Homework 3 Problem 11

#### Reggie's Topic: "Differential Geometry Representations of Thermodynamic Concepts"

I will investigate thermodynamics through a differential geometry lens, representing thermodynamic concepts as operations on manifolds. I do not know any specific details yet.

#### Kevin's Topic: "Controllability w/ Forms"

In my presentation I plan on explaining how to check controllability using one forms. I will then present an example of checking controllability using the forms method, and simultaneously do the example using the vector method that was learned in class to compare the two methods.

#### Antonis' Topic: "Tippe Top"

I will investigate the tippe top toy…

#### Andrew's Topic: "Integrating on Constraint Manifolds"

I plan to present a survey of methods for integrating trajectories on manifolds with constraints.

#### Andy's Topic: "3D Mapping and Computations"

In light of the problem I presented earlier this semester, I will be presenting no different ways to map and compute in 3D, including Euler angles, quaternions, and any other method that I stumble across that I feel would be a good complement to the material aforementioned. I will also cover the solution that I came up with to my previous problem (for those who are curious).

#### Melih's Topic: "Heavy top stability and nonsmooth Lagrangian mechanics"

My project presentation will be about heavy-top stability applied to my research project; rimless spoked wheel. I will introduce and exemplify the implications of differential geometry on dynamics (manifolds in dynamics, constraints, lagrangian and d'alembert, etc.).
Afterwards, by heavy-top stability, I will explain the stability properties of the spoked wheel in-between collisions. For the collisions, I am working on nonsmooth Lagrangian mechanics now and if I can concatenate swinging and colliding motions together, I will also mention about the nonsmooth Lagrangian briefly.

The rattleback is a toy whose shape resembles the hull of a boat. When spun in one direction, the toy will slow down, stop, wobble back and forth, and then reverse direction. When spun in the other direction, the toy spins, slows down, and stops as expected. I will try to investigate the behavior of this toy using the mechanics from this semester.

#### Jehanzeb's Topic: "Asynchronous Variational Integrators (AVI)"

I plan to present the theory of constructing AVIs, and the conservation properties such integrators obey.

#### Will's Topic: "Differential Flatness: No, no! Please don't inflate it."

"Roughly speaking differentially flat systems are systems whose entire set of solutions are in a smooth one-one correspondence with arbitrary curves in a space of dimension \$p\$, equal to the number of equations by which the system is underdetermined." [Rathinam, Thesis] I will introduce the definition of differential flatness and explain how to determine the flatness of a nonlinear system. I will also explain the usefulness of flatness and illustrate this by example.

#### Shu's Topic: "Application of Differential Geometry in Fluid Mechanics"

I am tring to present the dynamics of the elliptical body moving in an ideal fluid under the theory of integral Langrange-d'Alembert principle defining the Lagrangian for the system as the total kinetic energy of the body and surrounding fluid, which is important in the aquatic vehicles. I will also try to investigate how the manifold theory could be used in the hydrodynamics.

page revision: 13, last edited: 07 May 2007 01:27