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SUMMARY:Two New Developments concerning Noether's Two Theorems - Peter Olv
 er
DTSTART:20250317T160000Z
DTEND:20250317T170000Z
UID:TALK228268@talks.cam.ac.uk
CONTACT:Matthew Colbrook
DESCRIPTION:In her fundamental 1918 paper\, written whilst at Göttingen a
 t the invitation of Klein and Hilbert to help them resolve an apparent par
 adox concerning the conservation of energy in general relativity\, Emmy No
 ether proved two fundamental theorems relating symmetries and conservation
  laws of variational problems.  Her First Theorem\, as originally formulat
 ed\, relates strictly invariant variational problems and conservation laws
  of their Euler--Lagrange equations.  The Noether correspondence was exten
 ded by her student Bessel-Hagen to divergence invariant variational proble
 ms.  A key issue is when is a divergence invariant variational problem equ
 ivalent to a strictly invariant one.  Here\, I illustrate these issues usi
 ng a very basic example from her original paper\, and then highlight the r
 ole of Lie algebra cohomology in resolving this question in general.  This
  part includes some provocative remarks on the role of invariant variation
 al problems in the modern formulation of fundamental physics.\n \nNoether'
 s Second Theorem concerns variational problems admitting an infinite-dimen
 sional symmetry group depending on an arbitrary function.  I first recall 
 the two well-known classes of partial differential equations that admit in
 finite hierarchies of higher order generalized symmetries: 1) linear and l
 inearizable systems that admit a nontrivial point symmetry group\; 2) inte
 grable nonlinear equations such as Korteweg--de Vries\, nonlinear Schrödi
 nger\, and Burgers'.  I will then introduce a new general class: 3) underd
 etermined systems of partial differential equations that admit an infinite
 -dimensional symmetry algebra depending on one or more arbitrary functions
  of the independent variables.  An important subclass of the latter are th
 e underdetermined Euler--Lagrange equations arising from a variational pri
 nciple that admits an infinite-dimensional variational symmetry algebra de
 pending on one or more arbitrary functions of the independent variables.  
 According to Noether's Second Theorem\, the associated Euler--Lagrange equ
 ations satisfy Noether dependencies and are hence underdetermined and the 
 conservation laws corresponding to such symmetries are trivial\; examples 
 include general relativity\, electromagnetism\, and parameter-independent 
 variational principles.\n
LOCATION:Centre for Mathematical Sciences\, MR13
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