Differentially Constrained Dynamics

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Differentially Constrained Dynamics
with monetary applications to rolling coins

• Wayne Lawton
• Department of Mathematics National University of
  Singapore matwml_at_nus.edu.sg
• (65)96314907

the methods described herein are intended for
quantum gravitational speculation, the author
disclaims any financial responsibility for fools
attempts to apply these methods in Sentosa
Casinos
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Introduction
Our objective is to explain the physics behind
some of the results in the paper Nonholonomic
Dynamics by Anthony M. Bloch, Jerrold E. Marsden
and Dmitry V. Zenkov,, Notices AMS, 52(3), 2005.
Recall from the vufoils titled Connections.ppt
that a distribution (in the sense of Frobenius)
on an m-dim connected manifold M is defined by a
smoothly varying subspace c(x) of the tangent
space T_x(M) at x to M at every point x in M. We
note that dim(c(x)) is constant and define p m
dim(c).
Definition the Grassman manifold G(m,k) consists
of all k-dim subpaces of Rm, it is the
homogeneous space O(m)/(O(p)xO(m-p)) so has
dimension p(m-p).

Remark c can be described a section of the bundle
over M whose fibers is homeomorphic to G(m,p)
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Distribution Forms
c can be defined by a collection of p linearly
independent .
differential 1-forms .
If we introduce local coordinates .
then there exists a p x m matrix (valued function
of x) E .
that has rank p and .
hence we may re-label
the coordinate indices so that .
where B is .
an invertible p x p matrix and c is defined by
the forms .
where .
hence .
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Unconstrained Dynamics
The dynamics of a system with kinetic energy T
and forces F (with no constraints) is .
where .
we have .
For conservative .
where we define the Lagrangian .
For local coordinates .
we obtain m-equations and m-variables. .
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Holonomic Constraints
One method to develop the dynamics of a system
with Lagrangian L that is subject to holonomic
constraints .
is to assume that the constraints are imposed by
a constraint force F that is a differential
1-form that kills every vector that is tangent to
the (m-p) dimensional submanifold of the tangent
space of M at each point. This is equivalent to
DAlemberts principle (forces of constraint can
do no work to virtual displacements) and is
equivalent to the existence of p variables .
such that .
The (mp) variables (xs lambdas) are
computed from p constraint equations and the
m-equations given by .
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Example Particle on Inclined Plane
Here m 2, p 1 and for suitable coordinates .
where .
and .
is the (fixed) angle of the inclined plane.
Therefore .
and .
and .
and .
and .
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Nonholonomic Constraints
For nonholonomic constraints DAlemberts
principle can also be applied to obtain the
existence of .
such that .
where the mu-forms describe the velocity
constraints .
The (mp) variables (xs lambdas) are
computed from the p constraint equations above
and the m-equations
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Equivalent Form for Constraints
Since the mus and omegas give the same
connection we can obtain an equivalent system of
equations with different lambdas (Lagrange
multipliers)
On the following page we will show how to
eliminate the Lagrange multipliers so as to
reduce these equations to the form given in
Equation (3) on page 326 of the Nonholomorphic
Dynamics paper mentioned on page 2 of this paper.
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Eliminating Lagrange Multipliers
We observe that we can express .
hence we solve for the Lagrange multipliers to
obtain
and reduced (m-p) equations .
which with .
the p constraint equations determine the m
variables. .
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Let the Coins Roll
We consider a more general form of the rolling
coin problem described on p 62-64 of Analytical
Mechanics by Louis N. Hand and Janet D. Finch.
Here theta is the angle the radius R, mass m coin
makes with the y-axis, phi is the rotation angle
as it rolls along a surface described by the
graph of the height function z(x,y). So for this
problem m 4 (variables phi, theta, x, y), p2.
and constraints
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Exercise compare with Hand-Finch solution on p 64
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Ehresmann Connection
Locally on M the 1-forms .
define the distribution .
Hence they also define a fiber bundle .
where .
is an open subset of .
and .
Therefore .
can be identified with a horizontal .
subspace .
and this describes .
an Ehresmann connection .
on .
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When are the constraints holonomic ?
Our objective is to explain the relationship of
this Ehresmann connection to the differentially
constrained dynamics, in particular to prove the
assertion, made on the top right of page 326 in
paper by Bloch, Marsden and Zenkov, that its
curvature tensor vanishes if and only if the
distribution c is involutive or integrable. This
means that the differential constraints are
equivalent to holonomic constraints. Theorem 2.4
on page 82 of Lectures on Differential Geometry
by S. S. Chern, W. H. Chen and K. S. Lam, this
is equivalent to the condition
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Calculation
where .
where .
if and only if
if and only if
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Curved Coins
Let s compute the curvature for the rolling coin
system
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Curvature
We observe that each fiber of
is homeomorphic to
and that the 2-form on E
has values in the tangent spaces to the fibers.
Exercise Show that the expression above equals
the curvature, as defined on page 8 of
Connections.ppt, of the Ehresmann connection .
Reference Geometric Mechanics,Lagrangian
Reduction and Nonholonomic Systems by H. Cendra,
J. E. Marsden, and T. S. Ratiu, p. 221-273 in
Mathematics Unlimited - 2001 and Beyond,
Springer, 2001.
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