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Hamiltonian dynamics with nonconservative forces on extended phase space: An analytical mechanics derivation

This paper derives the full Hamiltonian equations of motion for nonconservative and nonautonomous dynamical systems on extended phase space using classic analytical mechanics, with specific focus on the momentum coordinate and generalized force associated with time.

Original author: Joseph T. A Peterson, Manoranjan Majji, John L. Junkins

Published 2026-11-02

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CC BY 4.0

Original author: Joseph T. A Peterson, Manoranjan Majji, John L. Junkins

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of how things move, from the swing of a pendulum to the orbit of a planet, scientists rely on a powerful framework called Hamiltonian mechanics. This approach treats a system's state as a collection of positions and momenta, evolving according to a specific energy function known as the Hamiltonian. When forces are conservative, meaning they conserve energy like gravity or a spring, the math is beautifully symmetrical and predictable. However, the real world is rarely so tidy. Friction, air resistance, and engine thrust are nonconservative forces that drain or add energy, breaking the perfect symmetry and making the standard equations harder to apply. Furthermore, many systems change their rules over time, such as a rocket burning fuel and losing mass, which introduces a direct dependence on the clock. To handle these complexities, physicists often turn to a concept called extended phase space. This technique treats time not just as a background clock, but as a coordinate on equal footing with position and momentum, effectively adding a new dimension to the mathematical map of the system. While these two ideas—handling messy forces and expanding the map to include time—are well known individually, they have rarely been combined into a single, coherent mathematical structure.

A team of researchers has now filled this gap by deriving a complete set of rules for how systems behave when both nonconservative forces and time-dependent changes are present, all within this expanded, multi-dimensional framework. Their work provides a rigorous method to describe the motion of complex systems, such as spacecraft navigating with thrusters or satellites affected by atmospheric drag, without losing the elegant mathematical structure that makes Hamiltonian mechanics so useful. The core of their discovery is a new set of equations that govern the evolution of these systems. These equations introduce a special new variable, a momentum associated with time itself, which acts as a bookkeeper for the system's energy changes. By including this variable, the researchers showed that even when external forces are pushing and pulling the system in unpredictable ways, the underlying mathematical structure remains intact and solvable.

The paper demonstrates that by treating time as a coordinate and adding its conjugate momentum, one can transform a difficult problem into a standard one. In the traditional view, changing the clock speed of a simulation—say, to make the math easier by slowing down time when things move fast—usually breaks the special symmetry of the equations. The researchers proved that by using their extended framework, this symmetry is preserved. They derived the full equations of motion for this new setup, showing exactly how the nonconservative forces, like friction or thrust, must be adjusted to fit the new, larger map. A key finding is that the total energy of this extended system, which includes the new time-momentum variable, remains constant throughout the motion, even when the original system is losing energy to friction. This conservation law acts as a guide, ensuring that the mathematical path stays true to the physical reality.

Crucially, the authors clarified a long-standing point of confusion regarding the relationship between this new time-momentum variable and the system's energy. They showed that the physical behavior of the original system is recovered only when the extended system is restricted to a specific condition where the new momentum equals the negative of the system's energy. This restriction is not a limitation on what the system can do; rather, it is a way of selecting the correct path from the infinite possibilities available in the larger mathematical space. When this condition is met, the complex, high-dimensional equations simplify perfectly to reproduce the known behavior of the original system, including the effects of all nonconservative forces.

The study also carefully distinguishes between different ways of handling time and forces. It shows that simply scaling the time variable in the old equations often destroys the mathematical structure, making the system harder to analyze. In contrast, the extended approach keeps the structure intact, allowing scientists to use powerful tools that were previously unavailable for systems with friction or changing parameters. The researchers did not just propose a new idea; they provided a complete derivation, starting from fundamental principles of virtual work and action, to prove that these equations are the correct description of reality. Their work confirms that the extended phase space is not just a theoretical curiosity but a necessary tool for accurately modeling the messy, time-dependent, and force-filled dynamics of the real world, offering a unified language for problems ranging from orbital mechanics to control theory.

Technical Summary: Hamiltonian Dynamics with Nonconservative Forces on Extended Phase Space

Problem Statement
While nonconservative forces and extended phase space formulations are individually well-established topics in classical Hamiltonian mechanics, their explicit combination within a unified analytical framework has been largely absent from the literature. Existing treatments often handle nonconservative forces via the Hamilton–d'Alembert principle on standard phase space or treat extended phase space (where time is a coordinate) primarily for conservative systems. Furthermore, while reparameterizing the evolution parameter (e.g., changing from time $t$ to a new parameter $\varepsilon$ ) is common in celestial mechanics and optimal control, direct conformal scaling of the equations of motion generally destroys the canonical Hamiltonian structure unless specific, restrictive conditions are met. This paper addresses the gap by deriving the full Hamiltonian equations of motion for nonautonomous, nonconservative dynamical systems formulated on an extended phase space, clarifying the role of the momentum conjugate to time, and systematically comparing different implementations of evolution-parameter transformations.

Methodology
The authors employ the framework of classic analytical mechanics, utilizing local coordinates and arbitrary symplectic (canonical) coordinates $(x, \pi)$ rather than restricting themselves to physical configuration-momentum pairs. The derivation proceeds through the following steps:

Review of Nonconservative Dynamics: The paper first establishes the equations of motion for nonconservative systems on standard phase space using the Hamilton–d'Alembert principle ( $\delta I + \int \delta W_{nc} dt = 0$ ). This yields generalized forces $\alpha_x$ and $\alpha_\pi$ derived from the virtual work of nonconservative forces.
Analysis of Direct Reparameterization: The authors examine the effect of directly scaling the evolution parameter via $dt = l d\varepsilon$ (where $l$ is a conformal factor). They demonstrate that for a general phase-space-dependent $l$ , this direct scaling results in "conformally Hamiltonian" dynamics that do not preserve the canonical symplectic structure on the original phase space.
Extended Phase Space Derivation: To preserve the Hamiltonian structure under general reparameterization, the authors introduce an extended phase space of dimension $2n+2$ , appending time $t$ as a generalized coordinate and introducing its conjugate momentum $\pi_t$ . They define an extended Hamiltonian $\bar{H} = l(H + \pi_t)$ .
Hamilton–d'Alembert Principle on Extended Space: The derivation is performed by reparameterizing the action integral and the virtual work term. A critical technical contribution is the rigorous handling of the virtual work term $\delta W_{nc}$ in the extended setting. The authors clarify that the variation $\delta$ in extended space (at fixed $\varepsilon$ ) differs from the variation at fixed time, leading to a specific form for the extended virtual work: $\tilde{\delta}W_{nc} = F_{nc} \cdot (\dot{t}\delta r - \dot{r}\delta t)$ .
Derivation of Equations of Motion: Applying the variational principle to the extended action yields the full set of extended equations of motion, including a new generalized force $A_t$ associated with the time coordinate.

Key Contributions and Results

Generalized Equations of Motion: The paper derives the complete set of $\varepsilon$ -parameterized canonical equations for the extended variables $(x, t, \pi, \pi_t)$ in the presence of arbitrary nonconservative forces. The equations are:
$\dot{x} = \frac{\partial \bar{H}}{\partial \pi} - A_\pi, \quad \dot{\pi} = -\frac{\partial \bar{H}}{\partial x} + A_x, \quad \dot{t} = \frac{\partial \bar{H}}{\partial \pi_t}, \quad \dot{\pi}_t = -\frac{\partial \bar{H}}{\partial t} + A_t$
where $A_x, A_\pi$ are scaled nonconservative forces, and $A_t$ is a specific generalized force term derived from the interaction between the nonconservative forces and the system's dynamics.
Conservation of the Extended Hamiltonian: A central result is the proof that the extended Hamiltonian $\bar{H} = l(H + \pi_t)$ is strictly conserved (an integral of motion) along the trajectories of the extended system, regardless of the presence of nonconservative forces or the explicit time-dependence of the original Hamiltonian.
The Zero Level Set ( $\Sigma_0$ ): The authors clarify the physical significance of the relation $\pi_t = -H$ . They demonstrate that the extended dynamics admit solution curves on any invariant level set $\bar{H} = \kappa$ . However, only the restriction to the zero level set ( $\Sigma_0$ , where $\bar{H}=0$ and thus $\pi_t = -H$ ) guarantees that the projected dynamics on the $(x, \pi)$ subspace reproduce the original physical dynamics (up to the prescribed reparameterization).
Structural Comparison: The paper provides a detailed comparison of four formulations:
1. Original $t$ -parameterized nonextended dynamics (Canonical).
2. Directly reparameterized nonextended dynamics (Generally non-canonical/conformally Hamiltonian).
3. Full extended dynamics (Canonical on $2n+2$ dimensions).
4. Extended dynamics restricted to $\Sigma_0$ (Physically relevant, but the constraint surface $\Sigma_0$ is odd-dimensional and thus cannot carry a symplectic structure itself; it is a restriction of the ambient canonical system).

Significance
The paper claims to fill a specific gap in analytical mechanics by explicitly developing the combination of nonconservative forces and extended phase space. Its significance lies in:

Systematic Preservation of Structure: It demonstrates that extending the phase space is a systematic method to implement general evolution-parameter transformations (conformal scaling) while retaining the canonical Hamiltonian form, a property lost in direct scaling methods.
Clarification of $\pi_t$ : It rigorously defines the momentum conjugate to time not merely as a mathematical artifact but as a coordinate whose dynamics are constrained to the zero level set to recover physical reality.
Unified Framework: It offers a consistent analytical framework for nonautonomous systems with nonconservative forces, applicable to arbitrary symplectic coordinates, which is particularly relevant for fields like celestial mechanics (regularization/linearization) and optimal control where time reparameterization is standard.

The authors maintain a modest tone, noting that if nonconservative forces are set to zero, their developments reduce to a concise account of the extended phase space construction for classical conservative systems, thereby unifying existing knowledge rather than overturning it.

Technical Summary: Hamiltonian Dynamics with Nonconservative Forces on Extended Phase Space

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