Search Results (18)

This paper presents an adaptive optimal consensus tracking control scheme for canonical nonlinear multi-agent systems (MASs) with unknown dynamics, employing an actor–critic reinforcement learning (RL) framework. The scheme integrates a sliding mode mechanism to suppress tracking errors and ensure consensus tracking between the followers and the leader. Additionally, optimal control is designed to find a Nash equilibrium in a graphical game. To address the intractability of obtaining an analytical solution for the coupled Hamilton–Jacobi–Bellman (HJB) equation, a policy iteration algorithm is utilized. Within this algorithm, a critic neural network (NN) approximates the gradient of the optimal value function, while an actor NN approximates the optimal control policy. Together, these networks form a compact actor–critic (AC) architecture that achieves optimal consensus tracking. Furthermore, the proposed method guarantees the boundedness of all closed-loop signals while ensuring consensus tracking. Finally, two simulations are conducted to verify the effectiveness and advantages of the proposed method. Full article

Deep coal mining encounters substantial challenges associated with diminished permeability and elevated gas pressure under multi-physics coupling conditions. This paper introduces an innovative methodology that integrates the staged high-pressure hydraulic fracturing of roof key strata with the static expansion fracturing of coal seams using solid expansive materials. The proposed technique facilitates the coordinated regulation of the stress and fracture fields, enabling efficient stress relief, long-term anti-reflection performance, and low-damage seam modification. By employing theoretical modeling based on Hamilton’s canonical equations, conducting experimental impact tests, and performing field validation, this research demonstrates that the synergistic strategy markedly alleviates stress concentration by more than 25%, increases permeability by a factor of 3.7, and enhances gas extraction efficiency while preserving fracture network stability. The results indicate sustainable gas drainage characterized by reduced decay coefficients, offering a reliable and efficient approach for strata pressure management and gas disaster mitigation in deep mining operations. This integrated solution contributes to both safety enhancement and resource recovery optimization under complex geo-mechanical conditions. Full article

Based on noncommutative relations and the Dirac canonical dequantization scheme, I generalize the canonical Poisson bracket to a deformed Poisson bracket and develop a non-canonical formulation of the Poisson, Hamilton, and Lagrange equations in the deformed Poisson and symplectic spaces. I find that both of these dynamical equations are the coupling systems of differential equations. The noncommutivity induces the velocity-dependent potential. These formulations give the Noether and Virial theorems in the deformed symplectic space. I find that the Lagrangian invariance and its corresponding conserved quantity depend on the deformed parameters and some points in the configuration space for a continuous infinitesimal coordinate transformation. These formulations provide a non-canonical framework of classical mechanics not only for insight into noncommutative quantum mechanics, but also for exploring some mysteries and phenomena beyond those in the canonical symplectic space. Full article

The covariant Hamilton–Jacobi formulation of electrodynamics is systematically derived from the first-order (Palatini-like) Lagrangian. This derivation utilizes the De Donder–Weyl covariant Hamiltonian formalism with constraints incroporating generalized Dirac brackets of forms and the associated polysymplectic reduction, which ensure manifest covariance and consistency with the field dynamics. It is also demonstrated that the canonical Hamilton–Jacobi equation in variational derivatives and the Gauss law constraint are derived from the covariant De Donder–Weyl Hamilton–Jacobi formulation after space + time decomposition. Full article

Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions, the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii. Full article

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton’s equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton’s equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton’s and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton’s, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton’s equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes. Full article

Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schrödinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used. Full article

It is shown that the complex Bernoulli differential equations admitting the supplementary structure of a Lie–Hamilton system related to the book algebra ${\mathfrak{b}}_2$ can always be solved by quadratures, providing an explicit solution of the equations. In addition, considering the quantum deformation of Bernoulli equations, their canonical form is obtained and an exact solution by quadratures is deduced as well. It is further shown that the approximations of $k^{th}$-order in the deformation parameter from the quantum deformation are also integrable by quadratures, although an explicit solution cannot be obtained in general. Finally, the multidimensional quantum deformation of the book Lie–Hamilton systems is studied, showing that, in contrast to the multidimensional analogue of the undeformed system, the resulting system is coupled in a nontrivial form. Full article

This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels. Full article

Because of the nonlocal and nonsingular properties of fractional derivatives, they are more suitable for modelling complex processes than integer derivatives. In this paper, we use a fractional factor to investigate the fractional Hamilton’s canonical equations and fractional Poisson theorem of mechanical systems. Firstly, a fractional derivative and fractional integral with a fractional factor are presented, and a multivariable differential calculus with fractional factor is given. Secondly, the Hamilton’s canonical equations with fractional derivative are obtained under this new definition. Furthermore, the fractional Poisson theorem with fractional factor is presented based on the Hamilton’s canonical equations. Finally, two examples are given to show the application of the results. Full article

In this paper we aim at presenting a concise but also comprehensive study of time-dependent (t-dependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations in order to formulate an explicitly time-dependent geometric Hamilton-Jacobi theory on lcs manifolds, extending our previous work with no explicit time-dependence. In contrast to previous papers concerning locally conformal symplectic manifolds, the introduction of the time dependency that this paper presents, brings out interesting geometric properties, as it is the case of contact geometry in locally symplectic patches. To conclude, we show examples of the applications of our formalism, in particular, we present systems of differential equations with time-dependent parameters, which admit different physical interpretations as we shall point out. Full article

This paper summarises a new framework of Stochastic Geometric Mechanics that attributes a fundamental role to Hamilton–Jacobi–Bellman (HJB) equations. These are associated with geometric versions of probabilistic Lagrangian and Hamiltonian mechanics. Our method uses tools of the “second-order differential geometry”, due to L. Schwartz and P.-A. Meyer, which may be interpreted as a probabilistic counterpart of the canonical quantization procedure for geometric structures of classical mechanics. The inspiration for our results comes from what is called “Schrödinger’s problem” in Stochastic Optimal Transport theory, as well as from the hydrodynamical interpretation of quantum mechanics. Our general framework, however, should also be relevant in Machine Learning and other fields where HJB equations play a key role. Full article

The equation for canonical gravity produced by Wheeler and DeWitt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton–Jacobi–Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Hamilton’s principal function to be explicitly dependent on this time locally. This choice results in a Wheeler–DeWitt equation with time. A working solution for the de Sitter minisuperspace is shown. Full article

The symplectic method for a thin piezoelectric plate problem is developed. The Hamiltonian canonical equation of thin piezoelectric plate is given by using the variational principle. By applying the separation of variables method, we can obtain symplectic orthogonal eigensolutions. As an application, the problem of a thin piezoelectric plate with full edges simply supported under a uniformly distributed load is discussed, and analytical solutions of the deflection and potential of a piezoelectric thin plate are obtained. A numerical example shows that the solutions converge very rapidly. The advantage of this method is that it does not need to assume the predetermined function in advance, so it has better universality. It may also be applied to the problem of thin piezoelectric plate buckling and vibrating. Full article

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset $U\subseteq M$, such that $\pi \left(U\right)$ has a manifold structure and ${\pi }_{\left|U\right.}:U\to \pi \left(U\right)$, the restriction to U of the canonical projection $\pi :M\to M/G$, is a surjective submersion. If $X_{\left|U\right.}$ is not vertical with respect to ${\pi }_{\left|U\right.}$, we show that such complete solutions solve the reconstruction equations related to $X_{\left|U\right.}$ and G, i.e., the equations that enable us to write the integral curves of $X_{\left|U\right.}$ in terms of those of its projection on $\pi \left(U\right)$. On the other hand, if $X_{\left|U\right.}$ is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of $X_{\left|U\right.}$ up to quadratures. To do that, we give, for some elements $\xi$ of the Lie algebra $\mathfrak{g}$ of G, an explicit expression up to quadratures of the exponential curve $exp\left(\xi t\right)$, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of $exp\left(\xi t\right)$ is valid for all $\xi$ inside an open dense subset of $\mathfrak{g}$. Full article