EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5478 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Conformable Hamiltonian Dynamics with Higher-Order Derivatives Yazen. M. Alawaideh1,∗, Aysel Ramazanova2, Hayat Issaadi3, Bashar. M. Al-khamiseh1, Muhammad Bilal4, Dumitru Baleanu5 1 MEU Research Unit, Middle East University, Amman, Jordan 2 University Duisburg-Essen 3 USTHB University, Operational Research Department, BP 32 El-Alia,Bab-Ezzouar, 16111, Algeria 4 Department of Physics, Shanghai University, Shanghai 200444, China 5 Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon Abstract. In this paper, we investigate higher-order calculus using the conformable derivative and integral. We use a fractional variant of the calculus of variations to obtain the Euler-Lagrange equation. Our route integral quantization approach streamlines the procedure by integrating solely over canonical coordinates qi, eliminating the requirement to integrate higher-order derivatives (qi = Dα t qi). In addition, we employ the conformable derivative to develop canonical conserved energy-momentum and Ostrogradsky’s Hamiltonian. Furthermore, we generalized the Hamilton formulation for higher order derivatives and applied this new formulation to obtained equations of motion for a one dimensional point particle. 2020 Mathematics Subject Classifications: 37J05, 70H20, 49S05, 26A33, 70G45 Key Words and Phrases: Left and right conformable fractional, Ostrogradsky’s Hamiltonian, Energy and momentum conservation, Euler-Lagrange, Conformable Higher-Order Derivatives, La- grange equation 1. Introduction Higher-Order Derivatives is proving to be invaluable across a spectrum of disciplines including chemistry, biology, and electronics [6, 20]. Recent research has showcased its util- ity in scaling phenomena [19, 23], classical mechanics, and mathematical modeling [25, 31]. Scholars have ventured into higher-order dynamical systems using Dirac’s restricted dy- namics [1, 2], examining the harmony between constraints and equations of motion [8], and advancing our understanding of systems with higher-order derivatives [9, 17]. Simplified ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5478 Email addresses: yazen.awaida@yahoo.com (Y. M. Alawaideh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 2 of 14 quantization techniques have been devised for both conservative and non-conservative sys- tems [16], with discrete systems receiving particular attention [4]. Recent inquiries have extended fractional equations to encompass fields like the Lee-Wick field [22], complex scalar fields [5], and the Dirac field [7]. Moreover, Noether’s theorem has been invoked to determine conserved quantities [3], while the application of Hamilton formalism with higher-order derivatives has facilitated the derivation of equations of motion, maintain- ing consistency with conventional mechanics [10]. The Riemann-Liouville derivative has emerged as a powerful tool for problem-solving [12], exhibiting convergence in the time domain as segment size increases. Many studies have advanced Hamilton-Ostrogradskii principle formulations for higher-order dynamical systems, including developments in sys- tems with higher-order derivatives and fractional derivatives, as well as innovations in path integral quantization [13, 15, 24, 29]. On the other hand, the Riemann-Liouville derivative and Caputo derivative do not obey the Leibniz rule and chain rule, which prevent us from applying these derivatives to the ordinary physical system with high-order derivatives. The conformable derivative was introduced in [18, 21]. This derivative obeys the Leibniz rule and the chain rule. The innovative approach, based on Leibniz and chain principles, gives identical derivatives of all orders under one, creating a flexible framework for studying higher-order derivatives. With increasing adoption by researchers, this definition broadens its applicability to include Hamiltonian formalism with high-order identical derivatives, ensuring independence from higher-order coordinate derivatives. This novel mathematical framework enables a more detailed depiction of dynamical systems characterized by com- plex behaviors such as memory effects, non-local interactions, and anomalous diffusion. Efforts focusing on fractional calculus within classical domains, mechanical systems, and variation problems aim to deepen our understanding of anomalous dynamics and non-local influences in physics and engineering. This paper aims to address the identified limitations by introducing a novel Hamiltonian formulation for systems involving higher-order deriva- tives. Our approach leverages a new definition of the conformable derivative that complies with both the Leibniz and chain rules, thereby enhancing its applicability. We anticipate that this proposed method will provide a wide range of accurate solutions to generalized differential equations with higher derivatives, effectively overcoming existing limitations. By integrating higher-order derivatives and fractional operators into our framework, we expect to create more adaptable models compared to those derived from traditional cal- culus. Furthermore, we intend to demonstrate that our Hamiltonian formulation can be developed independently of higher-order derivatives of the coordinates, which will sim- plify the analysis of complex dynamical systems and improve our understanding of their dynamics. The structure of this paper is as follows: Section 2 briefly discusses the definitions of conformable derivatives. Section 3 presents the Generalized Ostrogradsky’s Construction Form. Section 4 provides illustrative examples, and Section 5 introduces applications of conformable calculus along with suggestions for further research. Finally, the paper concludes with closing remarks. Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 3 of 14 2. Calculus of variations This section presents two distinct definitions of derivatives: left- and right-conformable derivatives (denoted CFDs). These definitions are integral to the Hamiltonian formulation and are employed in solving cases that result in equations of motion of various orders, such as (1/2, 1, 3/2, 2...). The left conformable derivative and the right-conformable derivative are defined and explored, along with their corresponding integrals, the left conformable integral (CFI) and the right conformable integral (CFI) [11, 18]. These concepts are funda- mental to understanding and applying higher-order calculus in various physical contexts. The right conformable derivative is defined as follows [11, 18]: Dα s|xf(t) = lim ϵ→∞ (t+ ϵ(t− s)1−α)− f(t) ϵ With the condition that s < t for the derivative to be valid for all t ≥ s. The left- conformable derivative is defined as follows [11, 18]: Dα t|s′f(t) = − lim ϵ→∞ (t+ ϵ(s ′ − t)1−α)− f(t) ϵ Noting that t ≥ s to avoid undefined expressions, as (t − s)α becomes negative in such cases. In the case where 0 < α ≤ 1, the left conformable integral (CFI) is defined as follows: Iαs|tf(t) = ∫ t s (ξ − s)1−αf(ξ)dξ Ensuring the integral limits are correctly defined from s to t. The right conformable integral (CFI) is defined as follows: Iα t|s′f(t) = ∫ s ′ t (s ′ − ξ)1−αf(ξ)dξ The relationships between the conformable derivatives (CFD) and integrals (CFI) are expressed as follows: Dα s|tI α s|tf(t) = f(t), Dα t|s′I α t|s′f(t) = f(t)− f(a), Next, we generalize to the case when s = 0, and use the notationDα s|t to denote (D α s|t = Dα t ) as follows : Dα s|xf(t) = lim ϵ→∞ (t+ ϵt1−α)− f(t) ϵ Where 0 < α ≤ 1 and Dα x represents a right CFD. The definition of CFD can be reformu- lated using l’Hôpital’s rule as follows [30]: Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 4 of 14 Dα s|tf(t) = f ′ (t)t1−t, Assuming f is differentiable, which is necessary for equation 2 to hold true. 3. Generalized Ostrogradsky’s Construction Form This section explores the Lagrangian and Hamiltonian formulations of mechanical sys- tems characterized by higher-order partial derivatives. It defines the configuration space using generalized coordinates and derives equations of motion through the Euler-Lagrange equations [14, 28]. Lagrangian Formulation We begin with a configuration space defined by n generalized coordinates q(t), Dα s|tq(t) and D2α s|tq(t), and D2α s|tαq(t). The equations of motion arise from the Euler-Lagrange equations, represented as : Jα [t] = Iα0|t0L(qi(t, ϵ α), Dα s|tqi(t, ϵ α), D2α s|tqi(t, ϵ α)) Where qi(t, ϵ α) = q(t) + ϵη(t) Dα t qi(t, ϵ α) = Dα t q(t) + ϵDα t η(t) and η(0) = η(t0) = 0 Subsequently, the need for an external value can be expressed as follows: [∂ϵS]ϵ=0 = Iα0|t0 [∂ϵL(qi(t, ϵ), D α t qi(t, ϵ), D α t D α t qi(t, ϵ))] = ∫ t 0 tα−1 [ ∂L ∂qi ∂qi ∂ϵ + ∂L ∂Dα t qi ∂Dα t qi ∂ϵ ∂L ∂Dα t D α t qi + ∂Dα t D α t qi ∂ϵ ] dt = ∫ t 0 tα−1 [ ∂L ∂qi η + ∂L ∂Dα t qi Dα t η + ∂L ∂Dα t D α t qi Dα t D α t η ] dt By employing integration by parts, we obtain the following expression: 0 = ∫ t 0 tα−1 [ ∂L ∂qi −Dα t ( ∂L ∂Dα t qi ) + ( ∂L ∂Dα t D α t qi )] η(t)dt This results in the fractional Euler-Lagrange equation. ∂L ∂qi −Dα t ( ∂L ∂Dα t qi ) + ( ∂L ∂Dα t D α t qi ) = 0 (1) By comparing Equations (A.3) and (A.4), we observe that the differential representation of the Hamiltonian requires identifying the partial derivatives of H with respect to qi , P1 , P2 ,and t. Using this comparison, we derive the Hamiltonian equations of motion as follows (see appendix A): Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 5 of 14 • First equation of motion: Since the term Dα s|tqid(P1) in equation (A.3) should correspond to the term ∂H ∂P1 dP1 in equation (A.4), we obtain : Dα s|tqi = ∂H ∂P1 , • Second equation of motion: Similarly, the term D2α s|tqid(P2) should match the term ∂H ∂P2 dP2 in Equation (A.4), leading to: D2α s|tqi = ∂H ∂P2 , • Third equation of motion: Moving to the term involving dqi in Equation (A.3), we observe that it should be equivalent to the term (Dα s|tP1+D2α s|tP2)dqi in Equation (4). Thus, we find: ∂H ∂qi = −Dα s|tP1 +D2α s|tP2, • Fourth equation of motion: The term ∂H ∂(t) represents the temporal change of the Hamiltonian, which is related to the temporal change of the term as : ∂H ∂t = − ∂l ∂t′ It should be highlighted that the results deduced from the above equations are closely aligned with what is observed in classical field theory, especially when dealing with integer orders within equations of motion. L = L(qi, D α t qi, D 2α t qi, D 3α t qi, · · · , Dnα t qi, t) Integrating the Lagrangian values with respect to time yields the action, which is a func- tional that describes the path followed in configuration space, represented as: S = ∫ L(qi, D α t qi, D 2α t qi, D 3α t qi, · · · , Dnα t qi, t)dt, The evolution of the classical system is determined by solving the Euler–Lagrange equa- tions of motion, which are derived as follows : n∑ i=0 (−1)iDiα t ( dL dqi ) = 0 qi = Diα t q i = 1, 2, 3, · · · The Hamiltonian formulation is defined as follows: n∑ i=1 piqi − L, qi = Diα t q, i = 1, 2, 3, · · · When calculating the total differential of the Hamiltonian, we obtain the following result. dH = n∑ i=1 pid(D iα t qi)+(Diα t qi)dpi− ∂L ∂(Diα t qi) d(Diα t qi)− ∂L ∂qi , qi = Diα t q, i = 1, 2, 3, · · · (2) Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 6 of 14 To provide an alternative perspective on the generalized momenta pi, which is linked to the expression qi = Diα t q, let’s examine the following formulation. We can uniquely present the generalized momenta by utilizing the equation qi = Diα t q : pi = ∂L ∂qi = ∂L ∂Diα t q (3) After substituting the momentum values into (3), we obtain at the following outcome : dH = n∑ i=1 Diα s|tqd(pi)− ∂L ∂q dq − ∂L ∂t dt, (4) By employing the Euler-Lagrange (1), we obtain at the following result: −∂L ∂q = n∑ i=1 (−1)iDiα s|t( ∂L ∂qi ) = n∑ i=1 (−1)i( ∂ipi ∂ti ) i = 1, 2, 3, · · · , n. (5) When we substitute (5) with (4), the outcome is as follows: dH = n∑ i=1 pid(D iα s|tqd(pi)) + (−1)i( ∂ipi ∂ti )dq −Diα s|tLdt. (6) This indicates that the Hamiltonian can be expressed in the following manner: H = H(q, Pi, t) i = 1, 2, 3, · · · , n. The total differential of this function is represented as follows: dH = ∂H ∂q ∂q + n∑ i=1 ∂H ∂qi d(pi) + ∂H ∂t dt, (7) When we compare (6) and (7), we can derive the following set of Hamilton’s equations of motion. ∂H ∂q = n∑ i=1 (−1)ipi, Diα s|t = ∂H ∂pi , To further establish the validity of our method for handling high-order derivatives, we will provide two examples of generalized conformable derivative concepts. These examples will serve to introduce and clarify the application of these concepts in supporting our approach. In this context, the path integral quantization is given as follows : K = ∫ dpdqei ∑n k=1(Pkq k−H(q,qk,t)dt For quadratic H(q, qk, t) in terms of Pk, and after integration over the momenta Pk we obtain : Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 7 of 14 K = ∫ dqei ∫ Ldt One should notice that the path integral quantization is obtained as an integration over the canonical coordinates qi without any need to integrate over higher order derivatives (qi, ql) as given by Ostrogradskii formulation. 4. Generalized Conformable Fractional Higher-Order Derivatives Classical mechanics describes the motion of particles and systems under forces, often described by a Lagrangian function. Recent interest in incorporating fractional derivatives has led to the development of fractional mechanics, using conformable derivatives. This work builds upon fractional mechanics and conformable derivatives to derive a generalized form of Euler-Lagrange equations for higher order conformable derivatives, establishing energy and momentum conservation in the presence of time independence and translational invariance of the Lagrangian [26]. Let’s begin by revisiting some fundamental concepts regarding theories involving particles in one dimensional motion. These theories are described by a Lagrangian function denoted as ”L” which depends on variables like position (x), velocity (Dα s|tx), and acceleration (D2α s|tx)) at a given time (t). δL− δt ∂L ∂t = δqi ∂L ∂qi + δDα t qi ∂L ∂Dα t qi + δD2α t qi ∂L ∂D2α t qi = δqi ( ∂L ∂qi − d dt ∂L ∂Dα t qi + d2 dt2 ∂L ∂D2α t qi ) + d dt ( δqi ∂L ∂Dα t qi + δDα t qi ∂L ∂D2α t qi − δqi d dt ∂L ∂D2α t qi ) . Therefore, the equation of motion is as follows: ∂L ∂qi −Dα t ∂L ∂Dα t qi +D2α t ∂L ∂D2α t qi = 0 (8) This aligns with the Euler-Lagrange outcome (refer to (1)). Through the utilization of fractional calculus in the preservation principles of classical mechanics, with a particu- lar emphasis on fractional energy and momentum, we establish what we term fractional mechanics. Within this framework, we reassess the equation of motion. Diα s|t ( L−Dα t qi ∂L ∂Dα t qi −D2α t qi ∂L ∂D2α t qi +Dα t qi d dt ∂L ∂Dα t qi ) = ∂L ∂t When the Lagrangian remains constant over time, indicating its time independence, it implies the existence of a conserved quantity known as energy. Consequently, the total Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 8 of 14 energy remains constant as time progresses. E = Dα s|tqi ∂L ∂Dα s|tqi +D2α s|tqi ∂L ∂D2α s|tqi −Dα s|tqi d dt ∂L ∂Dα s|tqi − L In the presence of translational invariance within the Lagrangian ∂L ∂qi = 0, we anticipate the conservation of momentum. Considering the equation for the equation of motion, we introduce the momentum px with the following expression: pqi = L ∂Dα s|tqi − d dt ∂L ∂D2α s|tqi This results in equation (11) being expressed as follows: D2α s|tpqi = ∂L ∂qi Therefore, when the Lagrangian exhibits translational invariance, the conservation of mo- mentum in the x-direction is expressed as px follows. Consequently, the energy can be defined as: E = Dα s|tqipqi +D2α s|tqi ∂L ∂D2α s|tqi − L Example 1. let us consider the Lagrangian [22] L = 1 2 ax(D2α t x)2 − 1 2 bx(Dα t x) 2 By applying the Euler-Lagrange equation 1 with respect to the independent variable qi, we obtain: 3 2 α(D2 s|tqi) 2 + bqiD 2 s|tqi + 2α(D s|t 1 qi)q (3) i + 1 2 b(D s|t 1 qi) 2 + aqiq (4) i = 0. (9) The momenta p1 and p2 are given as: p1 = ∂L ∂Dα s|tx = −bxDα t x. p2 = ∂L ∂D2α s|tx = axD2α s|tx. The Hamiltonian density can be written as: H = p1D α s|tq + p2D 2α s|tq − L = p21 2bx + p22 2ax (10) Additionally, the equations of motion according to Hamilton’s formalism are: Dα s|tx = −p1 bx . Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 9 of 14 D2α s|tx = p2 ax . H ∂x = −Dα s|tp1 +Dα s|tp2. Using the equation above, we obtain the following outcome: 3 2 a(D2α s|tqi) 2 + bqiD 2α s|tqi + 2a(Dα s|tqi)q (3) i + 1 2 b(Dα s|tqi) 2 + aqiq (4) i = 0 (11) The above equation is exactly the same as the equation that has been derived by Euler- Lagrange 9 in fractional form. For α = 1, we get: 3 2 a(D2 s|tqi) 2 + bqiD 2 s|tqi + 2a(D1 t qi)q (3) i + 1 2 b(D1 s|tqi) 2 + aqiq (4) i = 0 (12) Using the conformable derivative [29] and assuming D2 s|tqi = q̈l, D 1 s|t = q̇l 3 2 a(q̈l) 2 + bqiq̈l + 2a(q̇l)q (3) i + 1 2 b(q̇l) 2 + aqiq (4) i = 0 (13) It is worth noting that the results in Equation 13 are consistent with those found in Muslih et al [27]. Besides, the path integral is given by K = ∫ dqe i( 1 2 ax(D2α s|tx) 2− 1 2 bx(Dα s|tx) 2)dt (14) The path integral (14), is an integration over the canonical coordinate x, without any need to any integration over the velocity x = Dα s|tx as given by Ostrogradskii formulation. 5. Higher-Order Calculus: Applications and Recommendations for Future Work Understanding conformable calculus is crucial due to its high potential value, espe- cially in higher order derivatives, Hamiltonian systems, nonlinear motion, and differential equations. The Hamiltonian formalism for fractional differential equations is essential for comprehending the dynamics of intricate physical systems, particularly at the nanoscale. This approach enables the exploration of fractional derivatives of point particles, offering valuable insights into the movements of nanoscale materials and particles. These practical implications highlight the importance of derivatives in understanding complex systems in engineering and applied physics. This method helps researchers delve deeper into the stability, oscillation, and periodicity of systems exhibiting memory effects or long-range interactions. By incorporating the Hamiltonian method into the study of conformable derivatives, we can achieve a more comprehensive understanding of how systems respond to external influences and disturbances. Further research on conformable derivatives and generalized point particles is vital for advancing our understanding of these complex sys- tems. The insights gained from such studies have the potential to revolutionize fields like nanotechnology, biomechanics, and advanced engineering, offering innovative solu- tions and a deeper comprehension of the fundamental principles governing these intricate phenomena. Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 10 of 14 6. Conclusion In this research, we investigated two systems using the conformable version of the calculus of variations to derive the fractional Euler-Lagrange equation. It applies this calculus to the conformable version of classical mechanics, introducing the conformable Lagrangian and deriving the equation of motion. The path integral quantization was obtained directly as an integration over the canonical coordinate qi without the need to integrate over the variable Dα t qi. The classical equations of motion obtained in this work perfectly matched those obtained through the Lagrangian formulation. The Hamiltonian Method for Generalized Conformable Differential Equations with Higher-Order Derivatives offers a structured framework for studying nonlinear dynamics and invariance principles in complex systems involving controlled Lagrangians with higher-order derivatives. This research enhances theoretical applications in this domain, demonstrating invariance results concerning state variables and exploring a natural Hamiltonian formulation for composite higher derivative theories involving time derivatives. 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[31] G M Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Physics Reports, 371:461–580., 2002. Appendix In this appendix, we shift from the Lagrangian approach to the Hamiltonian approach, we introduce paired generalized momenta (pi and Φ) associated with and associated in conjunction with their respective generalized coordinates. pi = Dα s|t ∂L ∂Dα s|tqi +D2α s|t ∂L ∂(D2α s|tqi) πi = ∂L ∂(Dα s|tqi) We expand the phase space to include the canonical variables (qi, pi) and their associ- ated counterparts (ql, πi), where (ql = Dα s|t). Within this augmented phase space, the Hamiltonian is expressed as follows: H = piqi + πiD 2α s|tqi − L This implies that the Hamiltonian can be expressed as a function, as follows: H = H(qi, pi, qi, πi, t) In the generalized case: L = L(qi, D α s|tqi, D 2α s|tqi, · · · , D nα s|t qi, t) Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 13 of 14 Considering the variation principle, it is evident that the Euler–Lagrange equations gov- erning the motion of the system are expressed as follows : n∑ i=0 (−1)iDiα s|tqi( dL dqi ) = 0 qi = Diα s|tq i i = 0, 1, 2, · · · To streamline the representation of these equations through Ostrogradskii’s method and the introduction of paired generalized momenta (pi, πi) linked to the generalized coordi- nates (qi, D α s|tqi) the equations can be reformulated as follows : pk−1 i = ∂L ∂(Dα xλ Dα s|xλ qi) − d dt( ∂L ∂(∂Dqk+1 i ) ), k = 0, 1, 2, · · · , n− 1, πi = dL dqn . The phase space, defined by the canonical variables (qi and pk−1 i )and their associated counterparts (qi k and πi) encompasses the generalized coordinates (qi k and qk+1 i ). The formulation of the Hamiltonian is as follows : H = n−1∑ k=1 pkq k + πiq n i − L This implies that the Hamiltonian can be represented as follows: H = H(qi, p k−1 i , qk−1 i , πi, t), Now, the Hamiltonian formalism, denoted as H, is obtained by employing the Legendre transformation in the following manner: H = P1D α s|tqi + P1D 2α s|tqi − L, Determining the total differential of this Hamiltonian leads us to the following outcome. dH = P1d(D α s|tqi) + Dα s|tqid(P1) + P2d(D α s|tD α s|tqi) + D2α s|tqid(P2) − ∂L ∂qi − ∂L ∂Dα s|tqi Dα s|tqi − ∂L ∂D2α s|tqi D2α s|tqi, (A.2) The generalized momenta P1 and P2 corresponding to Dα s|tqi and D2α s|tqi can be defined as follows: P1 = ∂L ∂(Dα s|tqi) , P2 = ∂L ∂(D2α s|tqi) , Substituting the values of momenta P1 and P2 into (2), we get : dH = P1d(D α s|tqi) +Dα s|tqid(P1) + P2d(D α s|tD α s|tqi) +D2α s|tqid(P2)− ∂L ∂qi , By applying (1) of the Euler-Lagrange equation, we arrive at the following conclusion: dH = P1d(D α s|tqi)+Dα s|tqid(P1)+P2d(D 2α s|tqi)+D2α s|tqid(P2)+ ∂L ∂t −(Dα s|tP1−D2α s|tP2)dqi, (A.3) This indicates that the Hamiltonian can be represented in the following as follows: H = H(qi, P1, P2, t) Y. M. Alawaideh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5478 14 of 14 The total differential for this function can be expressed as: dH = ∂H ∂qi dqi + ∂H ∂P1 dP1 + ∂H ∂P2 dP2 + ∂H ∂t dt (A.4)