🔢 mathematics

Port-Hamiltonian systems with energy and power ports

This paper extends the port-Hamiltonian framework by introducing energy ports through the augmentation of the Lagrangian submanifold with external variables, thereby characterizing constrained Hamiltonian systems with input-dependent Hamiltonians.

Original authors: Kaja Krhac, Bernhard Maschke, Arjan van der Schaft

Published 2026-09-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kaja Krhac, Bernhard Maschke, Arjan van der Schaft

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

Imagine a world where every machine, from a simple pendulum to a complex power grid, is viewed not just as a collection of moving parts, but as a system constantly trading energy with its surroundings. In physics, scientists have long used a powerful framework called port-Hamiltonian theory to map out these exchanges. Think of this framework as a universal ledger that tracks how energy flows in and out of a system. It distinguishes between the internal state of the system—where the energy is stored—and the "ports," which are the specific points where the system connects to the outside world. Traditionally, these ports were understood as pairs of variables that multiply together to give power, much like voltage and current in an electrical circuit. This approach has been incredibly successful at describing how open systems interact, ensuring that the total energy is conserved as it moves between different components. However, this traditional view has a blind spot: it struggles to describe systems where the rules governing the energy storage itself can change based on external inputs, or where the system is constrained by hidden rules that limit its movement.

A team of researchers from Germany, France, and the Netherlands has now expanded this framework to close that gap. In their recent work, they introduced a new type of connection point called an "energy port." Unlike the standard power ports that measure the flow of energy, these new energy ports represent the external variables that actually shape the system's energy landscape. By adding these variables directly into the mathematical description of the system, the researchers found they could model a much broader class of physical realities. Specifically, they showed that this new approach perfectly describes constrained Hamiltonian systems—complex setups where the system is forced to follow specific rules, and where the energy function depends on inputs that the system does not control. This includes systems with "gauge degrees of freedom," where the system has more ways to move than there are equations to define them, a common feature in advanced physics like relativity and electromagnetism.

The core of this discovery lies in how the researchers treated the geometry of the system. In the traditional view, the relationship between a system's position and its momentum is often described by a single, smooth function. But many real-world systems are more complicated; their rules can be multi-valued or depend on parameters that shift. The authors utilized a mathematical tool known as a Morse family, which acts like a flexible template capable of describing these complex, multi-valued relationships. By attaching a space of external variables to this template, they created what they call a "port-Lagrangian submanifold." This new structure allows the system's energy to depend on inputs in a way that was previously difficult to capture. For instance, in a system with constraints, the energy isn't just a fixed value; it is a function that changes as the constraints are applied. The new framework treats these constraints not as obstacles to be solved, but as integral parts of the system's definition, appearing naturally as part of the energy port.

To demonstrate the power of this new definition, the researchers applied it to two distinct types of physical problems. First, they looked at input-output systems where the energy depends linearly on external inputs, such as a mechanical system being pushed by a variable force. In this scenario, the energy port represents the force itself, and the framework correctly calculates how the system's energy changes as that force varies. Second, and perhaps more significantly, they applied the theory to constrained systems, specifically those with gauge degrees of freedom. These are systems where the laws of physics allow for multiple descriptions of the same state, a situation that arises in the study of particles moving at relativistic speeds. The researchers showed that the total energy of such a system, which includes terms for these arbitrary constraints, fits perfectly into their new model. They used the example of a massive relativistic particle, a fundamental object in physics, to show how the framework handles the complex interplay between the particle's motion and the constraints imposed by the laws of relativity.

The results confirm that this extended framework is not just a theoretical curiosity but a robust way to describe the most general forms of constrained Hamiltonian systems. The authors proved that by augmenting the state space with external variables, they could describe systems where the Hamiltonian—the function that defines the total energy—depends on inputs. This means that the energy balance equation, which tracks how energy enters and leaves the system, must now include a term that accounts for changes in the constitutive relations themselves. In simpler terms, the system's energy doesn't just change because it moves or because energy flows in; it also changes because the very rules defining its energy are shifting. This insight unifies the description of open systems, constrained systems, and systems with external inputs under a single, coherent geometric structure.

The significance of this work is that it provides a unified language for describing complex physical interactions that were previously difficult to model together. By distinguishing between power ports, which measure flow, and energy ports, which measure the influence of external variables on the system's internal rules, the researchers have created a more versatile tool for engineers and physicists. Whether designing a control system for a robot or modeling the behavior of a particle in a high-energy collider, the ability to mathematically capture how external inputs reshape the system's energy landscape is crucial. The paper does not claim to solve every problem in physics, but it does offer a precise and general method for handling the specific class of problems where constraints and inputs are inextricably linked. Through the lens of this new framework, the complex dance of constrained systems becomes a clear, calculable interaction between the system's state and the external world.

Technical Summary: Port-Hamiltonian Systems with Energy and Power Ports

Problem Statement
The paper addresses a gap in the finite-dimensional port-Hamiltonian (pH) framework, which traditionally models open systems using a Dirac structure to define "power ports" (dual variable pairs representing energy flow) and a Lagrangian submanifold to define constitutive relations. While standard pH theory handles systems where the Hamiltonian is a function solely of the state space, it lacks a unified geometric description for systems where the Hamiltonian depends explicitly on external variables (inputs) and where constraints exist (constrained Hamiltonian systems). Specifically, the existing framework does not naturally distinguish between variables that drive energy changes via constitutive relations (inputs) and those that facilitate energy exchange with the environment (power ports).

Methodology
The authors extend the pH framework by augmenting the state space with a space of external variables. The core methodology relies on the geometric theory of Lagrangian submanifolds in cotangent bundles, described via Morse families (parameterized families of functions).

Extension of State Space: The state space $X$ is extended to $X \times \mathbb{R}^m$ , where $\mathbb{R}^m$ represents external variables. This augmentation extends the cotangent bundle $T^*X$ to $T^*(X \times \mathbb{R}^m)$ .
Definition of Energy Ports: A new pair of conjugated variables, termed energy ports $(\nu^p, \varepsilon^p)$ , is introduced on the Lagrangian submanifold. Unlike power ports, which are dual variables satisfying a power-conserving duality pairing, energy ports are external variables appearing in the constitutive relations.
Restricted Morse Families: The authors define a "restricted Morse family" $K: X \times \mathbb{R}^m \times \mathbb{R}^k \to \mathbb{R}$ . This family generates a port-Lagrangian submanifold $L_p$ . The rank condition on the derivatives of $K$ with respect to the external variables ensures the well-posedness of the energy ports.
Power Balance Reformulation: The time evolution of the system is governed by a quintuple $(X, F_p, \mathbb{R}^m, L_p, D)$ , where $D$ is a Dirac structure. The authors derive a modified power balance equation that includes an additional term accounting for the rate of change of energy due to variations in the constitutive relations (inputs), distinct from the standard power flow term.

Key Contributions

Formal Definition of Energy Ports: The paper introduces "energy ports" as a distinct class of variables from "power ports." Energy ports are not dual variables but are conjugated pairs associated with the space of external variables that parameterize the Hamiltonian.
Unified Geometric Framework: The authors demonstrate that constrained Hamiltonian systems (specifically those with gauge degrees of freedom) and input-output systems can be described within a single geometric framework using port-Lagrangian submanifolds generated by restricted Morse families.
Characterization of Constrained Systems: The paper shows that Dirac's extended Hamiltonian for constrained systems (linear in Lagrange multipliers) constitutes a Morse family. This allows constrained systems to be viewed as pH systems where the constraints act as parameters in the generating function.
Generalization of Input-Output Systems: The framework accommodates input-output systems where the Hamiltonian depends on inputs. It distinguishes between systems linear in inputs (where inputs are energy ports) and non-linear cases.

Results

General Case: In the most general case, the extended framework describes constrained Hamiltonian systems where the Hamiltonian function depends on inputs. The dynamics are determined by the intersection of the Dirac structure and the port-Lagrangian submanifold.
Power Balance: The derived power balance equation is:
$\frac{d\tilde{H}}{dt} + \dot{\varepsilon}^p_i \nu^p_i + e^p_j f^p_j = 0$
where $\tilde{H}$ is a modified energy function, $\nu^p$ are the energy port variables (inputs), $\varepsilon^p$ are their conjugates, and $(e^p, f^p)$ are the power port variables. The term $\dot{\varepsilon}^p_i \nu^p_i$ represents the energy change due to the variation of constitutive relations.
Specific Examples:
- Input-Output Systems: Systems linear in inputs are shown to have Lagrangian submanifolds generated by restricted generating functions on $X \times \mathbb{R}^m$ .
- Constrained Systems: Systems with gauge degrees of freedom (e.g., a massive relativistic particle) are modeled as pH systems where the constraints define the critical set of the Morse family.
- Constrained Input-Output Systems: The framework successfully models a massive relativistic particle in an external electromagnetic field, treating the field as an input and the mass-shell constraint as a gauge condition.

Significance and Claims
The paper claims that by augmenting the Lagrangian submanifold with external variables, the port-Hamiltonian framework gains the ability to directly access external variables appearing in the Hamiltonian function. This extension allows for a geometric characterization of constrained Hamiltonian systems (as defined by Dirac) where the Hamiltonian depends on inputs, and whose outputs are quantities conjugated to these inputs.

The authors emphasize that the multi-valued nature of Lagrangian submanifolds (via Morse families) is crucial for describing constrained systems, as it allows for a parameterized family of Hamiltonian functions where each member is physically viable. The introduction of energy ports provides a rigorous distinction between variables that drive the system's internal energy structure (constitutive relations) and those that facilitate energy exchange with the environment (power ports), thereby generalizing the pH theory to a broader class of open and constrained systems.

Technical Summary: Port-Hamiltonian Systems with Energy and Power Ports

More like this