Spartan Daily February 9, 2011

Volume 136, Issue 7https://scholarworks.sjsu.edu/spartandaily/1114/thumbnail.jp

Spartan Daily May 9, 2011

Volume 136, Issue 51https://scholarworks.sjsu.edu/spartandaily/1158/thumbnail.jp

Spartan Daily March 9, 2011

Volume 136, Issue 23https://scholarworks.sjsu.edu/spartandaily/1130/thumbnail.jp

9/9/2011

https://surface.syr.edu/scis_news/1102/thumbnail.jp

Boston University Music Organizations, December 9, 2011

This is the concert program of the Boston University Music Organizations performance on Friday, December 9, 2011 at 8:00 p.m., at the Boston University Concert Hall, 855 Commonwealth Avenue, Boston, Massachusetts. Works performed were Toccata Marziale by Ralph Vaughan Williams, Hammersmith by Gustav Holst, Festivo by Edward Gregson, Flag of Stars by Gordon Jacob, Adagio K. 488 by Wolfgang Amadeus Mozart, The Hebrides Overture, Op. 26 by Felix Mendelssohn, Fantasia on a Theme of Thomas Tallis by Ralph Vaughan Williams, and Egmont Overture by Ludwig van Beethoven. Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

On the Integrability of Tonelli Hamiltonians

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n-dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the size of its Mather and Aubry sets. As a byproduct we point out the existence of non-trivial common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one.Comment: 19 pages. Version accepted by Trans. Amer. Math. So