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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.


{{:gtspring2009:siminos.png?23}} This is not precisely the same as Predrag's request. The slice in their case is fixed and this eventually leads to failure when the slice becomes parallel to the group orbit, see appendix A2 on limitations. They propose to choose a new slice if this happens, while I think Predrag's request is to choose a new slice at each timestep. Another issue is numerical efficiency and stability in integrating the PDE which they don't discuss.  --- //[[siminos@gatech.edu|Evangelos Siminos]] 2009-02-24 14:49//




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