Search Results (9)

This study evaluates structure-preserving neural network architectures for learning holonomically constrained mechanical dynamics in Cartesian coordinates. In contrast to methods using reduced coordinates, the full ambient phase space ${\mathbb{R}}^{2n}$ is retained with explicit algebraic constraints $C_i\left(\mathbf{q}\right)=0$ to provide a test bed for constraint-aware learning. The Constraint-Aware Hamiltonian Neural Network (CA-HNN) is proposed, which augments the standard HNN with a dedicated multiplier network ${\lambda }_{\varphi }(\mathbf{q},\mathbf{p})$ for Lagrange multipliers and a composite loss function evaluated on predicted rollouts. The theoretical framework is grounded in the geometry of constrained Hamiltonian systems: the extended phase space ${\mathbb{R}}^{2n+m}$ carries a degenerate antisymmetric structure where an m-dimensional kernel encodes constraint directions, while the symplectic structure emerges on the $2(n-m)$-dimensional reduced manifold $\Sigma$. It is proven that the physical Hamiltonian is conserved on the constraint surface under augmented flow. Benchmarks on a pendulum ($C=x^2+y^2-l^2$), double pendulum, and bead on a parabola ($C=y-x^2$) demonstrate that CA-HNN reduces constraint violations $\parallel C\left(\mathbf{q}\right)\parallel$ by $5\times$ to $2400\times$ compared to standard HNNs. While the best energy conservation is achieved by PINNs, these findings clarify the roles of architectural inductive bias, constraint augmentation, and soft physics regularization. Full article

Under the dual carbon objectives, wind power penetration has accelerated markedly. However, the inherent volatility and insufficient peak regulation capability in energy storage allocation hamper efficient grid integration. To address these challenges, this paper presents a hybrid storage capacity configuration method that combines Symplectic Geometry Mode Decomposition (SGMD) with Particle Swarm Optimization (PSO). SGMD provides fine-grained, multi-scale decomposition of load–power curves to reduce modal aliasing, while PSO determines globally optimal ESS capacities under peak-shaving constraints. Case-study simulations showed a 25.86% reduction in the storage investment cost compared to EMD-based baselines, maintenance of the state of charge (SOC) within 0.3–0.6, and significantly enhanced overall energy management efficiency. The proposed framework thus offers a cost-effective and robust solution for energy storage at renewable energy plants. Full article

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of the reduction of a Hamiltonian system with known integrals of motion. Full article

We introduce QP manifolds that capture the generalised geometry of type IIA string backgrounds with Ramond–Ramond fluxes and Romans mass. Each of these is associated with a BPS brane in type IIA: a D2, D4, or NS5-brane. We explain how these probe branes are related to their associated QP-manifolds via the AKSZ topological field theory construction and the recent brane phase space construction. M-theory/type IIA duality is realised on the QP-manifold side as symplectic reduction along the M-theory circle (for branes that do not wrap it); this always produces IIA QP-manifolds with vanishing Romans mass. Full article

Let $(X,G,{\omega }_1,{\omega }_2,\left\{{\eta }^t\right\})$ be a manifold with a bi-Poisson structure $\left\{{\eta }^t\right\}$ generated by a pair of G-invariant symplectic structures ${\omega }_1$ and ${\omega }_2$, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds $(R_{L^i},G^i,{\omega }_1^i,{\omega }_2^i,\left\{{\eta }_i^t\right\})$, $i=1,2$ such that (1) $R_{L^i}$ is a submanifold of an open dense subset $X_{\left(H\right)}\subset X$; (2) symplectic structures ${\omega }_1^i$ and ${\omega }_2^i$, generating a bi-Poisson structure $\left\{{\eta }_i^t\right\}$, are $G^i$- invariant and coincide with restrictions ${\omega }_1{|}_{R_{L^i}}$ and ${\omega }_2{|}_{R_{L^i}}$; (3) the canonically defined group $G^i$ acts properly and locally freely on $R_{L^i}$; (4) orbit spaces $X_{\left(H\right)}/G$ and $R_{L^i}/G^i$ are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on $X_{\left(H\right)}$ and $G^i$-invariant functions on $R_{L^i}$ are isomorphic as Poisson algebras with the bi-Poisson structures $\left\{{\eta }^t\right\}$ and $\left\{{\eta }_i^t\right\}$ respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group. Full article

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented. Full article

The vibration signal induced by bearing local fault has strong nonstationary and nonlinear property, which indicates that the conventional methods are difficult to recognize bearing fault patterns effectively. Hence, to obtain an efficient diagnosis result, the paper proposes an intelligent fault diagnosis approach for rolling bearing integrated symplectic geometry mode decomposition (SGMD), improved multiscale symbolic dynamic entropy (IMSDE) and multiclass relevance vector machine (MRVM). Firstly, SGMD is employed to decompose the original bearing vibration signal into several symplectic geometry components (SGC), which is aimed at reconstructing the original bearing vibration signal and achieving the purpose of noise reduction. Secondly, the bat algorithm (BA)-based optimized IMSDE is presented to evaluate the complexity of reconstruction signal and extract bearing fault features, which can solve the problems of missing of partial fault information existing in the original multiscale symbolic dynamic entropy (MSDE). Finally, IMSDE-based bearing fault features are fed to MRVM for achieving the identification of bearing fault categories. The validity of the proposed method is verified by the experimental and contrastive analysis. The results show that our approach can precisely identify different fault patterns of rolling bearings. Moreover, our approach can achieve higher recognition accuracy than several existing methods involved in this paper. This study provides a new research idea for improvement of bearing fault identification. Full article

Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction. Full article

Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity. Full article