Integrable Hamiltonian Systems on Complex Lie Groups

Paperback (Print)
Used and New from Other Sellers
Used and New from Other Sellers
from $59.00
Usually ships in 1-2 business days
(Save 10%)
Other sellers (Paperback)
  • All (2) from $59.00   
  • New (1) from $59.00   
  • Used (1) from $130.99   
Close
Sort by
Page 1 of 1
Showing All
Note: Marketplace items are not eligible for any BN.com coupons and promotions
$59.00
Seller since 2007

Feedback rating:

(445)

Condition:

New — never opened or used in original packaging.

Like New — packaging may have been opened. A "Like New" item is suitable to give as a gift.

Very Good — may have minor signs of wear on packaging but item works perfectly and has no damage.

Good — item is in good condition but packaging may have signs of shelf wear/aging or torn packaging. All specific defects should be noted in the Comments section associated with each item.

Acceptable — item is in working order but may show signs of wear such as scratches or torn packaging. All specific defects should be noted in the Comments section associated with each item.

Used — An item that has been opened and may show signs of wear. All specific defects should be noted in the Comments section associated with each item.

Refurbished — A used item that has been renewed or updated and verified to be in proper working condition. Not necessarily completed by the original manufacturer.

New
Brand new. We distribute directly for the publisher. This paper is a study of the elastic problems on simply connected manifolds $M_n$ whose orthonormal frame bundle is a Lie ... group $G$. Such manifolds, called the space forms in the literature on differential geometry, are classified and consist of the Euclidean spaces $\mathbb{E}^n$, the hyperboloids $\mathbb{H}^n$, and the spheres $S^n$, with the corresponding orthonormal frame bundles equal to the Euclidean group of motions $\mathbb{E}^n\rtimes SO_n(\mathbb{R})$, the rotation group $SO_{n+1}(\mathbb{R})$, and the Lorentz group $SO(1,n)$.The manifolds $M_n$ are treated as the symmetric spaces $G/K$ with $K$ isomorphic with $SO_n(R)$. Then the Lie algebra $\mathfrak{g}$ of $G$ admits a Cartan decomposition $\mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with $\mathfrak{k}$ equal to the Lie algebra of $K$ and $\mathfrak{p}$ equal to the orthogonal comlement $\mathfrak{k}$ relative to the trace form. The elastic problems on $G/K$ concern the solutions $g(t)$ of a left invariant differential systems on $G$ $\textfrac{dg}{dt}(t)=g(t)(A_0+U(t)))$ that minimize the expression $\textfrac{1}{2}\int_0^T (U(t),U(t))\,dt$ subject to the given boundary conditions $g(0)=g_0$, $g(T)=g_1$, over all locally bounded and measurable $\mathfrak{k}$ valued curves $U(t)$ relative to a positive definite quadratic form $(\, , \,)$ where $A_0$ is a fixed matrix in $\mathfrak{p}$.These variational problems fall in two classes, the Euler-Griffiths problems and the problems of Kirchhoff. The Euler-Griffiths elastic problems consist of minimizing the integral $\textfrac{1}{2}\int_0^T\kappa^2(s)\,ds$ with $\kappa (t)$ equal to the geodesic curvature of a curve $x(t)$ in the base manifold $M_n$ with $T$ equal to the Riemannian length of $x$. The curves $x(t)$ in this variational problem are subject to certain initial and terminal boundary conditions. The elastic problems of Kirchhoff is more general than the problems of Euler-Griffiths in the sense that the quadratic form $(\, , \,)$ that defines th Read more Show Less

Ships from: Boonsboro, MD

Usually ships in 1-2 business days

  • Canadian
  • International
  • Standard, 48 States
  • Standard (AK, HI)
  • Express, 48 States
  • Express (AK, HI)
Page 1 of 1
Showing All
Close
Sort by

More About This Textbook

Overview

This paper is a study of the elastic problems on simply connected manifolds $M_n$ whose orthonormal frame bundle is a Lie group $G$. Such manifolds, called the space forms in the literature on differential geometry, are classified and consist of the Euclidean spaces $\mathbb{E}^n$, the hyperboloids $\mathbb{H}^n$, and the spheres $S^n$, with the corresponding orthonormal frame bundles equal to the Euclidean group of motions $\mathbb{E}^n\rtimes SO_n(\mathbb{R})$, the rotation group $SO_{n+1}(\mathbb{R})$, and the Lorentz group $SO(1,n)$. The manifolds $M_n$ are treated as the symmetric spaces $G/K$ with $K$ isomorphic with $SO_n(R)$. Then the Lie algebra $\mathfrak{g}$ of $G$ admits a Cartan decomposition $\mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with $\mathfrak{k}$ equal to the Lie algebra of $K$ and $\mathfrak{p}$ equal to the orthogonal comlement $\mathfrak{k}$ relative to the trace form. The elastic problems on $G/K$ concern the solutions $g(t)$ of a left invariant differential systems on $G$ $$\frac{dg}{dt}(t)=g(t)(A_0+U(t)))$$ that minimize the expression $\frac{1}{2}\int_0^T (U(t),U(t))\,dt$ subject to the given boundary conditions $g(0)=g_0$, $g(T)=g_1$, over all locally bounded and measurable $\mathfrak{k}$ valued curves $U(t)$ relative to a positive definite quadratic form $(\, , \,)$ where $A_0$ is a fixed matrix in $\mathfrak{p}$. These variational problems fall in two classes, the Euler-Griffiths problems and the problems of Kirchhoff. The Euler-Griffiths elastic problems consist of minimizing the integral $$\tfrac{1}{2}\int_0^T\kappa^2(s)\,ds$$ with $\kappa (t)$ equal to the geodesic curvature of a curve $x(t)$ in the base manifold $M_n$ with $T$ equal to the Riemannian length of $x$. The curves $x(t)$ in this variational problem are subject to certain initial and terminal boundary conditions. The elastic problems of Kirchhoff is more general than the problems of Euler-Griffiths in the sense that the quadratic form $(\, , \,)$ that defines the functional to be minimized may be independent of the geometric invariants of the projected curves in the base manifold. It is only on two dimensional manifolds that these two problems coincide in which case the solutions curves can be viewed as the non-Euclidean versions of L. Euler elasticae introduced in 174. Each elastic problem defines the appropriate left-invariant Hamiltonian $\mathcal{H}$ on the dual $\mathfrak{g}^*$ of the Lie algebra of $G$ through the Maximum Principle of optimal control. The integral curves of the corresponding Hamiltonian vector field $\vec{\mathcal{H}}$ are called the extremal curves. The paper is essentially concerned with the extremal curves of the Hamiltonian systems associated with the elastic problems. This class of Hamiltonian systems reveals a remarkable fact that the Hamiltonian systems traditionally associated with the movements of the top are invariant subsystems of the Hamiltonian systems associated with the elastic problems. The paper is divided into two parts. The first part of the paper synthesizes ideas from optimal control theory, adapted to variational problems on the principal bundles of Riemannian spaces, and the symplectic geometry of the Lie algebra $\mathfrak{g},$ of $G$, or more precisely, the symplectic structure of the cotangent bundle $T^*G$ of $G$. The second part of the paper is devoted to the solutions of the complexified Hamiltonian equations induced by the elastic problems. The paper contains a detailed discussion of the algebraic preliminaries leading up to $so_n(\mathbb{C})$, a natural complex setting for the study of the left invariant Hamiltonians on real Lie groups $G$ for which $\mathfrak{g}$ is a real form for $so_n(\mathbb{C})$. It is shown that the Euler-Griffiths problem is completely integrable in any dimension with the solutions the holomorphic extensions of the ones obtained by earlier P. Griffiths. The solutions of the elastic problems of Kirchhoff are presented in complete generality on $SO_3(\mathbb{C})$ and there is a classification of the integrable cases on $so_4(\mathbb{C})$ based on the criteria of Kowalewski-Lyapunov in their study of the mechanical tops. Remarkably, the analysis yields essentially only two integrables cases analogous to the top of Lagrange and the top of Kowalewski. The paper ends with the solutions of the integrable complex Hamiltonian systems on the $SL_2(\mathbb{C})\times SL_2(\mathbb{C})$, the universal cover of $SO_4(\mathbb{C})$.

Read More Show Less

Product Details

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)