Journal of Nonlinear Science

, Volume 21, Issue 1, pp 3–32 | Cite as

Nonlinear Dynamics of the 3D Pendulum

  • Nalin A. Chaturvedi
  • Taeyoung Lee
  • Melvin Leok
  • N. Harris McClamroch
Open Access


A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the non-integrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, relative equilibria, invariant manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the rich and complex dynamics of the 3D pendulum.


Pendulum Rigid body Nonlinear dynamics Attitude Equilibria Relative equilibria Stability Chaos 

Mathematics Subject Classification (2000)

70E17 70K20 70K42 65P20 


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Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Nalin A. Chaturvedi
    • 1
  • Taeyoung Lee
    • 2
  • Melvin Leok
    • 3
  • N. Harris McClamroch
    • 4
  1. 1.Research and Technology CenterRobert Bosch LLCPalo AltoUSA
  2. 2.Department of Mechanical and Aerospace EngineeringFlorida Institute of TechnologyMelbourneUSA
  3. 3.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  4. 4.Department of Aerospace EngineeringThe University of MichiganAnn ArborUSA

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