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Geometry of Classical Mechanics

  • Nicholas M. J. WoodhouseEmail author
Chapter
  • 2.9k Downloads
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

The analysis of the dynamics of a mechanical system begins with the introduction of generalized coordinates to label the configurations. There is a great deal of freedom in the choice of coordinates and a good choice can greatly simplify the work. It is rarely the case, however, that a single coordinate system can be used for all possible configurations. For example, the polar coordinates used to label the configurations of the spherical pendulum (p. 67) are singular when the rod is vertical, in the sense that these configurations do not determine unique values of θ and φ. When θ = 0 or θ = π, we have
$$(x,y,z) = (0,0,1),\qquad \mbox{ or}\qquad (x,y,z) = (0,0,-1)$$
irrespective of the value of φ. In these special configurations, the coordinate labels are not unique. A similar problem arises when Euler angles (p. 132) are used to label the orientations of a rigid body. When the z-axes of the inertial frame and of the body frame are aligned, which happens when θ = 0 or θ = π, the orientation is completely determined by the values φ + ψ. It is unchanged when φ and ψ are replaced by φ + k and ψ − k, respectively. This is the ‘gimbal lock’ problem.

Keywords

Tangent Vector Classical Mechanic Poisson Bracket Symplectic Form Transformation Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2009

Authors and Affiliations

  1. 1.Mathematical Institute University of OxfordOxfordUnited Kingdom

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