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Contact canonoid maps and their conserved and dissipated quantities

This paper investigates conserved and dissipated quantities arising from contact canonoid maps and their extensions to singular smooth maps by constructing polynomial first integrals and smooth invariant endomorphisms that yield up to $n$ functionally independent trace invariants in a $(2n+1)$ -dimensional space, illustrated through various physical examples involving friction and damping.

Original authors: R. Azuaje

Published 2026-09-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: R. Azuaje

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the physical world, energy rarely stays still. While the idealized laws of physics often describe systems where energy is perfectly preserved, like a pendulum swinging forever in a vacuum, real objects almost always lose energy to their surroundings. A swinging door slows down due to air resistance; a rolling ball stops because of friction. Scientists describe these losing systems using a framework called contact Hamiltonian mechanics. In this view, the state of a system is not just a point in space, but a point in a higher-dimensional landscape that includes a measure of energy loss. Within this landscape, there are special paths called trajectories that the system follows. The challenge for physicists is to find quantities—specific numbers or values—that remain constant or change in a predictable way as the system moves along these paths. Finding these "conserved" or "dissipated" quantities is like finding a reliable compass in a storm; it allows scientists to predict the future behavior of complex, messy systems without having to calculate every single step of their journey.

A recent study by R. Azuaje at the Czech Technical University in Prague tackles the problem of finding these reliable quantities when the mathematical maps used to describe the system are imperfect or "singular." In mathematics, a singular map is one where information is lost or compressed, much like trying to flatten a crumpled piece of paper without tearing it; some details simply vanish. Traditionally, methods for finding conserved quantities required the map to be smooth and reversible, meaning no information could be lost. Azuaje's work breaks this barrier. The researcher developed a new set of rules that allow for the discovery of these valuable quantities even when the map is singular and information is lost. The study proves that by looking at how the system's energy loss changes under these imperfect transformations, one can still construct a set of invariants—values that stay the same or change in a known pattern. This is significant because it extends the ability to analyze complex, real-world systems that were previously too messy or degenerate to study with these specific tools.

The core of the discovery lies in a technique that treats the system's energy loss as a variable that can be balanced. Imagine the system as a machine where energy is constantly leaking out. The researcher showed that even if you apply a transformation that distorts the machine's shape or compresses its parts, you can still find a new "energy loss rate" that matches the original. By comparing the original energy loss with this new rate, the study constructs a polynomial—a mathematical expression built from adding and multiplying terms—that acts as a filter. This filter pulls out specific values that remain constant as the system evolves. Crucially, this method works even when the transformation is singular, meaning it works even when the map collapses dimensions or creates points where the usual rules of geometry break down. The study demonstrates that these constant values can be found by examining the "traces" of a specific geometric object, which is essentially a way of summing up the stretching and squeezing effects of the transformation.

To make this concrete, the paper applies these ideas to several physical scenarios. One example involves a free particle moving through a medium that slows it down, like a ball rolling through thick syrup. The researcher showed that even when the mathematical description of this motion is simplified or altered in a way that loses information, the new method can still identify quantities that stay constant. Another example looks at a damped harmonic oscillator, which is a mass on a spring that loses energy over time. Here, the study found that the new construction produces a set of values that remain unchanged, providing a stable way to track the system's behavior despite the friction. The research also explored motion under gravity with linear friction, such as a projectile moving through the air where drag is proportional to speed. In these cases, the method successfully identified independent conserved quantities, proving that the approach is robust across different types of physical forces.

A key finding of the paper is the relationship between two different ways of looking at the same problem. One way uses a polynomial expression to find the constants, while the other uses a geometric object called a tensor to find them. The study proves that these two methods are actually describing the same underlying reality. They encode the same information about the system's spectrum, or its fundamental frequencies of change. This equivalence is powerful because it means scientists can choose the tool that is easiest to use for a specific problem, knowing they will get the same answer. The paper also establishes a limit on how many independent constants can be found. In a system with a specific number of dimensions, there is a maximum number of these unique, non-redundant constants. The study shows that for a system with a certain complexity, this maximum is reached, meaning the method is as efficient as it can possibly be.

The work is not just theoretical; it includes explicit examples of singular maps where the method succeeds. For instance, the researcher constructed a map for a five-dimensional system that achieves the theoretical maximum number of independent constants. This demonstrates that the method is not limited to simple cases but can handle high-dimensional, complex scenarios. The study also addresses the issue of "dissipation rates," which is the speed at which energy is lost. Sometimes, a transformation changes the rate at which energy is lost. The paper provides a way to normalize the equations so that these different rates can be compared and used to find constants, even when the rates are not the same. This flexibility allows the method to be applied to a wider range of physical situations, including those where the energy loss is not uniform.

Ultimately, this research provides a new toolkit for understanding dissipative systems. By relaxing the strict requirement that mathematical maps must be perfect and reversible, the study opens the door to analyzing a broader class of physical phenomena. The ability to find conserved quantities in singular systems means that scientists can now model and predict the behavior of complex, real-world objects that were previously difficult to handle with these specific geometric techniques. The findings are presented as rigorous mathematical proofs, ensuring that the results are certain and reliable. The paper concludes by suggesting that these methods can be used to construct new maps for physical systems, offering a practical starting point for future investigations into how energy and motion interact in the presence of friction and other dissipative forces. The work stands as a solid extension of existing knowledge, bridging the gap between idealized mathematical models and the messy reality of physical systems.

Technical Summary: Contact Canonoid Maps and Their Conserved and Dissipated Quantities

Problem Statement
Contact Hamiltonian mechanics provides a geometric framework for dissipative systems where the Hamiltonian $H$ satisfies a dissipation law rather than a conservation law. While symplectic canonoid transformations (which preserve the Hamiltonian character of dynamics while altering the geometric structure) are well-studied, their contact counterparts face challenges, particularly regarding singular maps and the preservation of dissipation rates. Existing literature has largely focused on local diffeomorphisms and "good" contact hypotheses. This paper addresses the gap in formulating a rigorous definition for contact canonoid maps that may be singular (non-invertible) and distinguishing between the dissipation rates of the original contact form and its pullback. The goal is to construct conserved and dissipated quantities (first integrals) for such maps without requiring the map to be a diffeomorphism or to preserve the contact structure strictly.

Methodology
The author defines a smooth map $\Phi: M \to M$ as a contact canonoid map if the pullback of the contact one-form $\eta$ , denoted $\alpha = \Phi^*\eta$ , satisfies a specific Lie derivative condition along the Hamiltonian vector field $X_H$ :
$\mathcal{L}_{X_H}(\Phi^*\eta) = -b \Phi^*\eta$
where $b$ is a smooth function. This condition does not require $\alpha$ to remain a contact form (i.e., $\alpha \wedge (d\alpha)^n \neq 0$ is not assumed).

The methodology proceeds through two primary constructions to generate invariants:

Polynomial Construction: The paper utilizes a pencil of top-degree forms generated by a one-form $\beta$ that shares the same dissipation rate as $\eta$ . By expanding the form $(\eta + t\beta) \wedge (d\eta + t d\beta)^n$ in an auxiliary parameter $t$ , the coefficients are shown to be first integrals (conserved quantities).
- To handle cases where the pullback $\Phi^*\eta$ $Φ^{*} η$ has a different dissipation rate ( $b \neq a$ $b \neq = a$ ), two normalizations are introduced:
  - Contact-volume normalization: Defined on the regular locus where the Jacobian determinant is non-zero.
  - Hamiltonian normalization: Defined by $\beta = \frac{H}{K}\Phi^*\eta$ (where $K = -\alpha(X_H)$ ). This normalization is valid wherever $K \neq 0$ , accommodating singular points of the map and zeros of the original Hamiltonian $H$ .
Invariant Tensor Construction: On the set where $K \neq 0$ , the author constructs a smooth invariant endomorphism $N$ on the distribution $D = \ker(\eta)$ . This tensor is defined by the relation $d\eta(NY, Z) = -H d\sigma(Y, Z)$ , where $\sigma = -\alpha/K$ . The traces of powers of $N$ ( $I_k = \text{tr}(N^k)$ ) yield conserved quantities, while $D_k = H I_k$ yield dissipated quantities.

Key Contributions and Results

Generalized Definition: The paper establishes a definition for contact canonoid maps that applies to singular maps and does not require the existence of a Reeb vector field for the pullback form $\alpha$ .
Polynomial Invariants: It proves that for any one-form $\beta$ satisfying $\mathcal{L}_{X_H}\beta = -a\beta$ , the coefficients of the polynomial expansion of $(\eta + t\beta) \wedge (d\eta + t d\beta)^n$ are conserved. Multiplying these by $H$ yields dissipated quantities.
Tensorial Counterpart: A smooth invariant endomorphism $N$ is constructed on $\{K \neq 0\}$ . The paper demonstrates that the traces of $N^k$ are conserved and that $N$ vanishes smoothly at points where $H=0$ .
Spectral Equivalence: A central result (Proposition 6) establishes a precise relationship between the polynomial coefficients and the traces of the tensor $N$ . Specifically, for the Hamiltonian-normalized case, the polynomial $P(t)$ associated with the one-forms and the characteristic polynomial of the tensor are related by $P(t) = (1+t)Q(t)$ and $\det(\text{id}_D + tA) = Q(t)^2$ . This implies that the tensor construction encodes the same spectral information as the polynomial construction.
Upper Bound on Independence: In a manifold of dimension $2n+1$ , the construction yields at most $n$ functionally independent trace invariants.
Explicit Examples: The theory is illustrated through several mechanical systems:
- Shear transformations in Darboux coordinates.
- Damped free particle: Demonstrating a rank-two singular map where the unnormalized polynomial retains information lost in the normalized tensor, and a map smooth across the singular locus.
- Unequal dissipation rates: A transformation where the dissipation rate of the pullback differs from the original, successfully generating conserved quantities via normalization.
- Damped harmonic oscillator: Providing global canonoid transformations and explicit tensor formulas.
- Gravity with linear friction: Constructing conserved quantities for a 5D system.
- 5D Singular Map: An example in dimension 5 ( $n=2$ ) where a single tensor produces two functionally independent trace invariants, attaining the theoretical upper bound.

Significance and Claims
The paper claims to provide a unified framework for generating conserved and dissipated quantities for contact Hamiltonian systems under broad conditions, including singular maps and varying dissipation rates. The significance lies in:

Robustness: The constructions do not require the map to be invertible or the pullback to be a contact form.
Completeness: The Hamiltonian normalization allows the theory to extend to points where the original Hamiltonian vanishes, a region often excluded in standard treatments.
Equivalence: The demonstration that polynomial and tensorial constructions encode identical spectral information unifies two previously distinct approaches to finding invariants.
Practicality: The shear condition provides a direct method for constructing such maps via polynomial ansatzes.

The author notes that while the constructions generate invariants, determining their functional independence, involutivity, and the existence of contact Hamiltonian descriptions on the image of singular maps requires additional hypotheses not covered in this work. The paper concludes by emphasizing that the pullback definition permits the association of conserved quantities with contact canonoid maps even in the presence of singularities.

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