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Journal of Geometric Mechanics

2009 , Volume 1 , Issue 4

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On the number of weakly Noetherian constants of motion of nonholonomic systems
Francesco Fassò, Andrea Giacobbe and Nicola Sansonetto
2009, 1(4): 389-416 doi: 10.3934/jgm.2009.1.389 +[Abstract](92) +[PDF](377.5KB)
We develop a method to give an estimate on the number of functionally independent constants of motion of a nonholonomic system with symmetry which have the so called 'weakly Noetherian' property [22]. We show that this number is bounded from above by the corank of the involutive closure of a certain distribution on the constraint manifold. The effectiveness of the method is illustrated on several examples.
Variational principles for spin systems and the Kirchhoff rod
François Gay-Balma, Darryl D. Holm and Tudor S. Ratiu
2009, 1(4): 417-444 doi: 10.3934/jgm.2009.1.417 +[Abstract](99) +[PDF](310.1KB)
We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.
A catalog of inverse-kinematics planners for underactuated systems on matrix groups
Sonia Martínez, Jorge Cortés and Francesco Bullo
2009, 1(4): 445-460 doi: 10.3934/jgm.2009.1.445 +[Abstract](65) +[PDF](317.3KB)
This paper presents motion planning algorithms for underactuated systems evolving on rigid rotation and displacement groups. Motion planning is transcribed into (low-dimensional) combinatorial selection and inverse-kinematics problems. We present a catalog of solutions for all left-invariant underactuated systems on SE(2), SO(3), and SE(2)$ \times $ R classified according to their controllability properties.
Nonholonomic Hamilton-Jacobi equation and integrability
Tomoki Ohsawa and Anthony M. Bloch
2009, 1(4): 461-481 doi: 10.3934/jgm.2009.1.461 +[Abstract](150) +[PDF](789.5KB)
We discuss an extension of the Hamilton-Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton-Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton-Jacobi theory for unconstrained systems. The proof also helps us identify a geometric meaning of the conditions on the solutions of the Hamilton-Jacobi equation that arise from nonholonomic constraints. The major advantage of our result is that it provides us with a method of integrating the equations of motion just as the unconstrained Hamilton-Jacobi theory does. In particular, we build on the work by Iglesias-Ponte, de Léon, and Martín de Diego [15] so that the conventional method of separation of variables applies to some nonholonomic mechanical systems. We also show a way to apply our result to systems to which separation of variables does not apply.
Book review: Geometric mechanics and symmetry, by Darryl D. Holm, Tanya Schmah and Cristina Stoica
Miguel Rodríguez-Olmos
2009, 1(4): 483-488 doi: 10.3934/jgm.2009.1.483 +[Abstract](98) +[PDF](81.5KB)

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