) = 0$.
   • Result: Integral sections of the projected $k$-vector field lift to solutions of the HdDW equations.

2. $z$-Dependent Approach (Action-Dependent):

   • Base Manifold: Extended space $Q \times \mathbb{R}^k$.
   • Section: Sections $\gamma: Q \times \mathbb{R}^k \to J_{Q,k}$.
   • Condition: The image of $\gamma$ must be a maximally coisotropic submanifold.
   • HJ Equation: Formulated as a condition on the projected class of the $k$-vector field to account for the gauge freedom ($\ker \chi$). The equation involves a matrix of functions $C$ satisfying a trace condition (related to $h \circ \gamma$ for standard, and traceless for evolution).
   • Significance: This approach is more flexible, allowing for sections that do not satisfy the stricter $z$-independent conditions, and is essential for recovering standard contact HJ theory when $k=1$ without restrictive assumptions.

B. Handling Non-Regular and Affine Hamiltonians

• The authors analyze Hamiltonians that are affine in dissipative variables (common in physics) and show how they yield second-order PDEs independent of the specific choice of gauge vector field.
• They also demonstrate that quadratic dependence on dissipative variables is structurally viable, broadening the scope beyond the affine case usually found in literature.
• The theory accommodates non-regular Hamiltonians (where the Hessian is singular), showing how projected dynamics can still be defined.

4. Results and Applications

The theoretical framework is validated through four distinct physical examples:

1. Damped Telegrapher/Klein–Gordon Equation:

   • Models damped wave propagation (e.g., transmission lines).
   • The authors derive explicit $z$-independent and $z$-dependent complete solutions.
   • They show how a quadratic dependence on dissipative variables can model effective, state-dependent damping coefficients.

2. Dissipative Hunter–Saxton Equation:

   • A nonlinear wave equation arising in liquid crystal dynamics.
   • The $z$-dependent approach yields nonlinear explicit solutions (e.g., quadratic in time) that are difficult to obtain via the $z$-independent method.
   • Demonstrates the flexibility of the $z$-dependent formalism in handling non-integrable projected fields.

3. Simple Dissipative First-Order Field Model:

   • A non-regular model illustrating that different geometric dissipative mechanisms (Standard vs. Evolution) can produce the same projected dynamics on the configuration space while encoding dissipation differently in the contact variables.

4. Covariant EIT-type Thermodynamic Model:

   • Applies the framework to Extended Irreversible Thermodynamics (EIT).
   • Models entropy production and fluxes as independent variables in a relativistic setting.
   • Shows that the formalism naturally handles balance laws and constitutive relations, recovering standard thermodynamic equations as a particular case.

5. Significance and Outlook

• Unification: The paper unifies the treatment of dissipative field theories under a single geometric umbrella, recovering standard contact mechanics ($k=1$) and $k$-symplectic conservative theories as limiting cases.
• Resolution of Non-Uniqueness: By explicitly addressing the kernel of the morphism $\chi$, the authors provide a rigorous way to handle the non-uniqueness of Hamiltonian vector fields in $k$-contact systems, which was a major obstacle in previous formulations.
• New Integrability Concepts: The introduction of maximally coisotropic foliations as the natural geometric object for $z$-dependent integrability offers a new perspective on complete integrability in field theory.
• Future Directions: The authors suggest extending the theory to singular/constrained Hamiltonians, higher-order field theories, and exploring the relationship between $k$-contact, $k$-symplectic, and multisymplectic formalisms in the context of dissipative deformations.

In summary, this work establishes a robust, geometrically consistent Hamilton–Jacobi theory for non-conservative field theories, providing powerful tools to analyze dissipative systems ranging from wave equations to relativistic thermodynamics.