Thermodynamics of Information in Dissipative Projectile Motion: A Dual Lagrangian Approach Unifying Energy and Entropy Balances

1. Introduction

Variational principles, such as the Principle of Stationary Action, are foundational to modern physics, describing the evolution of conservative systems through Lagrangian and Hamiltonian formalisms. Standard classical approaches to dissipative mechanical systems typically treat non-conservative forces either phenomenologically, such as through the introduction of Rayleigh dissipation functions to maintain the Lagrangian formalism [1], or via external work-energy balances [2]. More recently, Unified Mechanics Theory has sought to integrate thermodynamics directly into continuum mechanics to model material degradation and fatigue in solid systems [3].

Dissipation is intrinsically linked to entropy generation in non-equilibrium thermodynamics [4,5]. Historically, classical frameworks established that isolated systems evolve toward equilibrium states that maximise entropy [6,7,8]. Seeking to describe non-equilibrium processes, Prigogine pioneered the Minimum Entropy Production Principle (MEPP). However, as Jaynes [9] and Landauer [10] noted, the validity of MEPP is generally restricted to linear regimes near equilibrium. Beyond these linear regimes, the exact role of entropy in macroscopic continuous dynamics remains historically contentious, often leading to theoretical misconceptions regarding whether real systems minimise or maximise dissipation [11], or prompting various Maximum Entropy Production approaches [7,8]. Addressing this duality, Lucia recently emphasised that whether an actual physical trajectory minimises or maximises entropy generation strongly depends on the choice of system boundaries and the specific dynamic constraints applied [12]. Building upon these foundational debates, thermodynamic optimization and advanced non-equilibrium formalisms have developed a rich history, exemplified by frameworks such as Bejan's entropy generation minimization [13] and Jou’s extended irreversible thermodynamics [14].

Concurrently, modern non-equilibrium statistical mechanics has established robust mathematical connections between non-conservative phase-space dynamics, Shannon entropy [15], and thermodynamic dissipation [16]. A cornerstone of this relationship is Landauer's principle [17], which posits that the logical or physical erasure of information inherently requires a minimum thermodynamic cost in the form of heat dissipated to the environment. This thermodynamics of information has been validated primarily in microscopic stochastic systems and computing networks [18].

Despite these independent advances across mechanics, thermodynamics, and information theory, a significant gap persists: formulating these principles into a strict variational calculus framework—where macroscopic continuous dissipation is embedded directly into a purely geometric entropic action—remains a theoretical challenge. Macroscopic projectile motion subject to quadratic aerodynamic drag provides an ideal, highly transient physical paradigm to confront this open problem. While traditional Newtonian mechanics successfully predicts the geometric trajectory, it treats air resistance merely as a phenomenological mechanical loss. This classical perspective offers limited insight into the underlying thermodynamic and informational nature of the flight as it evolves toward a steady-state dynamical attractor, i.e., the terminal velocity [1].

In this contribution, we reinterpret dissipative macroscopic mechanics through the unified lens of non-equilibrium thermodynamics and information theory. We propose that because a dissipative projectile undergoes a continuous, irreversible contraction of its kinematic phase space toward a singular terminal attractor, it is mathematically erasing its initial state information. This reveals a coupling where macroscopic aerodynamic drag can be understood through information theory: the continuous erasure of initial mechanical conditions exacts an inevitable thermodynamic cost. Following the variational methodology described in Feynman’s Lectures on Physics for the minimum action principle [19], we construct an extended Lagrangian where the fundamental variable of optimization is entropy rather than action. By setting Newtonian dynamics as a geometric constraint on the environment's natural tendency to maximise entropy production, we demonstrate that the resulting trajectory emerges as a singular saddle-point extremal. Ultimately, this framework establishes a conceptual duality between continuous kinematic information loss and physical heat dissipation, proving that classical mechanical drag inherently acts as the physical penalty for the continuous erasure of macroscopic information.

The remainder of this paper is organised as follows. Section 2 establishes the physical framework, defining the macroscopic energy balances and the thermodynamic boundaries of the coupled system. Section 3 develops the entropy variational formulation, introducing the augmented Lagrangian and demonstrating the Hamiltonian nature of the singular extremal. Section 4 formulates the dynamics in terms of information theory, linking the phase-space contraction to the continuous loss of kinematic information. Section 5 presents the numerical simulation results, illustrating both the asymptotic convergence to the steady-state attractor and the optimization of the transient parabolic flight.Section 6 discusses the conceptual implications of the unified Lagrangian and outlines the limitations of the model, while Section 7 provides the concluding remarks. Finally, the supplementary appendices provide the rigorous mathematical derivations supporting this framework, including the equations of motion in intrinsic coordinates (Appendix A), the detailed variational calculus (Appendix B), the Liouville-Shannon phase-space formulation (Appendix C), the entropy equations for the constant impact range scenario (Appendix D), an extension to the linear friction regime (Appendix E), and the generalised entropy Lagrangian formulation for continuous dissipative systems (Appendix F)..

2. The Physical Framework of Projectile Motion

To construct a thermodynamic model of projectile motion, we must first reframe the mechanical dissipative flight as a macroscopic, closed, non-isolated system coupled to an isothermal environment. This requires defining the thermal boundaries and characterising the non-equilibrium transient states the projectile traverses during its flight. In this section, we establish the fundamental kinematic constraints governing the dissipative trajectory and formulate the instantaneous entropy production rate. By analysing the macroscopic heat dissipation, we lay the groundwork to link the mechanical properties of the flight to the thermodynamic evolution of the system, setting the stage for the variational approach.

2.1. Classical Mechanics and the Macroscopic Energy Balance

Macroscopic projectile motion involves continuous energy dissipation, which breaks the geometric symmetry of idealised parabolic flight (as illustrated in Figure 1a). While low-speed microscopic particles (low Reynolds number) experience linear drag, macroscopic objects at higher speeds are modelled by quadratic aerodynamic drag [20]:

$$\begin{array}{c}\mathit{F}=\mathit{m}\mathit{g}-\mathit{kv}\mathit{v}\end{array}$$

(1)

This non-conservative force continuously dissipates mechanical energy into the environment, shortening the maximum range across different launch angles (Figure 1b). From a mechanical perspective, this dissipative motion is governed by an energy balance among kinetic energy (K), potential energy (U), and the heat dissipated along the path (Q), expressed as:

$$\begin{array}{c}K={\displaystyle \frac{1}{2}}m{v}^2\end{array}$$

(2)

$$\begin{array}{c}U=\mathit{mgh}\end{array}$$

(3)

$$\begin{array}{c}Q=\int k{v}^2\mathit{dl}\end{array}$$

(4)

where m is the mass, h the height, g the gravity acceleration, and l represents the path over which the energy dissipation occurs. While these standard mechanical frameworks successfully predict the motion of projectiles [21,22], our approach shifts the analysis from a traditional force-based description to an energy-based thermodynamic perspective. To build this variational framework, we first explicitly define the thermodynamic states and the thermal boundaries of the coupled system.

2.2. Thermodynamic System and States

We define the projectile and the surrounding air as a coupled thermomechanical system, as illustrated in Figure 2. The interaction between the solid body and the fluid is characterised by the aerodynamic drag coefficient, k. In non-equilibrium thermodynamics, it is critical to define the thermal boundary conditions. Here, we model the surrounding atmosphere as an infinite, isothermal heat bath at a constant temperature T. For macroscopic projectiles traveling at subsonic speeds, the aerodynamic heating of the solid body itself is negligible. Consequently, all the mechanical power dissipated by the drag force

$$\begin{array}{c}{P}_{\mathrm{diss}}=F_d·v=k{v}^3\end{array}$$

(5)

is entirely transferred as convective heat to the environment. This physical assumption allows us to define the instantaneous macroscopic entropy production rate as:

$$\begin{array}{c}\dot{S}={\displaystyle \frac{k{v}^3}{T}}\end{array}$$

(6)

A comprehensive thermodynamic description requires defining the state of the system. In standard equilibrium thermodynamics, a state is well-defined when macroscopic variables remain time-invariant. During the projectile’s flight, however, both speed and angle evolve continuously, placing the system in a transient, non-equilibrium state. Unlike conservative systems, dissipative trajectories lack time-reversal symmetry and continuously lose mechanical energy. If the projectile undergoes an unconstrained fall, this continuous dissipation eventually balances the gravitational force, driving the system toward a macroscopic steady state where it reaches terminal velocity,

$$\begin{array}{c}{v}_{\mathrm{t}}=\sqrt{{\displaystyle \frac{\mathit{mg}}{k}}}\end{array}$$

(7)

which is completely independent of the initial launch conditions. Consequently, this terminal state acts as a dynamical attractor in phase space. Because distinct trajectories continuously converge toward this attractor, the system’s phase-space volume contracts over time. From an information-theoretic perspective, this continuous phase-space contraction means that the system gradually loses memory of its initial kinematic state, resulting in a continuous erasure of initial condition information.

2.3. The Entropy Variational Formulation

Having established the continuous loss of kinematic information, we now seek the fundamental physical principle governing this path. By defining an entropy-based Lagrangian, we use variational calculus to find the feasible physical trajectories.

2.3.1. The Thermodynamic Entropy Functional

Starting from the instantaneous entropy production rate $\dot{S}$, the total entropy generated throughout the flight path is evaluated by integrating over time. To express the trajectory directly within the geometric domain, this time integration is transformed into the path-angle differential dθ, using the kinematic relations detailed in Appendix A. The total thermodynamic entropy functional is expressed as (detailed in Appendix B.1):

$$\begin{array}{c}∆S={\int }_{t_0}^{t_1}\dot{S}\mathit{dt}={\int }_{\alpha }^{{\theta }_f}{\displaystyle \frac{k}{T}}{\displaystyle \frac{v^4}{g\cos \theta }}\mathit{d\theta }\end{array}$$

(8)

where the integrand acts as our unconstrained Entropy Lagrangian,

$$\begin{array}{c}L\left(v,\theta \right)={\displaystyle \frac{k}{\mathit{Tg}\cos \theta }}{v}^4\end{array}$$

(9)

However, this unconstrained expression yields non-physical trajectories. As demonstrated in Appendix B.1, unconstrained optimization of this functional forces the tangential acceleration to be zero, restricting the system to a steady-state regime. Therefore, physical kinematic constraints must be introduced. Applying Newton’s second law in intrinsic coordinates (tangential and normal axes), we derive the constraint coupling velocity v, the angle θ, and the derivative of the velocity with respect to the angle, v^'(dθ) = dv/dθ. As derived in Appendix B.2, this constraint can be written as G(v, v’, θ) = 0, where:

$$\begin{array}{c}G\left(v,v^{\mathrm{\text{'}}},\theta \right)=v^{\mathrm{\text{'}}}-\left(v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\right)=0\end{array}$$

(10)

To extremise the total entropy subject to this kinematic restriction, we introduce a trajectory-dependent Lagrange multiplier, λ(θ) ). This multiplier acts as a local thermodynamic entropy potential, representing the thermodynamic cost of enforcing the Newtonian path. The augmented Lagrangian, $ℒ$^*(v,v^',θ,λ), takes the form:

$$\begin{array}{c}{\mathcal{L}}^{*}\left(v,v^{\mathrm{\text{'}}},\theta ,\lambda \right)={\displaystyle \frac{k}{\mathit{Tg}\cos \theta }}{v}^4+\lambda \left(\theta \right)\left[v^{\mathrm{\text{'}}}-\left(v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\right)\right]\end{array}$$

(11)

Applying the Euler-Lagrange equation to $ℒ$^* with respect to velocity v:

$$\begin{array}{c}{\displaystyle \frac{d}{\mathit{d\theta }}}\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v^{\mathrm{\text{'}}}}}\right)-\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v}}\right)=0\end{array}$$

(12)

yields the differential equation governing the evolution of. The stationary trajectory (dS = 0) requires:

$$\begin{array}{c}{\lambda }^{\mathrm{\text{'}}}\left(\theta \right)={\displaystyle \frac{4k{v}^3}{Tg\cos \theta }}-\lambda \left(\theta \right)\left[\mathit{tg}\theta +{\displaystyle \frac{3k{v}^2}{\mathit{mg}\cos \theta }}\right]\end{array}$$

(13)

where λ^'(θ) = dλ/dθ. This result shows that the evolution of the Newtonian trajectory is coupled to the thermodynamic cost of adhering to the dynamic geometric constraint. Here, λ(θ) can be interpreted as an entropic sensitivity factor quantifying the response of the entropy production rate to the kinematic drag constraint [14].

2.3.2. The Hamiltonian Nature of the Extremal

The generalised momentum corresponding to our spatial evolution variable q is defined as $p={\displaystyle \frac{\partial L^{*}}{\partial v\mathrm{\text{'}}}}=\lambda \left(\theta \right)$ . Applying the Legendre transformation, the Hamiltonian is given by H = v’ p - $ℒ$^*. Substituting our previous expressions (as detailed in Appendix B.4), the Hamiltonian evaluates to:

$$\begin{array}{c}H\left(v,\theta ,\lambda \right)=\lambda \left(\theta \right)\left(v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\right)-{\displaystyle \frac{k{v}^4}{\mathit{Tg}\cos \theta }}\end{array}$$

(14)

To determine the nature of this extremal, we evaluate the second variation of the entropy functional (the Legendre condition) and find that it vanishes identically (d²S = 0). In variational calculus, this vanishing second variation defines the trajectory as a singular extremal, representing a functional saddle point in entropy space. Physically, this implies that the trajectory balances the thermodynamic tendency to maximise entropy production with the rigid kinematic boundaries imposed by classical mechanics.

Subject to Newtonian constraints, the trajectory for any launch angle is characterised by maximum allowable energy dissipation and maximum entropy generation. This contrasts with the Principle of Least Action in non-dissipative systems. While the standard mechanical action balances kinetic and potential energies to find a minimum, the unconstrained entropic Lagrangian lacks this internal stabilization. As previously demonstrated by the naïve Lagrangian (9), attempting to optimise entropy without restrictions yields non-physical solutions. Imposing Newton's laws as a dynamic constraint yields the saddle point, ensuring the trajectory dissipates the maximum energy permitted by the geometric boundaries.

2.3.3. The Geometric Evolution of Entropy Production

A consequence of the variational framework is the dependence of the thermodynamic entropy production on the trajectory’s geometry. By substituting the analytical solution of the velocity profile into the entropy production rate equation ($\dot{S}={\displaystyle \frac{k{v}^3}{T}}$), the entropy generation can be formulated purely in terms of the instantaneous path angle q (as derived via the energy domain in Appendix A.4):

$$\begin{array}{c}\dot{S}\left(\theta \right)={\displaystyle \frac{k}{T}}{\left({\cos }^2\theta \left[{\displaystyle \frac{1}{v_0^2{\cos }^2\alpha }}-{\displaystyle \frac{1}{v_t^2}}\left(F\left(\theta \right)-F\left(\alpha \right))\right.\right]\right)}^{-3/2}\end{array}$$

(15)

Where a is the initial launch angle, v₀ is the initial velocity, v_t is the terminal velocity, and F(q) is a geometric function defined as:

$$\begin{array}{c}F\left(\theta \right)={\displaystyle \frac{1}{2}}(\sec \theta \tan \theta +\mathit{ln}\mid \sec \theta +\tan \theta \mid )\end{array}$$

(16)

This formulation maps the instantaneous entropy production rate directly onto the angular evolution of the path. Integrating this rate yields the total accumulated entropy. The dependence on F(q) reflects a non-linear entropy production during thces the vertical asymptote (  -90º) and the velocity converges to v_t, the entropy production rate stabilises at a constant terminal value (${\dot{S}}_t$). This formulation characterises the system's transition from an initial transient state to a steady dissipation profile.

2.4. The Information Framework and Information Loss

The convergence towards the terminal velocity is governed by a dynamic attractor. As shown in Figure 3, regardless of the initial launch conditions, the system’s phase space contracts toward this final state. This convergence implies a continuous loss of the system’s initial kinematic information.

To formalise this information loss, we evaluate the divergence of the state velocity field in the velocity phase space. Using the Liouville equation for a dissipative 2D system governed by quadratic drag, the rate of change of the continuous Shannon entropy (H_S) associated with the kinematic probability distribution is (see Appendix C for the full derivation) [23]:

$$\begin{array}{c}{\displaystyle \frac{d{H}_S\left(t\right)}{\mathit{dt}}}={\nabla }_{v}a=-{\displaystyle \frac{3k}{m}}v\left(t\right)\end{array}$$

(17)

Where a is the acceleration. Shannon entropy decreases as the phase space contracts. We define the rate of kinematic information lost by the system, $\dot{I}$, as the negative change in Shannon entropy:

$$\begin{array}{c}\dot{I}\left(t\right)={\displaystyle \frac{\mathit{dI}\left(t\right)}{\mathit{dt}}}=-{\displaystyle \frac{d{H}_S\left(t\right)}{\mathit{dt}}}={\displaystyle \frac{3k}{m}}v\left(t\right)\end{array}$$

(18)

Integrating this expression reveals that the total information lost by the system is proportional to the physical path length l(t) travelled by the projectile:

$$\begin{array}{c}I\left(t\right)={\displaystyle \frac{3k}{m}}l\left(t\right)\end{array}$$

(19)

In this framework, the terminal velocity attractor acts as an entropic potential well. The divergence of the velocity field drives a flow of microstates through phase space until the initial kinematic information is dissipated and the system settles into its macroscopic steady state.

By combining the entropy production rate $\dot{S}$ (6) with the information loss rate ($\dot{I}$), we find the relationship:

$$\begin{array}{c}\dot{S}\left(t\right)={\displaystyle \frac{m^3}{27k^{2}T}}{\dot{I}\left(t\right)}^3\end{array}$$

(20)

This cubic dependency demonstrates that dissipating thermodynamic heat into the environment is coupled to the loss of the system's initial kinematic information as it converges toward the attractor.

3. Simulation Results of the Model

The theoretical trajectories derived from the variational framework were evaluated using the physical parameters of a standard golf ball: mass of m = 45.93 g, diameter d = 4.267 cm, and a quadratic drag coefficient k = 2.1·10-4 kg/m. Numerical simulations and curve fitting were performed in MATLAB. Code generation and mathematical verification were assisted by the Gemini 3.1 Pro large language model.

3.1. Thermodynamic Entropy and Information Evolution Towards the Steady State

The evolution of thermodynamic entropy production $\dot{S}$ as a function of position as the system approaches the macroscopic steady state (terminal velocity) is illustrated in Figure 4a, as inferred from (15). The continuous loss of kinematic information during the flight, driven by energy dissipation, is depicted in Figure 4b, generated using (19). Because the terminal state is an asymptotic limit, numerical simulations were truncated when the projectile reached 99.9% of its theoretical terminal velocity.

The steady-state behaviour of the system is captured by evaluating the governing differential equation for λ(θ) (13) in its asymptotic limit. As the trajectory approaches the vertical asymptote (q →-90º) and the velocity converges to the terminal attractor (v→ v_t), the dynamic Lagrange multiplier approaches a constant terminal value:

$$\begin{array}{c}{\lambda }_t={\displaystyle \frac{2}{g}}{\dot{S}}_{\mathrm{t}}\end{array}$$

(21)

Where ${\dot{S}}_{\mathrm{t}}={\displaystyle \frac{k{v}_{\mathrm{t}}^3}{T}}$is the constant entropy production rate at terminal velocity. Furthermore, expressing this asymptotic limit in terms of the continuous kinematic information erasure rate $\dot{S}$ derived in Section 4 (Eq. 20), $\dot{S}\left(t\right)={\displaystyle \frac{m^3}{27k^{2}T}}{\dot{I}\left(t\right)}^3$, the terminal multiplier takes the informational form:

$$\begin{array}{c}{\lambda }_t={\displaystyle \frac{2m^3}{27g{k}^{2}T}}{\dot{I}}^3\end{array}$$

(22)

In this asymptotic limit, the trajectory reaches a macroscopic steady state. This condition requires that the constant physical heat dissipation into the environment (${\dot{S}}_{\mathrm{t}}$) is constrained by the continuous erasure of kinematic information $\dot{\left(I\right)}$. This terminal behaviour confirms the coupling between the macroscopic thermodynamic evolution and the phase-space contraction derived in Section 2.3. As it is observed in Figure 5, the dynamic Lagrange multiplier smoothly converges toward this constant terminal attractor regardless of the initial launch conditions.

3.2. The Transient State in the Classical Parabolic Motion

Standard projectile motion can be reinterpreted through non-equilibrium thermodynamics by analysing the classic parabolic flight as a transient process between launch and ground impact. Bounded by the initial launch angle () and the impact angle (_imp), this finite trajectory segment allows the total entropy production to be evaluated, thereby identifying the optimal launch angle for maximum horizontal range.

The total accumulated entropy (obtained by integrating the instantaneous rate $\dot{S}$), the impact angle condition (derived by setting the total vertical displacement to zero), and the maximum horizontal range (x_imp) are governed by the following integral expressions:

$$\begin{array}{c}{S}_{\mathit{total}}\left(\alpha \right)={\displaystyle \frac{m}{T{v}_t^2}}{\int }_{{\theta }_f}^{\alpha }{\displaystyle \frac{\mathit{d\theta }}{{\cos }^5\theta {\left[\frac{1}{v_0^2{\cos }^2\alpha }+\frac{\left(F\left(\alpha \right)-F\left(\theta \right)\right.}{v_t^2}\right]}^2}}\end{array}$$

(23)

$$\begin{array}{c}0={\int }_{\alpha }^{{\theta }_{\mathrm{imp}}}{\displaystyle \frac{\mathrm{tg}\theta {\sec }^2\theta }{1-\frac{v_0^2{\cos }^2\alpha }{v_{\mathrm{T}}^2}\left(F\left(\theta \right)-F\left(\alpha \right)\right.)}}\mathit{d\theta }\end{array}$$

(24)

$$\begin{array}{c}{x}_{\mathrm{imp}}=-{\displaystyle \frac{v_0^2{\cos }^2\alpha }{g}}{\int }_{\alpha }^{{\theta }_{\mathrm{imp}}}{\displaystyle \frac{{\sec }^2\theta }{1-\frac{v_0^2{\cos }^2\alpha }{v_{\mathrm{T}}^2}\left(F\left(\theta \right)-F\left(\alpha \right))\right.}}\mathit{d\theta }\end{array}$$

(25)

Although these expressions are exact, evaluating the impact conditions requires numerical integration. Figure 6 illustrates the total entropy production over the entire flight as a function of the launch angle. Furthermore, Figure 7 displays the maximum range, while Figure 8 shows the total entropy production rate along the trajectory. The zero-crossing of the derivative identifies the optimal launch angle for maximum range.

Finally, this dissipative parabolic motion can be described through the coupling of physical dissipation and information theory. Using the relationship between path length and information loss (19), the maximum range x_imp and the optimal launch angle can be explicitly expressed as functions of the system's kinematic information loss (I):

$$\begin{array}{c}{x}_{\mathrm{imp}}={\int }_0^{t_{\mathit{imp}}}{v}_x\mathit{dt}={\int }_0^{I_{\mathit{imp}}}{v}_0\mathit{cos}\alpha e^{-{\displaystyle \frac{I}{3}}}\left({\displaystyle \frac{m}{3k}}·{\displaystyle \frac{1}{v\left(I\right)}}\mathit{dI}\right)={\displaystyle \frac{m{v}_0\mathit{cos}\alpha }{3k}}{\int }_0^{I_{\mathit{imp}}}{\displaystyle \frac{e^{-\frac{I}{3}}}{v\left(I\right)}}\mathit{dI}\end{array}$$

(26)

$$\begin{array}{c}\mathrm{tg}{\alpha }_{\mathrm{opt}}={\displaystyle \frac{\frac{e^{-\frac{I_{\mathrm{imp}}}{3}}}{v\left(I_{\mathit{imp}}\right)}d{I}_{\mathit{imp}}}{\mathit{d\alpha }}}+{\int }_0^{I_{\mathit{imp}}}{\displaystyle \frac{e^{-I/3}·\frac{\partial v\left(I\right)}{\partial \alpha }}{{\left(v\left(I\right)\right)}^2}}\mathit{dI}\end{array}$$

(27)

Beyond optimizing for maximum range, this geometric formulation applies to bounded trajectories with a fixed impact distance (x_imp). By specifying the boundary conditions at the point of impact ( = _imp) (see Appendix D), the instantaneous entropy production rate can be expressed as:

$$\begin{array}{c}{\dot{S}}_{\mathit{constantrange}}\left(\theta \right)={\displaystyle \frac{k}{T{\cos }^3\theta }}{\left[{\displaystyle \frac{1}{v_{\mathrm{imp}}^2{\cos }^2{\theta }_{\mathrm{imp}}}}+{\displaystyle \frac{1}{v_t^2}}\left(F\left({\theta }_{\mathrm{imp}}\right)-F\left(\theta \right)\right.)\right]}^{-3/2}\end{array}$$

(28)

In this formulation, the entropy production rate remains strictly trajectory-dependent. The geometric difference F(_imp) – F() serves as a thermodynamic distance measuring the entropic path length between the current state and the final impact state. Consequently, specifying a single boundary condition at the impact point is sufficient to characterise the thermodynamic evolution of the trajectory.

4. Discussion

Entropy Profile along the Bounded Parabolic Flight.

In bounded parabolic trajectories, entropy production is distinctly non-homogeneous, concentrating at the kinetic boundaries of launch and impact while dropping significantly near the zenith. From a classical mechanics perspective, the optimal launch angle for maximum horizontal range is traditionally treated as a defining characteristic of the motion. However, evaluated through our framework, this optimum does not correspond to a global thermodynamic extremum of accumulated entropy. This discrepancy reveals a crucial conceptual insight: the traditional bounded flight is merely an incomplete thermodynamic transient. The ground impact acts as an arbitrary geometric constraint that abruptly truncates the system's natural phase-space evolution toward its dynamic attractor (the terminal velocity). Consequently, the maximization of horizontal range is not an intrinsic thermodynamic property of dissipative systems, but rather a parametric artefact—a kinematic compromise forced by the premature termination of the information-erasure process.

The Augmented Lagrangian: Unifying Thermodynamics and Information

The dual Lagrangian formulation connects macroscopic mechanics and thermodynamics. Using the previously derived cubic relationship, the macroscopic entropy production rate can be expressed purely in informational terms:

$$\begin{array}{c}{\dot{S}}_{\mathit{gen}}\left(t\right)={\displaystyle \frac{m^3}{27k^{2}T}}{\dot{I}\left(t\right)}^3\end{array}$$

(29)

The Newton’s law for the tangential component

$$\begin{array}{c}m{\displaystyle \frac{\mathit{dv}}{\mathit{dt}}}=-\mathit{mg}\sin \theta -k{v}^2\end{array}$$

(30)

can be entirely rewritten as an informational constraint governing the acceleration of information loss ($\ddot{I}$). Substituting the informational equivalences yields:

$$\begin{array}{c}{\displaystyle \frac{m^2}{3k}}\ddot{I}=-\mathit{mg}\sin \theta -k{\left({\displaystyle \frac{m}{3k}}\dot{I}\right)}^2\end{array}$$

(31)

$$\begin{array}{c}\ddot{I}+{\displaystyle \frac{1}{3}}{\dot{I}}^2+{\displaystyle \frac{3\mathit{kg}}{m}}\sin \theta =0\end{array}$$

(32)

Consequently, the augmented Lagrangian can be formulated in terms of information as:

$$\begin{array}{c}{L}_I^{*}\left(\dot{I},\ddot{I},\theta ,{\lambda }_I\right)={\displaystyle \frac{m^3}{27k^{2}T}}{\dot{I}\left(t\right)}^3+{\lambda }_I\left(t\right)\left(\ddot{I}+{\displaystyle \frac{1}{3}}{\dot{I}}^2+{\displaystyle \frac{3\mathit{kg}}{m}}\sin \theta \right)\end{array}$$

(33)

or alternatively, it can be rewritten in terms of the thermodynamic entropy production rate:

$$\begin{array}{c}{L}_S^{*}\left(\dot{S},\ddot{S},\theta ,{\lambda }_S\right)=\dot{S}+{\lambda }_S\left(t\right)\left({\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{T}{k}}\right)}^{{\displaystyle \frac{1}{3}}}{\displaystyle \frac{\ddot{S}}{{\dot{S}}^{\frac{2}{3}}}}+{\displaystyle \frac{k}{m}}{\left({\displaystyle \frac{T}{k}}\right)}^{{\displaystyle \frac{2}{3}}}{\dot{S}}^{{\displaystyle \frac{2}{3}}}+g\sin \theta \right)\end{array}$$

(34)

As illustrated in Figure 9, this equivalence links the macroscopic and microscopic domains. Kinematic information loss ($\dot{I}$), grounded in Shannon entropy, governs the geometric evolution of the macroscopic trajectory (Newton's laws). Conversely, thermodynamic entropy production ($\dot{S}$) captures the microscopic consequence: thermal dissipation into the surrounding environment. The augmented Lagrangian unifies both thermodynamic laws: the unconstrained Lagrangian drives entropy generation (Second Law), while the dynamic constraint imposes macroscopic energy conservation (First Law). In this context, the cubic relationship between thermodynamic dissipation and information erasure (29) can be interpreted as a macroscopic continuum generalization of Landauer’s principle [17]. While Landauer’s principle originally establishes a minimum thermodynamic threshold for discrete microscopic state erasure, this formulation demonstrates an analogous continuous mechanism: the contraction of macroscopic kinematic phase space imposes an unavoidable entropic cost on the environment.

This formulation addresses the limitations of Prigogine's Minimum Entropy Production Principle (MEPP), which is strictly valid only for linear, near-equilibrium steady states. An unconstrained variational approach that extremises only the macroscopic entropy production rate forces the system into an artificial steady state with zero tangential acceleration (Appendix B.1). The augmented Lagrangian resolves this by introducing Newtonian dynamics as a geometric constraint.

In this framework, the variational balance emerges as an internal tension within the dissipation process. In the informational Lagrangian ($L_I^{*}$), the dynamic constraint encapsulates an "informational inertia" ($\ddot{I}$) that prevents the instantaneous erasure of the kinematic state. Equivalently, the thermodynamic Lagrangian ($L_S^{*}$) reveals a "thermodynamic inertia" ($\ddot{S}$): the macroscopic system resists abrupt accelerations in its entropy production rate. The unconstrained Lagrangian embodies the natural tendency to maximise entropy, while these inertial constraints prevent instantaneous dissipation. The resulting saddle-point extremal represents a physical compromise: the system maximises dissipation within the limits permitted by its mechanical and thermodynamic resistance.

Furthermore, this formulation embeds the thermodynamic arrow of time directly into the mechanical action. Unlike conservative mechanics, where the Lagrangian is time-reversible and phase-space volume is conserved (Liouville's theorem), macroscopic aerodynamic drag induces a continuous contraction of the phase space toward the terminal velocity attractor. By embedding the information erasure rate ($\dot{I}$) and the entropy production rate ($\dot{S}$) into the augmented Lagrangian, the variational principle breaks time-reversal symmetry. The variational extremal is bounded to the forward-time direction, aligning classical mechanical trajectories with the irreversible contraction of phase space.

Toward a Generalised Variational Principle for Dissipative Systems

The insights derived from the dissipative projectile trajectory suggest a generalised variational formulation for continuous dissipative systems. While classical Lagrangian mechanics relies on Hamilton's Principle of Least Action for conservative forces ($\Delta \int \mathit{Ldt}=0$) for conservative forces, non-conservative dissipation has historically required phenomenological constructs like Rayleigh dissipation functions. By shifting the primary functional from mechanical action to macroscopic entropic action, the dynamics of a continuous dissipative system described by generalised coordinates q(t) can be encapsulated into a single augmented entropic Lagrangian ($L_S^{*}$):

$$\begin{array}{c}{L}_S^{*}=\dot{S}+{\lambda }_S\left(t\right)·\left(f\left(E\right)=0\right)\end{array}$$

(35)

where $\dot{S}(\mathit{q},\dot{\mathit{q}})$ represents the rate of entropy production driven by the environment (embodying the Second Law of Thermodynamics), $f\left(E\right)\equiv {\displaystyle \frac{d{E}_{\mathit{mec}}}{\mathit{dt}}}+T\dot{S}=0$ enforces the instantaneous energy balance (embodying the First Law of Thermodynamics), and l_S(t) is the dynamic thermo-mechanical Lagrange multiplier.

The application of this variational principle requires expressing the macroscopic entropy production rate, $\dot{S}$, as a function of the system's geometric or kinematic variables. As demonstrated by the explicit angular derivation of $\dot{S}\left(\theta \right)$ in (15), mapping this thermodynamic drive onto the spatial trajectory is the mathematical prerequisite to extremizing the entropic action over the physical path.

This formulation bridges mechanics and non-equilibrium thermodynamics through three key mechanisms:

1-

Thermodynamic Drive vs. Mechanical Constraint: The unconstrained term $\dot{S}$ drives the system toward maximal dissipation, whereas f(E) = 0 establishes the phase-space boundary imposed by energy conservation, preventing unbounded dissipation.

2-

Dynamical Coupling via l_S(t): Extremizing the augmented action $A_s={\int }_{t_0}^{t_1}{L}_S^{*}\mathit{dt}$ with respect to l_S(t) recovers the physical trajectory f(E) = 0. Concurrently, variation with respect to the generalised coordinates q(t) yields the governing equation for l_S(t), quantifying the dynamic entropic cost of satisfying mechanical laws.

3-

Saddle-Point Extremal Nature: Because $L_S^{*}$ is linear with respect to l_S(t), the second variation vanishes identically along the physical path (d²A_S = 0). Physical trajectories are neither absolute maxima nor minima of entropy production, but singular saddle points in the augmented functional space.

Equation (35) thus offers a unified bridge between Newtonian dynamics and non-equilibrium thermodynamics, showing that real dissipative paths emerge as constrained extrema of entropy generation (see Appendix F for the detailed mathematical derivations and proofs).

Model Limitations and Future Perspectives

While this framework models the thermodynamic and informational dynamics of dissipative motion, it relies on several classical simplifications. First, heat transfer to the environment is assumed to be instantaneous and isothermal. Second, the aerodynamic drag coefficient is treated as strictly constant. Third, the projectile is modelled as a point mass, thereby omitting rotational dynamics and aerodynamic effects such as the Magnus effect.

Addressing these physical approximations offers immediate computational extensions, such as incorporating velocity-dependent drag coefficients, non-isothermal thermal coupling, and rigid-body mechanics. Beyond these practical extensions, the primary theoretical opportunity lies in formal generalization. Although the framework readily adapts to other macroscopic regimes, such as linear Stokes drag (Appendix E), connecting this variational formulation with statistical mechanics—such as the Mori-Zwanzig projection operator formalism [24] or Unified Mechanics Theory [3]— presents a path to anchor macroscopic informational constraints within microscopic irreversibility. Despite current simplifications, this model provides a quantitative foundation linking trajectory geometry, macroscopic information erasure, and thermodynamic dissipation in non-conservative systems.

5. Conclusions

This paper presents a variational framework integrating non-equilibrium thermodynamics and information theory into dissipative macroscopic mechanics. Rather than treating quadratic aerodynamic drag merely as a mechanical energy loss, this approach demonstrates that dissipative trajectories are governed by dual augmented Lagrangians—formulated equivalently in terms of thermodynamic dissipation rates ($L_S^{*}$), and informational erasure rates ($L_I^{*}$), that continuously balance kinematic evolution with entropy production.

Within this framework, a conceptual duality emerges: kinematic information loss tracks the phase-space contraction toward a dynamic attractor, while entropy production quantifies the resulting thermal dissipation into the environment. In either Lagrangian formulation, the variational saddle point balances the drive toward maximum dissipation against the system's mechanical and thermodynamic inertia. Consequently, the optimal launch angle for maximum horizontal range does not correspond to a global thermodynamic extremum, but arises from a parametric optimization constrained by the geometric boundaries of the flight path.

Ultimately, this methodology extends Landauer’s principle beyond microscopic and computational systems, providing a variational framework to incorporate the thermodynamic limits of information erasure into macroscopic classical mechanics.

Appendix A.1: Force Balance and Dynamical Equations

Consider a projectile of mass m moving in a vertical plane subject to gravity

$$\begin{array}{c}{\mathit{F}}_{\mathrm{g}}=-\mathit{mg}\widehat{\mathit{j}}\end{array}$$

(A1)

and a quadratic aerodynamic drag force

$$\begin{array}{c}{\mathit{F}}_{\mathrm{d}}=-{\mathrm{kv}}^2{\widehat{\mathit{u}}}_{\mathrm{t}}\end{array}$$

(A2)

where v is the instantaneous velocity magnitude and k is the drag coefficient. Resolving the forces along the tangential and normal directions yields:

Tangential direction (û_t): Both drag and the tangential component of gravity oppose the velocity vector (during the ascent phase), leading to:

$$\begin{array}{c}m{\displaystyle \frac{\mathit{dv}}{\mathit{dt}}}=-\mathit{mgsin}\theta -k{v}^2\end{array}$$

(A3)

Normal direction (û_n): The normal component of gravity provides the centripetal acceleration, causing the trajectory angle (measured with respect to the horizontal) to decrease over time:

$$\begin{array}{c}\mathit{mv}{\displaystyle \frac{\mathit{d\theta }}{\mathit{dt}}}=-\mathit{mgcos}\theta \end{array}$$

(A4)

To express the trajectory purely in the geometric domain, we eliminate the time parameter dt by applying the chain rule. Dividing the tangential acceleration by the angular velocity yields the variation of the velocity with respect to the flight angle:

$$\begin{array}{c}{v}^{\text{'}}={\displaystyle \frac{\mathit{dv}}{\mathit{d\theta }}}={\displaystyle \frac{\frac{\mathit{dv}}{\mathit{dt}}}{\frac{\mathit{d\theta }}{\mathit{dt}}}}={\displaystyle \frac{-\mathit{mgsin\theta }-k{v}^2}{-\frac{\mathit{mgcos\theta }}{v}}}={\displaystyle \frac{\mathit{vsin\theta }}{\mathit{cos\theta }}}+{\displaystyle \frac{k{v}^3}{\mathit{mgcos\theta }}}=v\mathit{tg\theta }+{\displaystyle \frac{k}{\mathit{mgcos\theta }}}{v}^3\end{array}$$

(A5)

This relation serves both as the basis for the classical Bernoulli solution and as the strict kinematic constraint for the variational formulation in Appendix B.

Appendix A.2: Transformation to the Arc Length Domain

To transition from the time domain to the geometric and thermodynamic domains, we change the independent variable to the arc length l, utilizing the kinematic relation dl = v dt. This allows us to rewrite the spatial derivatives as follows:

$$\begin{array}{c}{\displaystyle \frac{\mathit{d\theta }}{\mathit{dt}}}=-v{\displaystyle \frac{\mathit{d\theta }}{\mathit{dl}}}\end{array}$$

(A6)

$$\begin{array}{c}m{v}^2{\displaystyle \frac{\mathit{d\theta }}{\mathit{dl}}}=\mathit{mgcos}\theta \to {\displaystyle \frac{\mathit{d\theta }}{\mathit{dt}}}=-{\displaystyle \frac{g\mathit{cos\theta }}{v}}\end{array}$$

(A7)

For the kinetic energy K=1/2 mv², the spatial rate of change along the path is given by:

$$\begin{array}{c}{\displaystyle \frac{\mathit{dK}}{\mathit{dl}}}={\displaystyle \frac{\mathit{dK}}{\mathit{dt}}}{\displaystyle \frac{\mathit{dt}}{\mathit{dl}}}=\mathit{mv}{\displaystyle \frac{\mathit{dv}}{\mathit{dt}}}{\displaystyle \frac{1}{v}}=m{\displaystyle \frac{\mathit{dv}}{\mathit{dt}}}=-\mathit{mg}\sin \theta -{\displaystyle \frac{2k}{m}}K\end{array}$$

(A8)

Similarly, the spatial rate of change of the trajectory angle q becomes:

$$\begin{array}{c}{\displaystyle \frac{\mathit{d\theta }}{\mathit{dl}}}={\displaystyle \frac{\mathit{d\theta }}{\mathit{dt}}}{\displaystyle \frac{\mathit{dt}}{\mathit{dl}}}=\left(-{\displaystyle \frac{g\cos \theta }{v}}\right){\displaystyle \frac{1}{v}}=-{\displaystyle \frac{g\cos \theta }{v^2}}=-{\displaystyle \frac{\mathit{mg}\cos \theta }{2K}}\end{array}$$

(A9)

Appendix A.3: Derivation and Solution of the Bernoulli Equation

By dividing the energy equation by the angle equation, we eliminate the explicit arc length parameter l and obtain a single differential equation coupling the kinetic energy directly to the trajectory angle:

$$\begin{array}{c}{\displaystyle \frac{\mathit{dK}}{\mathit{d\theta }}}={\displaystyle \frac{\frac{\mathit{dK}}{\mathit{dl}}}{\frac{\mathit{d\theta }}{\mathit{dl}}}}={\displaystyle \frac{-\mathit{mg}\sin \theta -\frac{2k}{m}K}{-\frac{\mathit{mg}\cos \theta }{2K}}}\end{array}$$

(A10)

Rearranging the terms yields a classic first-order non-linear Bernoulli differential equation for K(q):

$$\begin{array}{c}{\displaystyle \frac{\mathit{dK}}{\mathit{d\theta }}}-2K\mathit{tg}\theta ={\displaystyle \frac{4k}{m^{2}g\cos \theta }}{K}^2\end{array}$$

(A11)

To find the analytical solution, we introduce the standard substitution u =1/K, which implies ${\displaystyle \frac{\mathit{du}}{\mathit{d\theta }}}=-{\displaystyle \frac{1}{K^2}}{\displaystyle \frac{\mathit{dK}}{\mathit{d\theta }}}$. Dividing the Bernoulli equation by K² and substituting u transforms it into a linear first-order differential equation:

$$\begin{array}{c}{\displaystyle \frac{\mathit{du}}{\mathit{d\theta }}}+2u\mathit{tg}\theta =-{\displaystyle \frac{4k}{m^{2}g\cos \theta }}\end{array}$$

(A12)

The integrating factor for this equation is

$$\begin{array}{c}I\left(\theta \right)=\exp \left(\int 2\mathit{tg}\theta \mathit{d\theta }\right)={\sec }^2\theta ={\displaystyle \frac{1}{{\cos }^2\theta }}\end{array}$$

(A13)

Multiplying both sides by sec² q allows for direct Integration:

$$\begin{array}{c}{\displaystyle \frac{d}{\mathit{d\theta }}}\left(u{\sec }^2\theta \right)=-{\displaystyle \frac{4k}{m^{2}g\cos \theta }}{\sec }^2\theta =-{\displaystyle \frac{4k}{m^{2}g}}{\sec }^3\theta \end{array}$$

(A14)

$$\begin{array}{c}u{\sec }^2\theta =-{\displaystyle \frac{4k}{m^{2}g}}\int {\sec }^3\theta \mathit{d\theta }+C\end{array}$$

(A15)

Using the standard integral for sec³ , we define the geometric function F() as:

$$\begin{array}{c}F\left(\theta \right)=\int {\sec }^3\theta \mathit{d\theta }={\displaystyle \frac{1}{2}}\left(\sec \theta \mathit{tg}\theta +\ln \left|\sec \theta +\mathit{tg}\theta \right|\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\sin \theta }{\cos \theta }}+\ln \left|\mathit{tg}\theta +{\displaystyle \frac{1}{\cos \theta }}\right|\right)\end{array}$$

(A16)

Substituting F(q) back and reverting to the kinetic energy variable K = 1/u, we isolate the integration constant and arrive at the complete analytical solution:

$$\begin{array}{c}{\displaystyle \frac{1}{K\left(\theta \right){\cos }^2\theta }}=C-{\displaystyle \frac{2k}{m^{2}g}}\left({\displaystyle \frac{\sin \theta }{{\cos }^2\theta }}+\ln \left|\mathit{tg}\theta +\sec \theta \right|\right)\end{array}$$

(A17)

Multiplying through by cos²  , the final exact solution for the kinetic energy profile along the trajectory is:

$$\begin{array}{c}{\displaystyle \frac{1}{K\left(\theta \right)}}=C{\cos }^2\theta -{\displaystyle \frac{2k}{m^{2}g}}\left(\sin \theta +{\cos }^2\theta \ln \left|\mathit{tg}\theta +\sec \theta \right|\right)\end{array}$$

(A18)

where C is an integration constant uniquely determined by the initial launch conditions (K₀, ₀ =). This rigorous geometric solution ensures full mathematical consistency with the thermodynamic and informational derivations presented in the core of the manuscript. From this expression and using K=1/2 m v², the function v() can be found.

Appendix A.4: Analysis in the energy domain: entropy equations

Departing from the energy domain, we transit towards the thermodynamic domain. We now move to thermodynamics point of view. The entropy production can be inferred from the dissipative term:

$$\begin{array}{c}{F}_d=k{v}^2\end{array}$$

(A19)

The dissipated power

$$\begin{array}{c}{P}_d=F_{d}v=k{v}^{2}v=k{v}^3\end{array}$$

(A20)

Hence, entropy production can be written as

$$\begin{array}{c}\dot{S}={\displaystyle \frac{P}{T}}={\displaystyle \frac{k}{T}}{v}^3\end{array}$$

(A21)

And in terms of kinetic energy

$$\begin{array}{c}K=A{\dot{S}}^{{\displaystyle \frac{3}{2}}}\end{array}$$

(A22)

Where we use the geometric function $F\left(\theta \right)={\displaystyle \frac{1}{2}}\left(\sec \theta \tan \theta +\ln \mid \sec \theta +\tan \theta \mid \right)$ and $A={\displaystyle \frac{T}{k}}{\left({\displaystyle \frac{m}{2}}\right)}^{3/2}$. We finally obtain

$$\begin{array}{c}\dot{S}\left(\theta \right)={\displaystyle \frac{k}{T}}{\left({\cos }^2\theta \left[{\displaystyle \frac{1}{v_0^2{\cos }^2\alpha }}-{\displaystyle \frac{1}{v_t^2}}(F\left(\theta \right)-F\left(\alpha \right))\right]\right)}^{-{\displaystyle \frac{3}{2}}}\end{array}$$

(A23)

In this expression, entropy production is a function of the trajectory, given the initial conditions of launch speed v₀ and angle . It is worth pointing out the dependence with v_t, which is what we defined as thermodynamic stationary state.

Integrating, the total entropy for a given path as a function of the initial angle is

$$\begin{array}{c}{S}_{\mathit{total}}\left(\alpha \right)={\displaystyle \frac{m}{T{v}_t^2}}{\int }_{{\theta }_f}^{\alpha }{\displaystyle \frac{\mathit{d\theta }}{{\cos }^5\theta {\left[\frac{1}{v_0^2{\cos }^2\alpha }+\frac{(F\left(\alpha \right)-F(\theta )}{v_t^2}\right]}^2}}\end{array}$$

(A24)

Appendix B.1: The Unconstrained Entropy Functional

To contextualise the robustness of the proposed model, it is instructive to first consider a more direct, naive approach. One might intuitively attempt to define the Lagrangian directly as the entropy production rate, such that

$$\begin{array}{c}{\mathcal{L}}_{\mathrm{l}}\left(l,\dot{l},t\right)=\dot{S}={\displaystyle \frac{k}{T}}{v}^3={\displaystyle \frac{k}{T}}{\dot{l}}^3\end{array}$$

(A25)

If we attempt to find the variational extremal for this formulation, we set:

$$\begin{array}{c}\delta {\int }_{t_0}^{t_1}{\displaystyle \frac{k}{T}}{\dot{l}}^3\mathit{dt}=0\end{array}$$

(A26)

Applying the standard Euler-Lagrange equation,

$$\begin{array}{c}{\displaystyle \frac{d}{\mathit{dt}}}\left({\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{l}}}{\partial \dot{l}}}\right)-\left({\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{l}}}{\partial l}}\right)=0\end{array}$$

(A27)

Yields

$$\begin{array}{c}{\displaystyle \frac{d}{\mathit{dt}}}\left({\displaystyle \frac{\partial }{\partial \dot{l}}}{\displaystyle \frac{k}{T}}{\dot{l}}^3\right)=0\end{array}$$

(A28)

$$\begin{array}{c}{\displaystyle \frac{d}{\mathit{dt}}}\left(3{\displaystyle \frac{k}{T}}{\dot{l}}^2\right)={\displaystyle \frac{6k}{T}}\dot{l}·\ddot{l}={\displaystyle \frac{6k}{T}}v·a_{\mathrm{t}}=0\end{array}$$

(A29)

Assuming a non-zero velocity and finite system constants (k, T > 0), this mathematical condition strictly requires the tangential acceleration to be zero (a_t = 0).

Physically, this restricts the validity of the Lagrangian strictly to steady-state regimes with constant velocity. While such a formulation aligns with classical non-equilibrium concepts—analogous to the minimum entropy production principle for stationary states—it completely fails to capture the transient dynamics and non-zero accelerations inherent to the system studied in this work. This limitation clearly highlights the necessity of adopting the extended Lagrangian formulation presented in our results, which successfully accommodates the full dynamic behaviour of the system without being artificially constrained to a zero-acceleration state.

The total macroscopic entropy produced along the trajectory, from the initial launch angle to the final impact angle _imp, is obtained by integrating the instantaneous entropy production rate over time: $\Delta S=\int \dot{S}\mathit{dt}$.

Using the kinematic transformations derived in Appendix A (dt = dl/v and the angular dependency $\mathit{dl}=-{\displaystyle \frac{v^2}{g\cos \theta }}\mathit{d\theta }$, we rewrite the temporal integral in the geometric domain:

$$\begin{array}{c}∆S={\int }_{t_0}^{t_1}\dot{S}\mathit{dt}={\int }_{\alpha }^{{\theta }_f}{\displaystyle \frac{k}{T}}{v}^3{\displaystyle \frac{v}{g\cos \theta }}\mathit{d\theta }\end{array}$$

(A30)

where the integrand acts as our unconstrained macroscopic Entropy Lagrangian:

$$\begin{array}{c}L\left(v,\theta \right)={\displaystyle \frac{k}{\mathit{Tg}\cos \theta }}{v}^4\end{array}$$

(A31)

Appendix B.2: Geometric Constraints and the Augmented Lagrangian

If the system were completely free to extremise this entropy production, it would yield non-physical trajectories. The projectile's spatial path is strictly confined by Newtonian dynamics. From the force balance in Appendix A, we can isolate the velocity derivative $v^{\text{'}}={\displaystyle \frac{\mathit{dv}}{\mathit{d\theta }}}$ to define a strict geometric constraint G(v, v', ) = 0:

$$\begin{array}{c}G\left(v,v^{\text{'}},\theta \right)=v^{\text{'}}-\left(v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\right)=0\end{array}$$

(A32)

To find the true physical trajectory, we must extremise the total entropy subject to this kinematic restriction. We introduce a trajectory-dependent Lagrange multiplier, , which acts as an "entropy potential" representing the thermodynamic cost of adhering to the geometric constraint. The augmented Lagrangian, $ℒ$^*(v,v^',θ,λ) is defined as:

$$\begin{array}{c}{\mathcal{L}}^{*}\left(v,v^{\text{'}},\theta ,\lambda \right)={\displaystyle \frac{k}{\mathit{Tg}\cos \theta }}{v}^4+\lambda \left(\theta \right)\left[v^{\text{'}}-\left(v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\right)\right]\end{array}$$

(A33)

Appendix B.3: Application of the Euler-Lagrange Equation

The trajectory chosen by nature must satisfy the condition of stationary entropy (S = 0). By applying the Euler-Lagrange equation to the Augmented Lagrangian with respect to the velocity v:

$$\begin{array}{c}{\displaystyle \frac{d}{\mathit{d\theta }}}\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v^{\text{'}}}}\right)-\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v}}\right)=0\end{array}$$

(A34)

We compute the necessary partial derivatives:

$$\begin{array}{c}\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v^{\text{'}}}}\right)=\lambda \left(\theta \right)\end{array}$$

(A35)

$$\begin{array}{c}\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v}}\right)={\displaystyle \frac{4k{v}^3}{\mathit{Tg}\cos \theta }}-\lambda \left(\theta \right)\mathit{tg}\theta +\lambda \left(\theta \right){\displaystyle \frac{3k{v}^2}{\mathit{mg}\cos \theta }}\end{array}$$

(A36)

Substituting these back into the Euler-Lagrange equation yields the governing differential equation for the dynamic evolution of the thermodynamic multiplier:

$$\begin{array}{c}{\lambda }^{\text{'}}\left(\theta \right)=\mathit{d\lambda }/\mathit{d\theta }={\displaystyle \frac{4k{v}^3}{Tg\cos \theta }}-\lambda \left(\theta \right)\left[\mathit{tg}\theta +{\displaystyle \frac{3k{v}^2}{\mathit{mg}\cos \theta }}\right]\end{array}$$

(A37)

Multiplying all terms by cos , and taking the limit for → -90 º at the terminal velocity.

$$\begin{array}{c}0={\displaystyle \frac{4k{v}_t^3}{Tg}}+\lambda -3\lambda \end{array}$$

(A38)

And

$$\begin{array}{c}{\lambda }_t={\displaystyle \frac{2m{v}_t}{T}}\end{array}$$

(A39)

This asymptotic result has a profound physical interpretation. In variational thermodynamics, Lagrange multipliers act as generalised thermodynamic forces or potentials that enforce the system's dynamical constraints. Here, at the terminal velocity attractor, the history-dependent nature of the trajectory vanishes, and the entropy multiplier _t stabilises to a clean ratio between the terminal linear momentum (p_t = m v_t) and the isothermal temperature T. This reveals that in the steady state, the thermodynamic 'cost' or sensitivity of maintaining the physical trajectory is strictly quantised by the ratio of the macroscopic momentum to the scale of thermal fluctuations in the surrounding heat bath.

Appendix B.4: The Hamiltonian and the Singular Extremal

To fully characterise the nature of this extremal trajectory, we evaluate the Hamiltonian of the system. The generalised momentum corresponding to our spatial evolution variable is:

$$\begin{array}{c}p=\left({\displaystyle \frac{\partial {\mathcal{L}}^{*}}{\partial v^{\text{'}}}}\right)=\lambda \left(\theta \right)\end{array}$$

(A40)

Applying the Legendre transformation H = v' p - $ℒ$^* and substituting v' from the geometric constraint, the Hamiltonian takes the explicit form:

$$\begin{array}{c}H\left(v,\theta ,\lambda \right)=\lambda \left(\theta \right)\left(v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\right)-{\displaystyle \frac{k{v}^4}{\mathit{Tg}\cos \theta }}\end{array}$$

(A41)

Thus, deriving the Hamiltonian with respect to its variables, two equations are found:

Newton’s constrain

$$\begin{array}{c}{\displaystyle \frac{\mathit{dv}}{\mathit{d\theta }}}={\displaystyle \frac{\partial H}{\partial \lambda }}=v\mathit{tg}\theta +{\displaystyle \frac{k}{\mathit{mg}\cos \theta }}{v}^3\end{array}$$

(A42)

Euler- Lagrange equation for entropy

$$\begin{array}{c}{\displaystyle \frac{\mathit{d\lambda }}{\mathit{d\theta }}}=-{\displaystyle \frac{\partial H}{\partial v}}={\displaystyle \frac{4k{v}^3}{\mathit{Tg}\cos \theta }}-\lambda \mathit{tg}\theta -{\displaystyle \frac{3k{v}^2}{\mathit{mg}\cos \theta }}\lambda \end{array}$$

(A43)

So, the Hamiltonian provides a description of a dynamic coupled system in the phase space (v, ) where the velocity specifies how the object moves and entropy moment, , ensures that the trajectory which extremises the entropy is compatible with the loss of information.

When evaluating the necessary conditions for this extremal, the second variation of the entropy functional evaluates exactly to zero (²S = 0). This defines the trajectory as a "singular extremal"—a functional saddle point. Physically, this implies that the classical trajectory represents a critical balance between the thermodynamic imperative to maximise dissipation and the rigid kinematic boundaries imposed by Newtonian mechanics.

Appendix C.1 Phase Space Contraction

In a conservative (Hamiltonian) system, Liouville's theorem states that the phase space volume is conserved (${\nabla }_{\mathsf{\Gamma }}·\dot{\mathsf{\Gamma }}=0$), meaning the Shannon entropy of the probability distribution of states remains constant over time. However, in a dissipative system, energy is lost to the environment, and the system converges towards an attractor (the terminal velocity v_t), causing the phase space volume to contract. If we define the Shannon entropy of the continuous probability distribution ,t) in phase space as [26]:

$$\begin{array}{c}{H}_S=-\int \rho \ln \rho d\mathsf{\Gamma }\end{array}$$

(A44)

The rate of change of Shannon entropy, as the distribution is transported by the dynamical flow $\dot{\Gamma }$, is governed by the expected value of the divergence of the state velocity field:

$$\begin{array}{c}{\displaystyle \frac{d{H}_S}{\mathit{dt}}}=〈{\nabla }_{\mathsf{\Gamma }}·\dot{\mathsf{\Gamma }}〉\end{array}$$

(A45)

Appendix C.2 Phase Space Divergence for 2D Quadratic Drag

For a projectile moving in a 2D plane (x, y) under quadratic drag, the state vector in phase space is defined by its position and velocity coordinates:

$$\begin{array}{c}\Gamma ={\left(x,y,v_x,v_y\right)}^T\end{array}$$

(A46)

The time evolution of the state vector is given by the equations of motion:

$$\begin{array}{c}\dot{\mathsf{\Gamma }}={\left(v_x,v_y,a_x,a_y\right)}^T\end{array}$$

(A47)

The divergence of this vector field in phase space is:

$$\begin{array}{c}{\nabla }_{\mathsf{\Gamma }}·\dot{\mathsf{\Gamma }}={\displaystyle \frac{\partial v_x}{\partial x}}+{\displaystyle \frac{\partial v_y}{\partial y}}+{\displaystyle \frac{\partial a_x}{\partial v_x}}+{\displaystyle \frac{\partial a_y}{\partial v_y}}\end{array}$$

(A48)

Since the spatial derivatives of the velocity components are inherently zero [1] $\left({\displaystyle \frac{\partial v_x}{\partial x}}={\displaystyle \frac{\partial v_y}{\partial y}}=0\right)$ the phase space divergence simplifies entirely to the divergence of the acceleration in velocity space:

$$\begin{array}{c}{\nabla }_{\mathsf{\Gamma }}·\dot{\mathsf{\Gamma }}={\nabla }_{\mathrm{v}}·a={\displaystyle \frac{\partial a_x}{\partial v_x}}+{\displaystyle \frac{\partial a_y}{\partial v_y}}\end{array}$$

(A49)

Substituting the components of acceleration under 2D quadratic drag ($a_x=-{\displaystyle \frac{k}{m}}v{v}_x\mathrm{and}{a}_y=-g-{\displaystyle \frac{k}{m}}v{v}_y\mathit{where}v=\sqrt{v_x^2+v_y^2})$, we apply the chain rule to compute the partial derivatives:

$$\begin{array}{c}{\displaystyle \frac{\partial a_x}{\partial v_x}}=-{\displaystyle \frac{k}{m}}\left({\displaystyle \frac{\partial v}{\partial v_x}}{v}_x+v\right)=-{\displaystyle \frac{k}{m}}\left({\displaystyle \frac{v_x^2}{v}}+v\right)\end{array}$$

(A50)

$$\begin{array}{c}{\displaystyle \frac{\partial a_y}{\partial v_y}}=-{\displaystyle \frac{k}{m}}\left({\displaystyle \frac{\partial v}{\partial v_y}}{v}_y+v\right)=-{\displaystyle \frac{k}{m}}\left({\displaystyle \frac{v_y^2}{v}}+v\right)\end{array}$$

(A51)

Summing these terms yields the total phase space divergence:

$$\begin{array}{c}{\nabla }_{\mathrm{v}}·a=-{\displaystyle \frac{k}{m}}\left({\displaystyle \frac{v_x^2+v_y^2}{v}}+2v\right)=-{\displaystyle \frac{k}{m}}\left(v+2v\right)=-{\displaystyle \frac{3k}{m}}v\end{array}$$

(A52)

Thus, the Liouville-Shannon relation for the dissipative projectile in 2D is exactly:

$$\begin{array}{c}{\displaystyle \frac{d{H}_S}{\mathit{dt}}}=-{\displaystyle \frac{3k}{m}}v\left(t\right)\end{array}$$

(A53)

Appendix C.3 Connection to Kinematic Information and Path Length

As the phase space contracts, the Shannon entropy decreases, representing a gain in certainty (information) about the system's macroscopic state as it converges toward the attractor. Defining the kinematic information capacity I(t) as the negative change of Shannon entropy (I = -H_S), we obtain:

$$\begin{array}{c}{\displaystyle \frac{\mathit{dI}}{\mathit{dt}}}={\displaystyle \frac{3k}{m}}v\left(t\right)\end{array}$$

(A54)

Given that v(t) = dl/dt, where l(t) is the arc length along the trajectory, integrating this expression yields a direct linear relationship between the purged information and the physical distance travelled:

$$\begin{array}{c}I\left(t\right)={\displaystyle \frac{3k}{m}}l\left(t\right)\end{array}$$

(A55)

Appendix C.4 Augmented entropy Lagrangian

The informational Lagrangian

$$\begin{array}{c}{L}^{*}\left(\dot{I},\ddot{I},\theta ,{\lambda }_I\right)={\displaystyle \frac{m^3}{27k^{2}T}}{\dot{I}\left(t\right)}^3+{\lambda }_I\left(t\right)\left(\ddot{I}+{\displaystyle \frac{1}{3}}{\dot{I}}^2+{\displaystyle \frac{3\mathit{kg}}{m}}\sin \theta \right)\end{array}$$

(A56)

Can be rewritten in terms of thermodynamic entropy following the change of variables:

$$\begin{array}{c}v={\left({\displaystyle \frac{T}{k}}\dot{S}\right)}^{{\displaystyle \frac{1}{3}}}\end{array}$$

(A57)

The first derivative

$$\begin{array}{c}\dot{I}={\displaystyle \frac{3k}{m}}{\left({\displaystyle \frac{T}{k}}\right)}^{{\displaystyle \frac{1}{3}}}{\dot{S}}^{{\displaystyle \frac{1}{3}}}\end{array}$$

(A58)

And the second derivative

$$\begin{array}{c}\ddot{I}={\displaystyle \frac{k}{m}}{\left({\displaystyle \frac{T}{k}}\right)}^{{\displaystyle \frac{1}{3}}}{\dot{S}}^{{\displaystyle \frac{2}{3}}}\ddot{S}\end{array}$$

(A59)

Then

$$\begin{array}{c}{L}^{*}\left(\dot{S},\ddot{S},\theta ,{\lambda }_S\right)=\dot{S}+{\lambda }_S\left(t\right)\left({\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{T}{k}}\right)}^{{\displaystyle \frac{1}{3}}}{\displaystyle \frac{\ddot{S}}{{\dot{S}}^{\frac{2}{3}}}}+{\displaystyle \frac{k}{m}}{\left({\displaystyle \frac{T}{k}}\right)}^{{\displaystyle \frac{2}{3}}}{\dot{S}}^{{\displaystyle \frac{2}{3}}}+g\sin \theta \right)\end{array}$$

(A60)