lagrangian reduction - universidade do...

ROUTH REDUCTION AS AN EXAMPLE

OF LAGRANGIAN REDUCTION

KATARZYNA GRABOWSKA,

PAWET URBANSKI

arlkv : 1708.0976%1

CALCULATIONS IN COORDINATES

R"+m

> ( q ,x , qi , .x ) ->L( q ,× , .q , .x)eR2L

cyclic variables : q= 0

212R( q ,×,p , .x)= poi - L(q ,x,oj,x

.

) Jq=0

edward ]°nnpwu+np=¥q=0gifts ftp.t?j.3I=o

1831-1907 IP= a Ra( x. x. ) = aq.ca) - LA ,q .ca ) , .x )

IRouthian depends only on ( xii )

WHAT IS GEOMETRIC PICTURE BEHIND THESE CALCULATIONS ?

I E. Marsden ,TSRatiu , Jsoneurle , F. Cantrijn , B. Langerock ,T . Metsdag ,E. Garcia . Toriano Andre 's

,] . Vankevschaver

, Mcrampin . . .

WHY DO IT AGAIN ?

D .

BQ I do ,T*T*Q - TT*Q - T*TQ

0¥pj'cdHA*a))=D=xaYdLGaD# ° "T*Q TQ

a } + }I¥5u"itionEASY V INTERESTING

RELATION

? ??_?

??_?

? ?? ?

?µHats ? ? ?

TOOLS :

→ SYMPLECTIC GEOMETRY : SYMPLECTIC REDUCTIONS,

LAGRANGIAN

SUBMANIFOLDS,

GENERATING OBJECTS :

GENERATING FUNCTION FUNCTION ON

A SUBMANIFOLD

f :Q→R QaCf→R

df(Q)cT*Q { yeT*Q : VJETC ( df ,v>=< y ,v>]cT*QT LAGRANGIAN 7

SUBMANIFOLD

itGENERATING C={ MEM : drfcm) - 0 }FAMILY fM → R { yeT*Q : g*y(m)=dfCm) }etQ

SYMPLECTIC RELATIONS ARE LAGRANGIAN

SUBMANIFOLDS 11

g :B - Pz

graph (g) c(Pz×Pz , Wz - 02 )¥2,13

ARE COTANGENT BUNDLES

WE CAN GENERATE THE GRAPH

COMPOSING SYMPLECTIC RELATIONS

MEANS ADDING GENERATING OBJECTS

EXAMPLET * to # THTHQ SYMPLECTOMORPHISM

L :TQ→R GENERATED BY

DLCTQ )cT*TQ TQ×T*Q > TQ×aT*Q > ( rfp )1→ - ( p ,O ) ERT

LAGRANGIAN

SUBMANIFOLDWHAT GENERATES Va( OILCTQD ?

TQ×aT*Q scrip ) - LCV ) - ( p ,v ) e R* * Q

TOOLS :

→ GEOMETRY OF AFFINE VALUES : AV - BUNDLES,

AFFINE PHASE

SPACES

AFFINE PHASE SPACE :

QXR( T.it ) ~ ( 9. g) ⇐ > d(q . g) (g) =O

tgff-§¥r("

'"

drcq ) :=[ ( r ,qD

PZ={ OITCQ ) ] ← AFFINE BUNDLE OVER QSYMPLECTIC

/ [ MODELED ON T*QMANIFOLD

Wz=Z*Wa2 :P2→T*Q

g-FZ -7 TQ 15 A BUNDLE

ATIYAH ALGEBROID FZ = T #OF AFFINE VALUES

n

,VECTOR BUNDLE OVER Q

ROUTH REDUCTION RELATION

Q - CONFIGURATION MANIFOLD

X - VECTOR FIELD ON Q - SYMMETRY

L :TQ→lR dtX.lyGEOMETRIC VERSION

OF A LAGRANGIAN WITHCYCLIC VARIABLE

YIN FACT

,WHAT IS IMPORTANT IS THE DISTRIBUTION

0×=< 04×7IT IS ONE DIMENSIONAL DISTRIBUTION

,DOUBLE

VECTOR SUBBUNDLE OF

TTQ

Lead7¥aTQ

ROUTH REDUCTION RELATION VIA HAMILTONIAN MECHANICS

8-+ * t * a

#TT*Q ¥

T*TQy)o, ,T*QyaTQ-7112 ( p ,v ) HLA )

- ✓

to

T¥Q

→ IN HAMILTONIAN MECHANICS SYMMETRY is GIVEN BY

dt*X - COTANGENT LIFT OF XTHAMILTONIAN

VECTOR FIELD FOR T*Q2pl-)i×(p)=(p , X(Ta( p ) ) )→ PHASE TRAJECTORIES LIE IN LEVEL SETS OF I×

c×={ PET * Q : i×(p)=d } ~ WISOTROPIC SUBMANIFOLD , REDUCTIONWITH RESPECT TO Ca = DIVIDING BYT AFFINE SUBBUNDLE OFT*Q→Q LIFTED ( R ,t ) ACTION

WTH MODEL Co → Q

if =L MANIFOLD OF+ * Qsca -7 Pa Q→M=Q/p TRAJECTORIES

-\

/ OF X) gympllAFFINE BUNDLE OVER M =CT1c MANIFOLDMODELED ON T*M REDUCED AFFINE PHASE

SPACE

BQ XQT*T*Q c- TT*Q - T*TQ

µt*g HalB

✓

TQ

txpa c- TB - ? ? ?( np.ec , , . .eu , ,oµ gene ... ,gx . , . - aBY A FUNCTION EQUAL ZERO ON

T*Q×P.

> Caxpa f. ( Pitt > LG ) - < p ,v )Pa


HERE COMES AV - GEOMETRY !

Q×R : ( q ,v)~ ( yscq ) , ntsx )R ' 9 ' "

y |=nQFLOW Ot X Ys

Z&e= Q×lR/~ Z&→M[ qiiftt :=[q ,n+t]=[ys( g) irtttsa ]

BEPZ , Lv

BTxp c- TB - PFZa ×

WHAT is it ? ?

FZ &= TZ%pp) ATIYAH ALGEBROID

TZ,

→ TM AV - BUNDLEg-

FUNDAMENTAL VECTOR

FIELD OF R - ACTIONWETZ

. [w]ts= [ wts } ] ON Z×

PFZ.

- AFFINE DIFFERENTIALS OF SECTIONS OF FZ,

→ TM

ROUTHIAN is NOT A FUNCTION ONTM BUT A SECTIONOF AN AV - BUNDLE

AFFINE ANALOG OF A MAP T*QXaTQ 3 ( p ,v ) - ) ( p ,v > EIR

PZ, XMTM a ( p ,v ) 1- e FZ ,

>

2,

ftp=dr( m ) < pie > =[ Trcv ) ]

mm

✓

B a

T*pZa-

TPZ. - PFZ

,

-Txtpzaepfza

SYMTPLECTOMORPHISMGENERATED BY A SECTION OVER A

SUBMANIFOLD

Pzaxtmo Pzxmtm - FZ , ( p . ,w ) - > < Pa ,w >

C&×aTQ → R

TM×mCx×aTQ

- FZ×

| ( Rr ) - La - |v .P tm ( w ,P,o)1→ Le )

- + Cpa ,w >a

TTGENERATING FAMILY

WITH PARAMETERS PIV. .

BQ XQT*T*Q c- TT*Q - T*TQ

to t ? ?B aTxpz- TPZ. - PFZ×

QUESTIONS :

2.

15 IT POSSIBLE to g|µpL , ,=y TMXMCYQTQ

- FZ×

y Fm ( w ,p,o)l→LH- + Cpa ,w >

YES, BUT NOT ALWAYS TO ONE ROUTHIAN

SECTION → EXAMPLE

2. 15 IT POSSIBLE TO GO DIRECTLY FROM T*TQ TO PFZ× ?^

,YES

,BUT YO HAVE TO INCLUDE VALUES

OF GENERATING OBJECTS

↳LAGRANGIAN REDUCTION

ANSWER TO QUESTION 1 WITH AN EXAMPLE :

TQ -

FZ×TM×mG×aTQ - FZ×/ → / vi→L(v ) - + La ) - < pie > + Cpa ,w > TM

TQ 20h > WETM

FORMULA DOES NOT DEPEND ON P ,ANY PEG OVER to ( v ) IS OK !

EXAMPLE :

Q=R2a(×,y ) L :TQ > ( x ,y ,x.

,j ) 1- x. y.

-

y'

er M=R⇒( y )

X=2*f PARAMETER

ROUTHIAN FAMILY READS ( y , .x , g.) 1- x. y.

-

y'

. xx.

ER

T*Tm > { ( yijioyb ) : j=& , a= -2g }TT*Mo{ ( yip , jip ) : j=x , p= -2g } Xn(y,p)=&3J

- 2yFp

h(y,p)=Lp+y2

ANSWER TO QUESTION 2

RTHE RELATION T*TQ - DPFZ

,15 THE AFFINE PHASE LIFT OF THE

REDUCED TANGENT RELATION F(Q×lR ) → 72£ OF THE PROJECTION

Q×1R→Z:QXIR - Za ( q ,r)1→[q ,v]={ ( yscq ) , Ntsd ) } AV - BUNDLE MORPHISMTsaFCQXIRFTQHR - FZ , ( v ,t)i→[yt]={ ( Tyscv )+sX( q ) , ttxs ) ]AV - BUNDLE MORPHISM

ROUTH REDUCTION R IS PF }× . IT IS GENERATED BYNONTRIVIAL SECTION OF CERTAIN AV - BUNDLE

T*T*Q-D T* PZ,

PHASE LIFT GENERATED

T*Q3G→ PZ , ✓ By FUNCTON ZERO ON CLXMPZL

FINAL COMMENT ON ( NOT REALLY ) A GENERALISATION

LAGRANGIAN WITH SYMMETRY of,

X.L=O

DTX -3 OECDTX ) DLCTQ) c 0×05

weather.INT?hNesaur..3YIsEEutoINt0tva#

↳ONE CAN SHOW THAT IT DEFINES X UP TO

MULTIPLICATION BY NUMBERS

NO DISTINGUISHED PARAMETERIZATION OF {Cd } BY x. . .

lagrangian reduction - universidade do...

Documents