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Integrability and Dynamics of the n-Dimensional Symmetric Veselova Top

Abstract

We consider the n-dimensional generalization of the nonholonomic Veselova problem. We derive the reduced equations of motion in terms of the mass tensor of the body and determine some general properties of the dynamics. In particular, we give a closed formula for the invariant measure, indicate the existence of steady rotation solutions, and obtain some results on their stability. We then focus our attention on bodies whose mass tensor has a specific type of symmetry. We show that the phase space is foliated by invariant tori that carry quasiperiodic dynamics in the natural time variable. Our results enlarge the known cases of integrability of the multi-dimensional Veselova top. Moreover, they show that in some previously known instances of integrability, the flow is quasiperiodic without the need of a time reparametrization.

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Notes

  1. 1.

    Readers may be more familiar with using the connected component \(\text {SO}(n)\) as the configuration space, however allowing both components simplifies our exposition in Sect. 5; moreover, Arnold (1988, p. 133) suggests that \(\mathrm{O}(n)\) is the ‘correct’ configuration space.

  2. 2.

    For \(a, b\in \mathbb {R}^n\), we denote \(a\wedge b =ab^T-ba^T \in \mathfrak {so}(n)\).

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Acknowledgements

LGN and JM are grateful to the hospitality of the Department of Mathematics Tullio Levi-Civita of the University of Padova, during its 2018 intensive period “Hamiltonian Systems”. LGN acknowledges the Alexander Von Humboldt Foundation for a Georg Forster Research Fellowship that funded a visit to TU Berlin where the last part of this work was completed. The authors are thankful to Božidar Jovanović for comments on an early draft of this paper and for sharing the recent preprint Gajić and Jovanović (2018) with us.

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Correspondence to Luis C. García-Naranjo.

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This research was made possible by a Newton Advanced Fellowship from the Royal Society, No. NA140017.

Communicated by Anthony Bloch.

Appendices

A Field’s Theorem on Reconstruction

We briefly recall the language and main properties of (compact) group actions and equivariant vector fields, needed for the proofs of quasiperiodicity.

Suppose a (compact Lie) group G acts smoothly on a manifold M. If two points lie in the same orbit, their isotropy subgroups are conjugate. This motivates the partition or stratification of M into “orbit types,” where the orbit type stratum \(M_{(H)}\) consists of those points with isotropy subgroup (ISG) conjugate to H, or more precisely a stratum is a connected component of such points. Moreover, if \(M_H\) is (a connected component of) the set of points with ISG equal to H, then, up to connected components, \(M_{(H)} = G.M_H\) (the image of \(M_H\) under the group action). The orbit space M / G is similarly stratified by the orbit type; one denotes its corresponding stratum by \((M/G)_H\). All of these spaces \(M_{(H)}, M_H\) and \((M/G)_H\) are manifolds.

If in addition there is an equivariant dynamical system on M, then it descends to a dynamical system on M / G which respects each stratum. The question of reconstruction is, if one knows some properties of the dynamics on M / G what does this imply about the dynamics on M. We use the following simplified version of an important theorem due to Field (1980) (see also Field 2007, Chapter 8 or Cushman et al. (2010)).

Theorem A.1

(Field 1980) Consider a smooth equivariant dynamical system, with compact symmetry Lie group G acting freely on the phase space M. The dynamics passes down to the smooth orbit space M / G.

  1. (i)

    Let \(x\in M/G\) be an equilibrium point of the reduced equations. Then, the inverse image of x in M is a group orbit which is foliated by invariant tori of dimension at most the rank of G.

  2. (ii)

    Let \(\gamma \) be a periodic orbit of the reduced dynamics in M / G. Then, the inverse image of this curve is also foliated by invariant tori, but now of dimension at most \(\mathrm {rk}(G)+1\).

In both cases, the dynamics in the invariant tori is quasiperiodic.

In case (i) the dynamics on M is called a relative equilibrium, while in (ii) it is a relative periodic orbit. If the dynamics on M / G include an invariant quasiperiodic torus, then it is unknown except in special cases, see Fassò et al. (2015), what the corresponding reconstructed dynamics may be. The more general version of Field’s theorem does not require the action to be free, but this suffices for our purposes.

B Physical Inertia Tensors Satisfying the Hypothesis of Fedorov–Jovanović

Recall that Fedorov and Jovanović (2004, 2009) work under the assumption that there is an \(n\times n\) diagonal matrix A such that the inertia tensor \(\mathbb {I}:\mathfrak {so}(n)\rightarrow \mathfrak {so}(n)\) satisfies

$$\begin{aligned} \mathbb {I}(a\wedge b) = (Aa)\wedge (Ab),\quad (\forall a,b\in \mathbb {R}^n). \end{aligned}$$
(B.1)

In this appendix, we examine the feasibility of this condition within the family of physical inertia tensors. The following proposition, which was already suggested in Fedorov and Jovanović (2009), Jovanović (2010), shows that for \(n\ge 4\) a physical inertia tensor that satisfies (B.1) necessarily corresponds to an axisymmetric rigid body.

Proposition B.1

Let \(\mathbb {I}\) be a physical inertia tensor defined by \(\mathbb {I}(\Omega )=\mathbb {J}\Omega +\Omega \mathbb {J}\) (see (1.1)).

  1. (i)

    If \(n=3\) then the identity (B.1) holds for arbitrary \(a, b\in \mathbb {R}^3\) with

    $$\begin{aligned} A_i=\sqrt{\frac{(J_i+J_j)(J_i+J_k)}{J_j+J_k}}, \end{aligned}$$

    for \(\{i,j,k\}=\{1,2,3\}\), where \(\mathbb {J}=\mathop {\mathrm{diag}}\nolimits [J_1,J_2,J_3]\) (with \(J_i+J_j>0\) for \(i\ne j\)).

  2. (i)

    If \(n\ge 4\), there exists a diagonal matrix A satisfying (B.1) if and only if \(\mathbb {I}\) is the inertia tensor of an axisymmetric body. In this case, the body frame can be chosen in such way that \(\mathbb {J}\) is given by

    $$\begin{aligned} \mathbb {J}=\text{ diag }\left[ A_1A_2-\tfrac{1}{2} A_2^2,\tfrac{1}{2}A_2^2,\dots , \tfrac{1}{2} A_2^2\right] \end{aligned}$$
    (B.2)

    and \(A=\mathop {\mathrm{diag}}\nolimits [A_1, A_2, \dots , A_2]\), where \(A_1,A_2\in \mathbb {R}\) satisfy \(A_1\ge A_2/2>0\).

Proof

(i) is a calculation.

(ii) “Only if”: by selecting the body frame with \(f_1\) parallel to the symmetry axis of the body we have \(\mathbb {J}=(J_1,J_2,\dots ,J_2)\) with \(J_1\ge 0\) and \(J_2>0\). The statement follows by putting \(A_1=(J_1+J_2)/\sqrt{2J_2}\) and \(A_2=\sqrt{2J_2}\)

“If”: suppose that (2.3) holds and \(\mathbb {I}\) is a physical inertia tensor with corresponding diagonal matrix \(\mathbb {J}\). Then, there are real numbers \(A_1,\dots , A_n\) such that for each \(i\ne j\),

$$\begin{aligned} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad J_i+J_j=A_iA_j.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (E_{ij}) \end{aligned}$$

We claim that if the \(J_i\) satisfy these conditions, then at least \(n-1\) of the \(A_i\) coincide. In that case suppose (w.l.o.g.) \(A_2=\cdots =A_n\). Then, comparing the equations \(E_{1i}\) shows that all \(J_1+J_i\) coincide for \(i>1\), and hence \(J_2=\cdots =J_n\) as required.

To prove the claim, note that for any collection \(\{i,j,k,\ell \}\) of 4 distinct indices,

$$\begin{aligned} E_{ij}+E_{ik}-E_{jk} \text { becomes } 2J_i = A_iA_j+A_iA_k-A_jA_k, \end{aligned}$$

and similarly

$$\begin{aligned} E_{ij}+E_{i\ell }-E_{j\ell } \text { becomes } 2J_i = A_iA_j+A_iA_\ell -A_jA_\ell . \end{aligned}$$

Subtracting these shows

$$\begin{aligned} (A_i-A_j)(A_k-A_\ell )=0. \end{aligned}$$

Suppose \(J_1\ne J_2\) and apply this to the indices 1, 2, ij with \(i,j>2\). It follows that \(A_3=A_4=\dots = A_n\). Finally, if \(A_1=A_3\), we are done, while if \(A_1\ne A_3\), apply the previous reasoning to the indices 1, 3, 2, 4 and one can conclude that \(A_2=A_4\), and hence indeed \(A_2=\cdots =A_n\). \(\square \)

The usefulness of a condition of type (B.1) for the developments in Fedorov and Jovanović (2004, 2009) (also Jovanović 2009, 2010) seems to arise from the fact that an inertia tensor satisfying (B.1) maps the set of rank 2 matrices in \(\mathfrak {so}(n)\) into itself. For interest, although we make no use of this, we show that for \(n\ge 4\), the only physical inertia tensors with this property are axisymmetric.

Proposition B.2

Let \(\mathbb {I}\) be a physical inertia tensor defined by (1.1). If \(n\ge 4\), then \(\mathbb {I}\) maps the set of rank two matrices in \(\mathfrak {so}(n)\) into itself if and only if the body is axisymmetric.

Proof

Let us consider the case \(n=4\). Suppose \(\mathbb {I}\) maps the space of rank two matrices in \(\mathfrak {so}(4)\) into itself and that the body is not axisymmetric. Choose the body frame so that \(\mathbb {J}=\mathop {\mathrm{diag}}\nolimits [J_1, \dots , J_4]\) with \(J_1\ne J_3\), \(J_2\ne J_4\). A direct calculation shows that

$$\begin{aligned} \det (\mathbb {I}((f_1+f_3)\wedge (f_2+f_4)))=(J_1-J_3)^2(J_2-J_4)^2\ne 0, \end{aligned}$$

which shows \(\mathbb {I}((f_1+f_3)\wedge (f_2+f_4))\) has rank 4, reaching a contradiction. A similar argument shows this implication for \(n>4\).

The converse statement follows from part (ii) of the Proposition  B.1. \(\square \)

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Fassò, F., García-Naranjo, L.C. & Montaldi, J. Integrability and Dynamics of the n-Dimensional Symmetric Veselova Top. J Nonlinear Sci 29, 1205–1246 (2019). https://doi.org/10.1007/s00332-018-9515-5

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Keywords

  • Nonholonomic dynamics
  • Integrability
  • Quasi-periodicity
  • Symmetry
  • Singular reduction

Mathematics Subject Classification

  • 37J60
  • 70E17
  • 70E40
  • 58D19
  • 34A30