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The paper Geometry of the Prytz Planimeter described a mechanical instrument whose configuration space is an $S^1$-bundle with an $SU(1,1)$ action. That paper goes on to study the holonomies of various paths in the base space. Motion is described by a connection on this bundle.

I am interested because it's an example of a $G$-bundle appearing in classical mechanics. Are there other explicit classical mechanical systems that engineers study that exhibit the basic concepts of differential geometry so lucidly?

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Another possible example is the falling cat problem which has been studied among the others by Richard Montgomery. Cf.Gauge Theory of the falling cat

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A wonderful quote from the article. "The main point of these earlier works is that a dictionary can be developed between the gauge theory of the physicist's and mathematicians, and the problems occuring in the orientation control of deformable bodies. Briefly, in this dictionary the space of shapes of the body plays the role of the base space, or space-time in the physicist's gauge theory. Its tangent space is the space of controls. The state space, or configuration space of the body, is principal bundle of the theory. ... –  Tom LaGatta Dec 29 '12 at 8:11
    
... The gauge group is the group of rigid reorientations of the body. The gauge field summarizes the condition that the angular momentum be zero." –  Tom LaGatta Dec 29 '12 at 8:11
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Another couple of examples are the sphere rolling on the plane (or any surface, for that matter) without twisting or slipping, which is described by a connection on a principal SO(3)-bundle over the surface, and the motion of a trailer (or series of hitched trailers) being pulled by a tractor, which is described by a PSL(2,R)-connection (or a product of these) over the space of positions of the tractor. (This latter example is probably closely related to the planimeter problem described in Foote's article.)

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Take the planar three-body problem. Or, said a bit differently, take that 'cat' to consist of three point masses moving about in the plane -- a triangle! Fix the center of the mass at the origin by the usual trick. Take G = SO(2). Now take the quotient and you get the cone over the usual Hopf fibration. The points of the sphere in the base spacerepresent oriented similarity classes of triangles. This geometry is at the heart of much modern understanding of the planar three body problem. You can find references in my 2000 paper with Chenciner A remarkable periodic solution of the three-body problem in the case of equal masses and the geometry explained in some detail in the 1st few pages and in the appendix to my 1996 paper The geometric phase of the three-body problem. You can download these from http://count.ucsc.edu/~rmont/papers/list.html

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Take the planar three-body problem. Or, said a bit differently, take that 'cat' to consist of three point masses moving about in the plane -- a triangle! Fix the center of the mass at the origin by the usual trick. Take G = SO(2). Now take the quotient and you get the cone over the usual Hopf fibration. The points of the sphere in the base spacerepresent oriented similarity classes of triangles. This geometry is at the heart of much modern understanding of the planar three body problem. You can find references in my 2000 paper with Chenciner A remarkable periodic solution of the three-body problem in the case of equal masses' and the geometry explained in some detail in the 1st few pages and in the appendix to my 1996 paper The geometric phase of the three-body problem'. You can download these from http://count.ucsc.edu/~rmont/papers/list.html

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Just to record what I heard from another source, Geometric Control Theory has some promise. Many examples from classical mechanics examined from the point of view of connections on their phase space.

The text is Geometric Control of Mechanical Systems by Francesco Bullo and Andrew D. Lewis.

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