Skip to main content
SpringerLink
Log in

Geometry and Dynamics of Planar Linkages

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this article we present an approach to describing the geometry and curvature of the configuration spaces of a class of simple idealized planar linkages. This is based on determining the curvature of such configuration spaces canonically embedded into Euclidean space, and then the behaviour of the dynamics of such linkages can be understood via the associated geodesic flow. Our objective is to present a method which, in principle, can be applied to many different examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bangert V.: On the existence of closed geodesics on two-spheres. Internat. J. Math. 4(1), 1–10 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Confeld, I.P., Fomin, S.P., Sinai, Ya.G.: Ergodic Theory. Berlin: Springer, 1990

  3. Denker M., Philipp W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. and Dynam. Sys. 4(4), 541–552 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dolgopyat D.: On decay of correlations in Anosov flows. Ann. of Math. 147(2), 357–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Efimov N.V.: Impossibility a complete regular surface in Euclidean 3-space whose Gaussian curvature has a negative upper bound. Sov. Math. Dokl. 4(1), 843–846 (1963)

    MathSciNet  MATH  Google Scholar 

  6. Farber, M. Invitation to Topological Robotics. European Mathematical Society, Zurich, 2008

  7. Franks J.: Geodesics on S 2 and periodic points of annulus homeomorphisms. Invent. Math. 108(2), 403–418 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Hunt T.J., MacKay R.S.: Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity 16, 1499–1510 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Kachurovskii A.G.: Rates of convergence in ergodic theorems. Russ. Math. Surv. 51(4), 653–703 (1996)

    Article  MathSciNet  Google Scholar 

  10. Kapovich M., Millson J.: Universality theorems for configuration spaces of planar linkages. Topology 41(6), 1051–1107 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Katok A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes tudes Sci. Publ. Math. 51, 137–173 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kempe, A.B.: How to draw a straight line; a lecture on linkages. London: McMillan, 1977. (cf. http://www.archive.org/details/howtodrawstraigh00kemprich)

  13. Klotz Milnor T.: Efimov’s theorem about complete immersed surfaces of negative curvature. Adv. in Math. 8, 474–543 (1972)

    Article  MATH  Google Scholar 

  14. Manning A.: Topological entropy for geodesic flows. Ann. of Math. (2) 110(3), 567–573 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Margulis G.: Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funct. Anal. Appl. 3(4), 335–336 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ratner M.: The central limit theorem for geodesic flows on nn-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973)

    Article  MathSciNet  Google Scholar 

  17. Thurston W., Weeks J.: The Mathematics of three dimensional manifolds. Scientific American 251, 94–106 (1984)

    Article  Google Scholar 

  18. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79. Berlin: Springer, 1982

  19. Xu G., Bajaj C.: Curvature computations of 2-manifolds in \({\mathbb R^k}\) . J. Comput. Math. 21, 681–688 (2003)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Matemática, Universidade do Porto, Porto, 4169-007, Portugal

    M. L. S. Magalhães

  2. Department of Mathematics, Warwick University, Coventry, CV4 7AL, UK

    M. Pollicott

Authors
  1. M. L. S. Magalhães
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Pollicott
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Pollicott.

Additional information

Communicated by G. Gallavotti

Rights and permissions

Reprints and permissions

About this article

Cite this article

Magalhães, M.L.S., Pollicott, M. Geometry and Dynamics of Planar Linkages. Commun. Math. Phys. 317, 615–634 (2013). https://doi.org/10.1007/s00220-012-1521-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1521-0

Keywords

Search

Navigation