ISSN:
1941-4889
eISSN:
1941-4897
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Journal of Geometric Mechanics
December 2012 , Volume 4 , Issue 4
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2012, 4(4): 365-383
doi: 10.3934/jgm.2012.4.365
+[Abstract](3717)
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Abstract:
In continuation of [7] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i\big(Vol(f)\big)\ \bar{g}\big((P_i)_fh,k\big) vol(f^*\bar{g}) $$ on the space of immersions $Imm(M,N)$ and on shape space $B_i(M,N)=Imm(M,N)/{Diff}(M)$. Here $(N,\bar{g})$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\bar{g}$ is the induced Riemannian metric on $M$, $vol(f^*\bar{g})$ is the induced volume density on $M$, $Vol(f)=\int_M vol(f^*\bar{g})$, $\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $\Delta^{f^*\bar{g}}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Phi_i(Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.
In continuation of [7] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i\big(Vol(f)\big)\ \bar{g}\big((P_i)_fh,k\big) vol(f^*\bar{g}) $$ on the space of immersions $Imm(M,N)$ and on shape space $B_i(M,N)=Imm(M,N)/{Diff}(M)$. Here $(N,\bar{g})$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\bar{g}$ is the induced Riemannian metric on $M$, $vol(f^*\bar{g})$ is the induced volume density on $M$, $Vol(f)=\int_M vol(f^*\bar{g})$, $\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $\Delta^{f^*\bar{g}}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Phi_i(Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.
2012, 4(4): 385-395
doi: 10.3934/jgm.2012.4.385
+[Abstract](1836)
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Abstract:
We study a special class of endomorphism fields of the generalized tangent bundle ${\mathcal{T}}M:=TM\oplus T^*M$ of a smooth manifold $M$. An operator of this class is defined as follows: it has a vanishing Courant-Nijenhuis torsion and is diagonalizable (after a possible extension of scalars) with constant dimensions of its eigenspaces. Such an endomorphism field is called a semi-simple generalized Nijenhuis operator. The generalized paracomplex and complex structures give examples of such operators.
In this study, we distinguish two cases according to whether the operator has exactly two eigenvalues or has at least three elements in its spectrum. In the first case, we prove that either the operator is affinely related to a generalized complex structure, or it is equivalent to a pair of transverse Dirac structures on ${\mathcal{T}}M$. In the second case, the semi-simple generalized Nijenhuis operator is conjugate to a special kind of generalized Nijenhuis operator obtained from usual Nijenhuis tensors.
We study a special class of endomorphism fields of the generalized tangent bundle ${\mathcal{T}}M:=TM\oplus T^*M$ of a smooth manifold $M$. An operator of this class is defined as follows: it has a vanishing Courant-Nijenhuis torsion and is diagonalizable (after a possible extension of scalars) with constant dimensions of its eigenspaces. Such an endomorphism field is called a semi-simple generalized Nijenhuis operator. The generalized paracomplex and complex structures give examples of such operators.
In this study, we distinguish two cases according to whether the operator has exactly two eigenvalues or has at least three elements in its spectrum. In the first case, we prove that either the operator is affinely related to a generalized complex structure, or it is equivalent to a pair of transverse Dirac structures on ${\mathcal{T}}M$. In the second case, the semi-simple generalized Nijenhuis operator is conjugate to a special kind of generalized Nijenhuis operator obtained from usual Nijenhuis tensors.
2012, 4(4): 397-419
doi: 10.3934/jgm.2012.4.397
+[Abstract](2795)
+[PDF](484.7KB)
Abstract:
It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree $> 1$, which is locally homogeneous of degree $k$ with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.
It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree $> 1$, which is locally homogeneous of degree $k$ with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.
2012, 4(4): 421-442
doi: 10.3934/jgm.2012.4.421
+[Abstract](2924)
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Abstract:
This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\mathcal{T}^E E^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.
This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\mathcal{T}^E E^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.
2012, 4(4): 443-467
doi: 10.3934/jgm.2012.4.443
+[Abstract](2629)
+[PDF](339.7KB)
Abstract:
We study the Hess--Appelrot case of the Euler--Poisson system which describes dynamics of a rigid body about a fixed point. We prove existence of an invariant torus which supports hyperbolic or parabolic or elliptic periodic or elliptic quasi--periodic dynamics. In the elliptic cases we study the question of normal hyperbolicity of the invariant torus in the case when the torus is close to a `critical circle'. It turns out that the normal hyperbolicity takes place only in the case of $1:q$ resonance. In the sequent paper [16] we study limit cycles which appear after perturbation of the above situation.
We study the Hess--Appelrot case of the Euler--Poisson system which describes dynamics of a rigid body about a fixed point. We prove existence of an invariant torus which supports hyperbolic or parabolic or elliptic periodic or elliptic quasi--periodic dynamics. In the elliptic cases we study the question of normal hyperbolicity of the invariant torus in the case when the torus is close to a `critical circle'. It turns out that the normal hyperbolicity takes place only in the case of $1:q$ resonance. In the sequent paper [16] we study limit cycles which appear after perturbation of the above situation.
2012, 4(4): 469-485
doi: 10.3934/jgm.2012.4.469
+[Abstract](2478)
+[PDF](634.2KB)
Abstract:
It is well-known that a Lie algebroid $A$ is equivalently described by a degree 1 NQ-manifold $\mathcal{M}$. We study distributions on $\mathcal{M}$, giving a characterization in terms of $A$. We show that involutive $Q$-invariant distributions on $\mathcal{M}$ correspond bijectively to IM-foliations on $A$ (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on $\mathcal{M}$.
It is well-known that a Lie algebroid $A$ is equivalently described by a degree 1 NQ-manifold $\mathcal{M}$. We study distributions on $\mathcal{M}$, giving a characterization in terms of $A$. We show that involutive $Q$-invariant distributions on $\mathcal{M}$ correspond bijectively to IM-foliations on $A$ (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on $\mathcal{M}$.
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