Symplectic Superposition Solution of Free Vibration of Fully Clamped Orthotropic Rectangular Thin Plate on Two-Parameter Elastic Foundation

Based on the method of separation of variables in Hamiltonian system and superposition method, the series expansion solution of the free vibration problem of orthotropic rectangular thin plates (ORTPs) with four clamped edges (CCCC) on two-parameter elastic foundation is obtained. The original vibration problem is decomposed into two subproblems with two opposite sides simply supported, and the general solution of each subproblem is obtained by using the expansion of symplectic eigenvectors. Then by superposing these two general solutions, the series expansion solution of the original problem is obtained. The advantage of this method is that the solution process is carried out in symplectic space, and the validity of variable separation and symplectic eigenvectors expansion ensures the rationality of the solution process, while avoiding the presetting of the solution form. Finally, the correctness of symplectic superposition solution obtained in this paper is verified by calculating three concrete examples of fully clamped rectangular thin plates.

Gravitational Constraints on a Lightlike Boundary

We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the first- and second-class ones. In particular, in the 3+1-dimensional case, we show that the reduced phase space has two local degrees of freedom, instead of the usual four in the non-degenerate case.

Lagrangian Curve Flows on Symplectic Spaces

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and the span of γ,γx,…,γx(n−1) is a Lagrangian subspace of R2n. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R2n with respect to the symplectic group Sp(2n), (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s C^n(1)-KdV flows and A^2n−1(2)-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

Tulczyjew’s triplet for lie groups III: Higher order dynamics and reductions for iterated bundles

Given a Lie group G, we elaborate the dynamics on T+T+G and T+TG, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space TT+G, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.

Calculating the Dimension of the Universal Embedding of the Symplectic Dual Polar Space using Languages

The main result of this paper is the construction of a bijection of the set of words in so-called standard order of length $n$ formed by four different letters and the set $\mathcal{N}^n$ of all subspaces of a fixed $n$-dimensional maximal isotropic subspace of the $2n$-dimensional symplectic space $V$ over $\mathbb{F}_2$ which are not maximal in a certain sense. Since the number of different words in standard order is known, this gives an alternative proof for the formula of the dimension of the universal embedding of a symplectic dual polar space $\mathcal{G}_n$. Along the way, we give formulas for the number of all $n$- and $(n-1)$-dimensional totally isotropic subspaces of $V$.

Symplectic Geometry and Its Applications on Time Series Analysis

This chapter serves to introduce the symplectic geometry theory in time series analysis and its applications in various fields. The basic concepts and basic elements of mathematics relevant to the symplectic geometry are introduced in the second section. It includes the symplectic space, symplectic transformation, Hamiltonian matrix, symplectic principal component analysis (SPCA), symplectic geometry spectrum analysis (SGSA), symplectic geometry mode decomposition (SGMD), and symplectic entropy (SymEn), etc. In addition, it also briefly reviews the applications of symplectic geometry on time series analysis, such as the embedding dimension estimation, nonlinear testing, noise reduction, as well as fault diagnosis. Readers who are familiar with the mathematical preliminaries may omit the second section, i.e. the theory part, and go directly to the third section, i.e. the application part.

Joint Invariants of Linear Symplectic Actions

We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants.