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by Kevin Davis
Abstract
This paper serves as a brief overview into determining controllability using forms, using the concept of the derived flag. It is not original work done by me, but it is an...
Introduction
Systems of ordinary differential equations (ODEs) can be divided into two classes, determined and underdetermined. The latter class can be defined as a system of ODEs with dependent...
This paper is a brief introduction to dissipation induced instabilites. I will first providing several technical definitions of stability and try to illustrate them through examples. I use the...
by Reggie Weece
Abstract
Beginning with a brief thermodynamics review, differential geometry analysis of thermodynamic systems will yield equations describing reversible and irreversible...
I guess this page should be deleted. I've accidentally created two pages with the same name. The one that has the actual information can be accessed from the grey toolbar, under Projects.
—Antonis
by Antonis Vakis
Introduction
The Tippe Top is a toy that belongs to the general category of spinning tops. Spinning tops are toys that can be spun around an axis and balance on a point, relying on...
I discuss Asynchronous Variational Integrators and Multi-symplectic mechanics. Please see the files section for the full document.
by -Jehanzeb
Instructor email: scott@me598.com
by Shu
Abstract
In this paper, I present the dynamics of the elliptical body moving in an ideal fluid under potential theory and integral Langrange-d'Alembert principle defining the Lagrangian for...
Homework 1
Homework 2
Homework 3
Homework 4
[tex][pdf]
Problem 1
Problem 2
Problem 3
Problem 4
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Problem 6
Problem 7
Problem 8
Problem
Use Newtonian mechanics to derive the equation of motion
(1)
for the “snowboarder” described in class.
Solution
To derive the equation using Newtonian mechanics, we first draw a free...
Problem
Solution by: Andy Bosiljevac
Express the equation of motion
(1)
for the “falling chain” described in class in canonical Hamiltonian form.
Solution
Let: µ = mass per unit length
l =...
Problem
A mass is situated at the point in the plane; a mass is situated at the point . The two masses are joined by a linear spring with spring constant but are otherwise unconstrained. Assume...
Problem
Prove that the torus is a manifold.
Solution
First, we'll prove that is a manifold. We'll show that is the union of four compatible charts, namely and depicted above.
Let
with that...
Problem
Give an example of a map which is injective but not surjective. Give an example of a map which is surjective but not injective. Give an example of a map which is neither surjective nor...
Problem
Consider a rigid body in which is free to rotate in any direction about its center of mass, the latter fixed in space. The configuration manifold for this system i.e., the manifold...
Problem 7
An isomorphism between vector spaces is a linear bijection. Prove that if is a diffeomorphism between manifolds then is an isomorphism for each
Solution:
The following theorem is...
I'm doing this problem because I my email record has no one claiming it, and it's still empty here. I'm sorry if I've stolen your problem. -AndrewC
Problem
A distribution on a manifold assigns to...
[tex][pdf]
Consider with local coordinates and with local coordinates .
Problem 1
Problem 2
Problem 3
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Problem 8
Problem
Consider with local coordinates and with local coordinates .
Let be given in local coordinates by
(1)
If the point has coordinates and , compute .
Solution
This problem is asking,...
Problem:
Consider with local coordinates and with local coordinates . Consider the two form on . Compute .
Solution:
Let be given in local coordinates by
.
Then,
and, for the two form , the...
Problem
Consider with local coordinates and with local coordinates .
Let
(1)
for all .
Compute with respect to this inner product.
Solution
We have the relationships,
(2)
(3)
. Let
(4)
We...
Problem 4
Consider the vector field and the two form on . Compute .
Solution
To determine consider pairing with and an arbitrary vector field . Using the definitions of and we...
Problem 5
Compute the Lie derivative .
Solution
From Problem 4, and .
By Cartan's Magic Formula, , so we'll compute each term individually and then plug them in.
First we'll compute . Since is a...
Problem 6
Is closed? Exact? What about and ?
Solution
For the closedness, we should check whether ;
(1)
(2)
(3)
Thus, only is found to be closed.
For the exactness, we should check if form...
Solution by: Andy Bosiljevac
Problem:
Given:
(1)
Compute Div(X)
Solution:
(2)
Starting in parts,
(3)
, so
(4)
Next,
(5)
(6)
(7)
(8)
Now, taking the differential of...
Problem
Consider the control system
(1)
on , where and are real-valued control inputs. Is this system locally accessible near ? Small-time locally controllable near ?
Solution
The first thing to...
(tex) (pdf)
Problem 1
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Problem 3
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Problem
Show that an injective immersion need not be an embedding.
Solution
Let's say we've got a piece of string. We can bend the string so that it looks like a figure eight, with the two ends...
Problem
Give an example of a Lagrangian system that has a degenerate Lagrangian but a second-order Lagrangian vector field .
Melih's Solution:
Let
(1)
Using this, we can obtain a coordinate...
Select a topic for your final project and post a title and brief abstract to the homework wiki.
Reggie's Topic: "Differential Geometry Representations of Thermodynamic Concepts"
I will investigate...
Problem
The real projective plane is obtained by identifying opposite edges of a rectangle with one another as shown.
What is the smallest integer for which can be immersed in ? What is the...
Problem
Let be a finite-dimensional manifold. Show that the canonical one form on , defined such that
(1)
for and , satisfies for any one form .
Solution
We will think of as a map from to ....
Problem
Show that there can exist no constant nonzero Hamiltonian vector field on with the standard symplectic form . What if “Hamiltonian” were replaced with “locally...
Problem
Recall the bead on a rotating hoop described in class. Show that the canonical Hamiltonian equations obtained using the total energy in the system as the Hamiltonian disagree with the...
Problem
Consider the mechanical system with configuration manifold and Hamiltonian . Verify explicitly that flow along the corresponding Hamiltonian vector field is area-preserving.
Solution
To...
Problem
For the "falling chain" problem described in class, show that the Lagrangian is hyperregular. Compute the canonical one form , the canonical symplectic form , and the Liouville volume form...
Problem:
A mass moves in the plane, joined to the origin by a linear spring with spring constant and zero rest length. Assume the position of the mass to be described by its polar coordinates on...
Problem
Suppose that dissipation is introduced to the system in the previous problem with the Rayleigh dissipation function , where is a positive constant. Compute the corresponding exterior force...
(tex) (pdf)
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 1
Do there exist nonorientable Lie groups? If so, give an example. If not, why not?
Solution
All Lie groups are orientable. I can illustrate this by example. A Lie group acting on a form is...
Problem
A mass hangs on a massless rod with length from the center of a wheel with mass , rotational moment of inertia , and radius as shown:
Use Routh reduction to obtain the equations of...
Problem 3
The Lie group acts on such that . Show that the lifted action on leaves the Lagrangian from the preceding problem invariant, and compute the corresponding momentum map. Show that this...
Problem
Recall the “extensible barbell” example from class. The system's symmetry with respect to the group action
(1)
corresponded, via Noether's theorem, to a conservation law for the vector...
Problem
The scalar quantity is also conserved in the “extensible barbell” example. What group action corresponds to this conservation law?
Solution
Let be with the group action addition...
Problem
The Lie group acts on such that . Endowed with the symplectic form , is a symplectic manifold and this action is canonical. Show that this action has a nonequivariant momentum map given...
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Check the files for my paper and the paper it's based on.
Some links that students of ME598 may find useful:
A LaTeX tutorial
Fuck this… here's the .pdf:
https://netfiles.uiuc.edu/bosiljev/ME598
by Andy Bosiljevac
The Euler rotation theorem states that any three-dimensional rotation can be described by
a single rotation...
This paper introduces and exemplifies the applications of differential geometry on dynamics (manifolds in dynamics, constraints, lagrangian and d'alembert, etc.). After restating basic concepts,...
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Course Description
Beginning with a self-contained review of Lagrangian and Hamiltonian mechanics in local coordinates, this course will provide an introduction to the differential geometric...
There's Atiyah's Conjecture and Brockett's Condition,
Chorin's Projection and Kostant's Partition;
Arnol'd Diffusion is something to see,
But Marsden-dash-Weinstein Reduction's for me.
Fluid...
Homeworks
Homework 1
Homework 2
Homework 3
Homework 4
Projects
An Investigation of the Tippe Top
An Introduction to Thermodynamics Via One-Forms
An Introduction to Dissipation Induced...
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