Mechanical Connections. Wayne Lawton Department of Mathematics National University of Singapore email@example.com http://www.math.nus.edu.sg/~matwml (65)96314907. 1. 3-6 Earth, Tangents, Tubes, Beanies. Contents. 7-10 Rolling Ball Kinematics. 11-13 Nonholonomic Dynamics – Formulation.
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Department of Mathematics National University of Singapore
7-10 Rolling Ball Kinematics
11-13 Nonholonomic Dynamics – Formulation
14-22 Distributions and Connections
23-24 Nonholonomic Dynamics - Solution
25-26 Rolling Coin Dynamics
27 Symmetry and Momentum Maps
28 Rigid Body Dynamics
29 Boundless Applications
Page 1 of my favorite textbook [Halliday2001] grabs the reader with a enchanting sunset photo and the question: “How can such a simple observation be used to measure Earth?”
brain is !!!
The figure on page 44 in [Marsden1994] illustrates the parallel translation of a Foucault pendulum, we observe that the cone is a flat surface that has the same tangent spaces as the sphere ALONG THE MERIDIAN.
Radius = 1
Holonomy: rotation of tangent vectors parallel translated around meridian = area of spherical cap.
Tubes used for anatomical probing (imaging, surgery) can bend but they can not twist. So how do they turn?
Unit tangent vector of tube curve on sphere, Normal vector of tube tangent vector to curve
Tube in plane geodesic curve on sphere
No twist tangent vector parallel translated (angle with geodesic does not change) holonomy = area enclosed by closed curve.
if a ball rolls along the curve
Astonishingly, a unit ball can rotate about the z-axis by rolling without turning !
Here are the steps:
1. [0 0 -1] [pi/2 0 -1]
2. [pi/2 0 -1] [pi/2 -d -1]
3. [pi/2 -d -1] [0 d -1]
The result is a translation and
rotation by d about the z-axis.
Theorem [Lioe2004] If
where A = area bounded by u([0,T]).
Proof The no turning constraints give a connection
on the principle SO(2) fiber bundle
and the curvature of this connection, a 2-form on
with values in the Lie algebra so(2) = R, coincides
with the area 2-form induced by the Riemannian metric.
is a rotation trajectory
is a small trajectory variation
is defined by
Theorem [Lioe2004] If
is the shortest
trajectory with specified
the ball rolls along an arc of a circle in the plane P and u([0,T]) is an arc of a circle in the sphere. Furthermore, M(T) can be computed explicitly from the parameters of either of these arcs.
Potential Application: Rotate (real or virtual) rigid body by moving a computer mouse.
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 k dimensional submanifold of the tangent space of M at each point. This is equivalent to D’Alembert’s principle (forces of constraint can do no work to ‘virtual displacements’) and is equivalent to the existence of p variables
The 2m-k variables (x’s & lambda’s)
are computed from m-k constraint
equations and the m equations given by
For nonholonomic constraints D’Alemberts principle can also be applied to obtain the existence of
where the mu-forms describe the velocity constraints
The 2m-k variables (x’s & lambda’s)
are computed from the m-k constraint
equations above and the m equations
On vufoils 20 and 21 we will show how to eliminate (ie solve for) the m-k Lagrange multipliers !
synthetic geometry algebra
Calculus: fundamental theorems local global
Implicit Function Theorem for a smooth function F
Local (near p) foliation (partition into submanifolds) consisting of level sets of F (each with dim = n-m)
(global) foliation of O into 2-dim spheres
Definition A dim = k (Frobenius) distribution d on a manifold E is a map that smoothly assigns each p in E
A dim = k subspace d(p) of the tangent space to E at p.
Example A foliation generates a distribution d such that point p, d(p) is the tangent space to the submanifold containing p, such a distribution is called integrable.
Definition A vector field v : E T(E) is subordinate
to a distribution d (v < d) if
The commutator [u,v] of vector fields is the vector field uv-vu where u and v are interpreted as first order partial differential operators.
Theorem [Frobenius1877] (B. Lawson by Clebsch & Deahna) d is integrable iff u, v < d [u,v] < d.
Remark. The fundamental theorem of ordinary
diff. eqn. evey 1 dim distribution is integrable.
are differential 1-forms.
d is integrable iff
Proof See [Chern1990] – crucial link is Cartan’s formula
Remark Another Cartan gem is:
between manifolds with rank = dim B,
the vertical distribution d on E is defined by
and a connection is a complementary distribution c
This defines T(E) into the bundle sum
Theorem c is the kernel of a V(E)-valued connection
and image of a horizontal lift
denote the horizontal projection.
and points p, q in B then every path f from p to q in B defines a diffeomorphism (holonomy) between fibers
Proof Step 1. Show that a connection allows vectors in T(B) be lifted to tangent vectors in
T(E) Step 2. Use the induced bundle construction to create a vector field on the total space of the bundle induced by a map from [0,1] into B. Step 3. Use the flow on this total space to lift the map. Use the lifted map to construct the holonomy.
Remark. If p = q then we obtain holonomy groups.
Connections can be restricted to satisfy additional (symmetry) properties for special types (vector, principle) of bundles.
are vector field extensions.
Theorem This defintion is independed of extensions.
Theorem A connection is integrable (as a distribution) iff its curvature = 0.
Theorem A connection has holonomy = 0
iff its curvature = 0.
we introduce local coordinates
there exists a (m-k) x m matrix (valued function of p) E
with rank m-k and
hence we may re-label
the coordinate indices so that
where B is
an invertible (m-k) x (m-k) matrix and c is defined by
if and only if
if and only if
Since the mu’s and omega’s define the same distribution we can obtain an equivalent system of equations with different lambda’s (Lagrange multipliers)
On the next page we will show how to eliminate the Lagrange multipliers so as to reduce these equations to the form given in Eqn. (3) on p. 326 in [Marsden2004].
is a momentum map if
is a Poisson manifold
with a left Hamiltonian action by a Lie group G with Lie
with linear dual
is the left-invariant vector field on
by the flow
The reduced space
is a PM.
Theorem [Marsden1990,1994] If H : P R is G-inv. then it induces a Hamiltonian flow on the red. space.
is the pullback under right translation. The Hamiltonian
is a positive definite self-adjoint inertial operator, and
is a fiber bundle whose connection (canonical 1-form on the symplectic manifold P) gives dynamic reconstruction from reduced dynamics.
Theorem [Ishlinskii1952] (discovered 1942) The holonomy of a period T reduced orbit that enclosed a spherical area A is
Falling Cats, Heavy Tops, Planar Rigid Bodies, Hannay-Berry Phases with applications to adiabatics and quantum physics, molecular vibrations, propulsion of microorganisms at low Reynolds number, vorticity free movement of objects in water, PDE’s – KDV, Maxwell-Vlasov, …
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