It is an element in a certain normal bordism group. Ian Agol Ian Agol Ryan Budney Ryan Budney The problem might be that a homotopy of immersions of S does not necessarily yield an immersion of Sx[0,1]. One needs that the derivative in [0,1]-direction is linearly independent of the derivatives in the S-direction.
Let me edit my answer a bit to make the key step less "insiderish". But I changed my answer. It's only a partial response to your question, not really everything you were looking for. What do you mean by "formal extension" -- is that the 1-parameter family of immersions with normal vector field? But it does not answer the question of whether a given immersion can be so extended. It is if the disk has positive codimension, and this is a key step in proving Smale-Hirsch.
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Featured on Meta. Unicorn Meta Zoo 9: How do we handle problem users? What do you want to do in September for MO's tenth anniversary? Linked 7. Related 2. Question feed. Inserting into the geodesic equations, and taking into account the symmetric and anti-symmetric terms, we see that 1. In this formula we already observe the interplay between the Lie bracket stucture and the metric.
Cartan's viewpoint bypasses the traditional methods in constrained dynamics where Lagrange multiplier terms are added to represent the constraint forces just to be eliminated afterwards ; see Blajer and references therein. Moreover, when using the Euler-Lagrange equations, the system of ODEs comes in implicit form unsuitable for being easily integrated numerically. In Cartan's approach all this information is embodied in the Christoffel symbols or equivalently in the structure coefficients. Cartan's approach provides an algorithmic way to derive the equations of motion for nonholonomic systems:.
Building up on the example in Cartan , section 11 we start up the derivation of the equations of motion for "Caplygin's sphere''.
Details of the derivation and a theoretical analysis will be provided elsewhere. The configuration space is 2 x SO 3. It is assumed that the center of mass coincides with the geometric center.
Simple Morphisms in Algebraic Geometry
We follow Arnol'd's notation Arnol'd , where capital letters denote vectors as seen from the body. We denote by the angular velocity viewed in the space frame, and the angular velocity as viewed in the body frame, which we may assume attached at the principal axis of inertia. Intrinsically speaking, this corresponds to left translation in the Lie group SO 3. Thus in space. Some comments are in order. Firstly, the component corresponds to pivoting around the contact point, and therefore is arbitrary.
We observe that 2. It is very important in our context is to observe that the constraints define a distribution E in Q which is both right SO 3 -invariant and 2 -invariant. This brings us immediatelly to the issue of integrability of nonholonomic systems, which was introduced in Koiller and extensivelly discussed in the colletanea by Cushman and Sniatycki , also see Bates and Cushman, There is a conflict between the left SO 3 invariance of the Hamiltonian 2. The adapted basis is the dual basis of the incidentally, we provide a correction to the coefficients given in Cartan, Another choice could be using a , as the free parameters,.
We now outline the procedure for the general case also integrable using Cartan's programme. To organize the calculations, we write the left invariant forms in SO 3 as. The last entry follows the alphabet, and the reader will forgive us for mixing up the notation in the left hand side. It is worthy, however, to proceed as intrinsically as possible. Let f 1 , f 2 , f 3 the right-invariant vectorfields in SO 3 forming the dual basis for the , ,. The constraint distribution E is annihilated by the 1-forms. Surprisingly, the system is still integrable.
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See Arnol'd et al. To orthonormalize, we need the dual basis of the , , , that is, the left invariant vectorfields F 1 , F 2 , F 3 such that. To compute the inner product we revert to the basis of left-invariant vectorfields via 2.
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Direct and inverse "development'' of frames and curves were so obvious to Cartan and for that matter, also to Levi-Civita that he they did not bother to give details. Actually, inverse parallel transport seems closest to their way of thinking. We elaborate these concepts, exhibiting explicitly Theorem 3. A frame for E q o T q o Q can be transported along a curve c t in Q.
The "novelty'' here as stressed by Cartan : c t is an arbitrary curve in Q , that is t does not need to be tangent to E. In fact, we are led to the linear time-dependent system of ODEs. In particular, an orthonormal frame at E q o is transported to E c t and remains orthonormal.
Vector Fields and Other Vector Bundle Morphisms - A Singularity Approach
We have. L EMMA 3. P ROOF. Let U t a curve of frames in m. As before, it is not assumed that t is tangent to E c t. The curve t in m E q o , is called the hodograph of c t to m. The curve c t is called the development of t.
Then consider the vectorfield in x O m given by. Indeed, by the previous item,.
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What if we had used a different frame on U? We would get a system of ODEs. We can upgrade this construction to provide a parallel frame along c t , by declaring 0.
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This gives. Consider a mechanical system with kinetic energy T and external forces F written in contravariant form, we lower indices using the metric so that F TQ , subject to constraints defined by the distribution E. The non-holonomic dynamics is given by. Let be the hodograph of c. Let be the parallel frame along c t obtained in Theorem 3. This approach can be helpful for setting up numerical methods, and in some cases reducing the non-holonomic system to a second order equation on m.
We also observe that F can represent non-holonomic control forces actuating over the system, as those studied in Krishnaprasad et al. In this section and the next we discuss the question of whether two non-holonomic connections D and on E have the same geodesics. The corresponding dual frame satisfies. Using matrix notation is not only convenient for the calculations, but also to set up the equivalence problem Gardner Consider the linear group G of matrices of the form. For sub-Riemannian geometry, see Montgomery In non-holonomic geometry we are lead to a more difficult equivalence problem see section 6.