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<- [[:chaosbook]]

====== Chapter: Continuous symmetries ======

(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#continuous|Continuous symmetries]])  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-12//

Enter the latest posts at the bottom - flows better. 

===== Kuramoto-Sivashinski O(2) quotienting =====

==== Literature ====

{{:gtspring2009:siminos.png?23}} Two continuous symmetry reduction papers of interest:
{{:gtspring2009:research_projects:siminos:rowley_and_marsden_-_2000_-_reconstruction_equations_and_the_karhunen-loeve_ex.pdf|Rowley and Marsden 2000}} and 
{{:gtspring2009:research_projects:siminos:no3404.pdf|Rowley et.al. 2003}}. The procedure looks like a trick Arnol'd suggests in "Ordinary differential equations" book.
It works "locally" but how are the pieces glued together?  --- // 2009-02-22 16:20//

{{:gtspring2009:pc.png|}} can you give page number reference to Arnol'd "Ordinary differential equations" suggestion ?   --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-22 16:20//  {{:siminos.png?23}} $6.2 --- //[[siminos@gatech.edu|Evangelos Siminos]] 2009-02-24 14:49//

{{:gtspring2009:pc.png|}} Notes on {{:gtspring2009:research_projects:siminos:rowley_and_marsden_-_2000_-_reconstruction_equations_and_the_karhunen-loeve_ex.pdf|Rowley and Marsden 2000}}:

The main result is a method for the reconstruction of traveling KL modes from their corresponding symmetry-reduced modes by finding reconstruction equations for the relative equilibrium moving frame variable <latex>x-c t</latex>. This is applied to KL (or POD) long time averaged data, in cases where the average drifts with a constant velocity  <latex> c </latex>.

In
the //geometric phase literature// one gets dynamical equations on the phase
space modulo the symmetry group (these are called the //reduced equations// on the //reduced phase space//) and the
problem is then how to put back into the dynamics the missing group, or //phase variables//. These additional equations
are usually called the //reconstruction equations//.

{{:chaosbook:chaosbook:rowmar-slice.jpg|}} Whenever one has equivariant dynamics on //M//, one gets a well-defined dynamical system on the quotient (or orbit) space <latex>M/G</latex> which consists, in our case, of the space in which two functions related by a translation are
identified. When //M// is an inner product space and the group action is by isometries, there is a natural way to identify,
at least locally in function space, the quotient space with a subspace of //M//; namely we pick a point <latex>u_0 \in M </latex> and
look at the affine space through the point <latex>u_0</latex> orthogonal to the group orbit through that point. 
They call this affine
space a //slice// and denote it by <latex>S_{u_0}</latex>. This is precisely the same as Predrag's request that reduced equations be obtained by projecting out the component of velocity field that points along the group theory orbit.





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