59 research outputs found
A Lie algebroid framework for non-holonomic systems
In order to obtain a framework in which both non-holonomic mechanical systems
and non-holonomic mechanical systems with symmetry can be described, we
introduce in this paper the notion of a Lagrangian system on a subbundle of a
Lie algebroid.Comment: 18 page
Routh Reduction by Stages
This paper deals with the Lagrangian analogue of symplectic or point
reduction by stages. We develop Routh reduction as a reduction technique that
preserves the Lagrangian nature of the dynamics. To do so we heavily rely on
the relation between Routh reduction and cotangent symplectic reduction. The
main results in this paper are: (i) we develop a class of so called magnetic
Lagrangian systems and this class has the property that it is closed under
Routh reduction; (ii) we construct a transformation relating the magnetic
Lagrangian system obtained after two subsequent Routh reductions and the
magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full
symmetry group
Towards a prototype of a spherical tippe top
Among spinning objects, the tippe top exhibits one of the most bizarre and counterintuitive behaviours. The commercially available tippe tops basically consist of a section of a sphere with a rod. After spinning on its rounded body, the top flips over and continues spinning on the stem. It is the friction with the bottom surface and the position of the center of mass below the centre of curvature that cause the tippe top to rise its centre of mass while continuing rotating around its symmetry axis (through the stem). The commonly used simplified mathematical model for the tippe top is a sphere whose mass distribution is axially but not spherically symmetric, spinning on a flat surface subject to a small friction force that is due to sliding.
Adopting a bifurcation theory point of view we reach a global geometric understanding of the phase diagram of this dynamical system. According to the eccentricity of the sphere and the Jellet invariant (which includes information on the initial angular velocity) three main different dynamical behaviours are distinguished: tipping, non-tipping, hanging (i.e. the top rises but converges to an intermediate state instead of rising all the way to the vertical state). Subclasses according to the stability of relative equilibria can further be distinguished. Since our concern is the degree of confidence in the mathematical model predictions, we applied 3D-printing and rapid prototyping to manufacture a ’3-in-1 toy’ that could catch the three main characteristics defining the three main groups in the classification of spherical tippe tops as mentioned above. This ’toy’ is suitable to validate the mathematical model qualitatively and quantitatively
Routh reduction for singular Lagrangians
This paper concerns the Routh reduction procedure for Lagrangians systems
with symmetry. It differs from the existing results on geometric Routh
reduction in the fact that no regularity conditions on either the Lagrangian
or the momentum map are required apart from the momentum being a
regular value of . The main results of this paper are: the description of
a general Routh reduction procedure that preserves the Euler-Lagrange nature of
the original system and the presentation of a presymplectic framework for Routh
reduced systems. In addition, we provide a detailed description and
interpretation of the Euler-Lagrange equations for the reduced system. The
proposed procedure includes Lagrangian systems with a non-positively definite
kinetic energy metric.Comment: 34 pages, 2 figures, accepted for publicaton in International Journal
of Geometric Methods in Modern Physics (IJGMMP
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