REPORT ON THE THESIS OF JASON ZIMMERMAN
DAVID SINGER
The subject of Jason Zimmerman’s thesis is the generalization of the Rolling
Sphere problem to n dimensions, n > 2. The Rolling sphere problem is a beautiful
example of a problem in the calculus of variations with a non-holonomic constraint.
It is quite old; there is a reference to the problem in [2], in an exercise on page 185,
as well as a discussion of the rolling hoop problem (pages 14-15), and he refers to
a discussion in a 1936 book by Macmillan, [3].
Problems with non-holonomic constraints have caused a good deal of difficulty
over the years due to some very subtle technicalities. One approach to their study
is through the use of the Pontrjagin Maximum Principal in the theory of optimal
control; this is the approach taken by Zimmerman. An alternative approach uses
exterior differential systems; see, for example, [1], where the problem of a surface
rolling on another surface is studied.
The most serious difficulty in dealing with such problems arises from the fact
that abnormal minimizers may exist. In a non-holonomic problem, a non-integrable
subbundle C of the tangent bundle of a manifold M is given. If p and q are two
points in a manifold M , let ΩC (p, q) denote the space of paths tangent to the
subbundle and joining p to q. The problem is to minimize some functional over
ΩC (p, q). A difficulty arises if the space is not a manifold, in which case a minimum
may arise having nothing to do with the specific functional.
In nice cases, a solution arising as an abnormal minimizer also turns out to satisfy
the Euler-Lagrange equations corresponding to normal minimizers, and thus they
are local minimizers. Unfortunately, it is not necessarily the case that a minimizer
is locally minimizing. The first counterexample was produced by Montgomery
[4], who noted that several authors had incorrectly claimed that all minimizers
satisfy the EL equations. At present, according to Montgomery, it is not yet known
whether all minimizers have to be smooth curves.
In view of these subtleties, it is clear that the problem addressed in Zimmerman’s
thesis is not a trivial one. One of his major results, section 2.4, is a careful proof
that for the generalized rolling sphere problem all minimizers are projections of
normal minimizers. As I have indicated above, this is by no means an obvious
result. It is possible that his proof may generalize to a wider class of problems,
although I am not able to judge this.
The major results are the proofs of complete integrability for n = 3 and n = 4.
The case of n = 3 is not surprising, as it only required finding the Hamiltonian
H1 . The case of n = 4 presented considerably more technical difficulty, and the
result is very satisfying and a little mysterious. Interesting examples of completely
integrable systems are of considerable interest, particularly since they seem in many
cases to arise in connection with integrable partial differential equations. The Euler
Date: May 28, 2002.
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DAVID SINGER
elastica and its generalization, the Kirchhoff rod, for example, play the role of soliton
solutions to the nonlinear Schrödinger equation.
There are two places in the thesis where I believe modifications are necessary.
The first occurs on pages 22 and 23. Theorem 2.6 is seriously flawed in its statement.
The notation used in the statement is not defined until later in the thesis, and it
was not until I had read quite a bit further that I could make sense of it. There
is no connection given in the hypothesis between the element (a, q) in L∗ and the
matrix M . Thus the statement 2) of the theorem makes no sense. Furthermore, the
matrix Aij is not defined, though it appears to be the same matrix as Bij . In the
statement of 3), M appears to be an element of L∗ . At some point the identification
between L and L∗ is made, but it is often hard to keep track of which object is
being studied when. (I note that there is a statement on page 25 after Definition
2.9 which appears to define M . This is, unfortunately, preceded by a definition of
A which refers to Aij as its coordinates.
The second difficulty occurs on pages 28 and 29. I have really struggled with
the last statement of Theorem 2.12, concerning the abnormal extremals. As this is
one of the significant results of the paper, this needs to be clarified. Specifically,
an abnormal extremum does not arise from the variational problem (at least, not
a priori). Therefore, I do not understand how the equation
dP
= [Ωu , K]
dt
has been derived (although I am confident that it is correct). A clear explanation
of this equation is vital.
Assuming that the two items above are addressed satisfactorily, I recommend
that the thesis be accepted in partial fulfilment of the requirements for the degree
of Doctor of Philosophy.
References
[1] R.Bryant & L. Hsu, Rigidity of Integral Curves of Rank 2 Distributions, Invent. Math, 114
(1993), 435-461.
[2] H. Goldstein, Classical Mechanics, Reading, Addison-Wesley, 1950.
[3] W.D. Macmillin, Dynamics of Rigid Bodies, 1936. (Dover reprint, 1960).
[4] R. Montgomery, Abnormal Minimizers, SIAM J. Control and Optimization, 32 (1994), 16051620.
Department of Mathematics,Case Western Reserve University, Cleveland, OH 441067058
E-mail address: [email protected]
MINOR CORRECTIONS
page 13 A consistent stylistic defect is illustrated in the proof of Proposition 2. A
reference is given to ([27],[38]). The first is a book and the second is a
forty page paper. As a practical matter, such citations are useless. It is
really necessary to cite page numbers. In some cases, this defect is extremely
annoying: extreme examples are to be found on page 23, where a theorem is
stated and its proof is given as appearing in four references, three of which
are books.
page 13 The factor of 2 in the Cauchy–Schwartz inequality should not be there.
page 14 The last statement in the proof of Proposition 1.3 is misleading. It is true
that if (G, d) is complete then (G, dQ ) is also complete. This follows immediately from the fact that a sequence which is Cauchy with respect to dQ
is Cauchy with respect to d. (It is not necessary to cite two references for
this.) However, it does not follow from the statement that the metrics are
topologically equivalent. Completeness is not a topological invariant.
page 24 The quantity F(x) is not defined, making the definition 2.8 difficult to understand. As stated, the symmetry appears to apply pointwise.
page 32 There is a minus sign missing from the formula (*).
page 46 In Theorem 3.6, the curve ξ(t) is in the cotangent bundle of G, not in L∗ .
It would also help to remind the reader of the relationship between a and A,
what P is, etc.
page 49 The last formula on the page appears to be missing a factor of k from the


first term. I did not understand the appearance of K(B
ij ) and K(Pij ) in the
formula. I could not get from that line to the first line of page 50. I am sure
this is all elementary, but I simply can’t follow it.
page 51 In proposition 3.9, the term Xni should be X(n+1)i .
page 53 In the second line, the pair of norm bars should be replaced by parentheses,
since it is a scalar quantity.
page 75 Something is missing from the second line following Definition A.3. concerning
overlapping bundle charts. More importantly, the description of a control
system is flawed. It is true that locally a control system can be viewed as a
family of vector fields parametrized by controls. However, to assume that this
is true globally is, I believe, too restrictive. It certainly rules out the possibility
of control systems on manifolds which do not have global nonvanishing vector
fields, for example. The existence of the global mapping F : M × U −→
T (M ) does not follow from the usual local definition. (In the case under
consideration in this dissertation, however, M is a Lie group and T (M ) is a
trivial bundle, so this does not materially effect the rest of the thesis.)
pp. 65,71 This concerns the references to the “Euler-Griffiths elastic problem” (EGEP).
Zimmerman describes this problem on page 65 as that of finding a curve which
RT
minimizes 0 k 2 (s)ds. Later, he suggests on page 71 that there may be connections between the RSP in higher dimensions and the EGEP in higher
dimensions. One of the well-known results about the EGEP is that all solutions lie in a three-dimensional subspace, so that one gets no new solutions in
going to higher dimensions. Perhaps there is a connection between RSP and
some version of the Kirchhoff problem in higher dimensions.
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