# American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

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## Journal of Geometric Mechanics

September 2022 , Volume 14 , Issue 3

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2022, 14(3): 381-408 doi: 10.3934/jgm.2021031 +[Abstract](695) +[HTML](211) +[PDF](979.21KB)
Abstract:

We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.

2022, 14(3): 409-426 doi: 10.3934/jgm.2022006 +[Abstract](438) +[HTML](137) +[PDF](425.62KB)
Abstract:

We consider a smooth \begin{document}$2n$\end{document}-manifold \begin{document}$M$\end{document} endowed with a bi-Lagrangian structure \begin{document}$(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$\end{document}. That is, \begin{document}$\omega$\end{document} is a symplectic form and \begin{document}$(\mathcal{F}_{1},\mathcal{F}_{2})$\end{document} is a pair of transversal Lagrangian foliations on \begin{document}$(M, \omega)$\end{document}. Such a structure has an important geometric object called the Hess Connection. Among the many importance of Hess connections, they allow to classify affine bi-Lagrangian structures.

In this work, we show that a bi-Lagrangian structure on \begin{document}$M$\end{document} can be lifted as a bi-Lagrangian structure on its trivial bundle \begin{document}$M\times\mathbb{R}^n$\end{document}. Moreover, the lifting of an affine bi-Lagrangian structure is also affine. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on \begin{document}$M\times\mathbb{R}^n$\end{document}. This lifting can be lifted again on \begin{document}$\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$\end{document}, and coincides with the initial dynamic (in our sense) on \begin{document}$M\times\mathbb{R}^n$\end{document}. By replacing \begin{document}$M\times\mathbb{R}^{2n}$\end{document} with the tangent bundle \begin{document}$TM$\end{document} or cotangent bundle \begin{document}$T^{*}M$\end{document} of \begin{document}$M$\end{document}, results still hold when \begin{document}$M$\end{document} is parallelizable.

2022, 14(3): 427-446 doi: 10.3934/jgm.2022007 +[Abstract](470) +[HTML](103) +[PDF](390.58KB)
Abstract:

Consider a locally compact, second countable and connected subcartesian differential space with finite structural dimension. We prove that it admits embedding into a Euclidean space. The Whitney embedding theorem for smooth manifolds can be treated as a corollary of embedding for subcartesian differential space. As applications of our embedding theorem, we show that both smooth generalized distributions and smooth generalized subbundles of vector bundles on subcartesian spaces are globally finitely generated. We show that every algebra isomorphism between the associative algebras of all smooth functions on two subcartesian differential spaces is the pullback by a smooth diffeomorphism between these two spaces.

2022, 14(3): 447-471 doi: 10.3934/jgm.2022014 +[Abstract](424) +[HTML](67) +[PDF](1471.42KB)
Abstract:

In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby 'modified' equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.

2022, 14(3): 473-490 doi: 10.3934/jgm.2022015 +[Abstract](297) +[HTML](148) +[PDF](394.39KB)
Abstract:

In this paper, we present a new general system of equations describing the steady motion of atmosphere with uniform density in ellipsoidal coordinates, which is derived from the general governing equations for viscous fluids. We first show that this new system can be reduced to the classic Ekman equations. Secondly, we obtain the explicit solution of the Ekman equations in ellipsoidal coordinates. Thirdly, for the viscosity related to the height, we obtain the solution of the classical problem with zero acceleration at the bottom of Ekman layer. Finally, the uniqueness and dynamical properties of solution are demonstrated.

2021 Impact Factor: 0.737
5 Year Impact Factor: 0.713
2021 CiteScore: 1.3