1. Introduction

Classical mechanics, as delineated by Newton and Laplace, hinges on the notion of time's reversibility. Yet, it doesn't grapple with the feasibility of reverse motion. Pertinently, anomalies sprang forth in heat experiments where heat perpetually transitions from warmer to colder bodies, never vice versa. This undeniable occurrence cast doubts on the mechanistic theory, marking the dawn of our comprehension of irreversible phenomena. Newtonian mechanics, in its essence, surmises that an observer can discern an equilibrium state of a mechanical system and can apportion it an energy and particle number.

Ergontropic Dynamics proposes a unique vantage point. Rather than deriving irreversible thermodynamics from its reversible microscopic counterpart, our approach bridges dynamics and thermodynamics, threading through the intricate realms of energy, work, and entropy. Anchoring this theory is the second law of thermodynamics, interpreted as a quintessential form of universal causality. Additionally, this entry delves into the nuances of rotational/vortical motion and its correlation to energy dissipation. In addition, the gradient of free energy that appears naturally in the present formulation, pivotal in thermodynamics and dynamics, serves as the "driving force" behind a myriad of physical processes. A potential difference, this gradient propels systems towards spontaneous reactions, aiming to diminish the energy variance. When the gradient reaches zero, the system attains equilibrium, signifying maximum stability under the prevailing conditions. The direction of spontaneous processes is dictated by this gradient, often observed as processes tending toward states with reduced free energy. Such principles also underpin coupled reactions, like ATP hydrolysis in biology, where the free energy released drives other energy-requiring reactions. Furthermore, the stability of various phases of a substance—solid, liquid, or gas—is influenced by their respective free energies at specific temperatures and pressures, with the lowest free energy phase being the most stable. Integrating the gradient of free energy into the main equation of dynamics bridges thermodynamic properties, such as stability and spontaneity, with dynamic behavior, providing a comprehensive framework to predict system evolution.

Implementing an information-theoretic scaffold, we gaze at entropy through the lenses of energy and other system variables. The drive to maximize entropy steers us to the cardinal equations of (electro)dynamics. Woven into this approach is the sacrosanct principle of energy and angular momentum conservation. We leverage the method of Lagrange multipliers to ascertain extremal values of system functions. The intricate connection between entropy and energy unfurls: energy emerges as the 'bookkeeper,' whilst entropy coordinate the evolutionary rhythm of processes. Furthermore, this formulation touches upon the intriguing notion of extended charged spheres and their inherent momentum. It postulates that the definitions tethered to point-like particles might not always hit the mark. Concluding, we frame the dynamics for both charged and neutral particles, spotlighting the indispensable role entropy and energy play in shaping system behavior.

2. Formulation

In the new framework, the fundamental equation of dynamics is written in the form

\begin{equation} m\frac{dv}{dt} = -m\frac{\partial \phi}{\partial r} - \frac{\partial}{\partial r}\left(\frac{J^2}{2I} - \omega \cdot J - F\right) \end{equation}

and it captures the dynamics of a particle within the framework of Ergontropic Dynamics. Each term in this equation represents a distinct influence on the particle's acceleration.

The first term \(-m\frac{\partial \phi}{\partial r}\) accounts for the effects of external forces and their associated potential \(\phi\) on the particle's motion. The negative gradient of the potential with respect to \(r\) describes how the particle responds to changes in this potential.

The second term \(-\frac{\partial}{\partial r}\left(\frac{J^2}{2I} - \omega \cdot J - F\right)\) is particularly intriguing within the Ergontropic Dynamics framework. This term encapsulates the intricate interplay between angular momentum (\(J\)), angular velocity (\(\omega\)), and an additional free energy (\(F\)) acting on the system. The derivative with respect to \(r\) indicates how changes in space affect this combined force term. This equation highlights the crucial role that angular momentum and velocity play in determining the particle's acceleration in this context, and as well free energy sources, always feeding irreversibility ^[1],^[2][3].

Central to the ergontropic framework is the seamless integration of energy and entropy into equations of motion. This integration begets an additional term, aptly named the "topological torsion current," ^[4] which encapsulates a system's innate propensity to elevate its entropy while influencing its energy dynamics. The inclusion of this term augments the equations of motion, capturing the intertwined interplay of energy tendency to minimal distribution and entropy maximization.

One of the key facets of ergontropic dynamics is its departure from the isolated treatment of translational and vortical motions. Traditional mechanics tends to examine these motions independently in a clear lack of consistency, whereas ergontropic dynamics harmonizes these two phenomena, describes them simultaneouly in the same logical, consistent framework, paving the way for a unified description of a diverse array of physical processes. This integration extends the framework's adaptability, rendering it applicable across a spectrum of subjects ranging from planetary motions to fluid dynamics and plasma behaviors^[3]. As a result, ergontropic dynamics provides a unified language to describe a multitude of physical phenomena, offering a more comprehensive understanding of the interconnected nature of the natural world.

3. Applications

Ergontropic dynamics introduces a new perspective on the study of planetary dynamics. The development of the topological torsion current reveals novel features of the equilibrium conditions in spinning electromagnetic or gravitational systems. The topological torsion current is described by the modified dynamical equation of motion that results from the equilibrium analysis. Notably, this topological current provides, via a vector potential, a novel link between linear momentum and angular motion. This relationship has broad ramifications that could change how we perceive the basic dynamics of rotating systems, offering fresh avenues for exploration in the realm of celestial mechanics ^[4], providing a compelling solution to a perplexing anomaly observed in spacecraft behavior during close planetary flybys – the anomalous acceleration. The topological torsion current, arising from ergontropic dynamics, emerges as a prime candidate to explain this anomalous acceleration. Its unanticipated role in shaping motion challenges existing paradigms and offers a direct and straightforward explanation for a long-standing puzzle ^[4]. This revelation underscores the potential of ergontropic dynamics to illuminate and resolve longstanding mysteries in the realm of astrophysics and space exploration. Emergent phenomena in nature often exhibit unexpected order. A notable example is the idea that a more intricate network of interactions between elements can result in fewer components needed to give rise to particular phenomena. This perspective, once proposed by Verlinde and others, argued for the emergence of forces from entropy considerations. However, this view was already emphasized in 2002 ^[5], when we consider the gravitational force's as an emergent force, or maybe in a more specific formulation, as an instance of direct conversion to force from an entropy-gradient. By investigating a particle in circular motion influenced by an entropic force, we can derive equations analogous to those in classical mechanics, but based on a thermodynamic foundation. This approach, known as Ergontropic Dynamics, not only retrieves well-known equations like Newton's gravitational law but does so from an entirely different vantage point, shedding new light on long-established physical principles.

If the entropy gradient in phase space does not reach equilibrium, the phase space density may not have a time derivative of zero, i.e., dρ/dt may not be zero. This leads to the concept of the set of reachable states of a system forming a compressible "fluid" in phase space. This observation provides additional insight into Hamiltonian dynamics and suggests further investigation in the areas of statistical physics and fluid dynamics. Indeed, this finding sheds light on the limitations of the Liouville theorem and has practical applications in areas such as ray stacking, stochastic cooling, and Rabi oscillations, to name a few ^[6]. Strategies to improve engine efficiency through innovative materials, design concepts, and alternative energy sources are explored in the framework explained briefly in this entry, highlighting the use of nanomaterials and surface engineering to create hydrophobic or other types of surfaces for harnessing entropy-gradient forces, in particular classifying two major mechanisms to drive engines, the thermal and the information-driven engine ^[7]. 

In thermodynamic equilibrium, the canonical momentum for each particle is provided by the familiar expressions detailed in standard textbooks. However, when the system is not in equilibrium, there are gradients in momentum space influenced by entropy. This can be represented by the following equation:

\begin{equation} \mathbf{P} = M \mathbf{v} + q \mathbf{A} + mT \frac{\partial S}{\partial p} \end{equation}

In this equation, the last term represents the gradient of entropy in a nonequilibrium scenario. Here, S stands for the entropy when the system is out of equilibrium.

For a deeper understanding, we can define S as a transformed function. The equation for S is:

\begin{equation}\label{1} S = \sum_i ^N S_i \left( E_i - \frac{p_i^2}{2m} -\frac{J_i^2}{2 I_i} - q_i V_i + q_i(\mathbf{A}_i \cdot \mathbf{v}_i)\right). \end{equation}

This equation incorporates, besides the usual components, a novel term regarded as 'entropic momentum'. This 'entropic momentum' is crucial as it represents the information exchanged with the medium from the system's perspective. It may lead to a breaking of symmetry in the action-to-reaction law, as discussed in several instances in Ref. ^[8].

4.

Ergontropic Dynamics and Variational Problems: A Unified Approach

In 1686, Newton addressed the problem of the shape of a rotationally symmetric body of least resistance, marking an early foray into variational calculus. However, the formal advent of the calculus of variations began a decade later in 1696 with Johann Bernoulli's presentation of the brachistochrone problem. Rather than relying on minimum-time assumptions or Euler-Lagrange equations, integrating the concepts of energy and entropy—central to thermodynamic equations—allows for a unified approach to mechanics. This integration offers an expanded framework for variational problems, such as the brachistochrone in a uniform field. Within a Cartesian frame and using the present formulation, the cycloid equations $x(t)=R(\Omega t-sin\Omega t)$ and y(t)=−g/Ω² *(1−cosΩt)+RcosΩt can be derived, highlighting the cycloid as the path of fastest descent. Notably, the core dynamic equation, encapsulates the natural propensity to minimize energy and maximize entropy, offering a more intuitive approach compared to traditional Lagrangian formulations.

Ergontropic dynamics opens a new chapter in our understanding of the intricate coordination of the natural world with its complex interaction between energy and entropy. Due to the unification of fundamental concepts, it transforms electrodynamics and classical mechanics, integrating dissimilar phenomena together into one coherent system. Ergontropic dynamics opens the door for a deeper and more thorough knowledge of the intricacies of the physical cosmos by linking energy and entropy. Through this novel perspective, we gain a deeper appreciation for the symbiotic relationship between these foundational concepts that underpin the behavior of the universe.

5. Conclusion

In summary, ergontropic dynamics stands as a groundbreaking paradigm shift, fostering a deeper connection between energy, entropy, and the behavior of physical systems. Its unification of translational and vortical motions introduces a more holistic representation of the natural world. Through the lens of ergontropic dynamics, the subtle interplay of energy and entropy unveils novel relationships, elucidates perplexing anomalies, and promises a richer understanding of the underlying fabric of reality. As we embrace this novel paradigm, we embark on a journey that challenges conventional wisdom and propels us towards a deeper and more profound comprehension of the intricate tapestry of the cosmos.  For further exploration, whether for academic research or pedagogical pursuits, a comprehensive repository of related publications is readily available in the references or in the ArXivs.