Integration of tensor fields

Transcription

1 Integrtion of tensor fields V. Retsnoi Abstrct. The im of this pper is to introduce the ide of integrtion of tensor field s reerse process to the Lie differentition. The definitions of indefinite nd definite integrls for tensor fields re similr to the nlogous definitions for integrble functions in undergrdute differentil clculus. In our definition the definite integrl of function is lso function not number nd the definite integrl of generl tensor field is lso tensor field of the sme type. A few geometricl exmples included in the text clrify the topic being discussed. M.S.C. 1: 34A3 37C1 16W5. Key words: Vector field; flow; Lie deritie; integrtion of tensor field. 1 Introduction One of the most significnt tools in differentil geometry nd globl nlysis in continuous enironment mechnics nd dynmicl systems is the notion of ector field. Let us list some bsic concepts relted to ector fields which we use throughout this rticle: trjectories nd flows interction of flows phse portrit drgging of tensor fields including functions ector fields nd differentil forms [] long flow Lie derities nd integrtion of tensor fields. As it ws mentioned in the Abstrct the min purpose of the present pper is to introduce the ide of integrtion of tensor field s reerse process to the Lie differentition. In we recll some well-known fcts bout Lie derities on mnifold in the form conenient in wht follows. It is known tht the min property of the Lie deritie is its independence of chnge of coordintes. The Lie differentition technique is deeloped in [1] where computtion formuls re deried in nonholonomic bsis. The nonholonomy object J.A. Schouten [3] ppering in clcultion formuls is consequence of intercting of non-commuting bsis opertors nd llows to pply this technique to the theory of Lie groups. In prticulr the structure constnts re precisely the nonholonomy object of the left- or right-inrint bsis in Lie group. In 3 nd 4 we define indefinite nd definite integrls of tensor field nd explin the geometricl mening of integrls by simple exmples. In prticulr in 4 re clculted the integrls of shift nd rottion opertors in the flows of rottions on the plne nd in the spce. BSG Proceedings Vol. 13 pp c Blkn Society of Geometers Geometry Blkn Press 13.

2 Integrtion of tensor fields 97 Lie deritie on mnifolds Let M be smooth n-dimensionl mnifold nd let X be smooth ector field 1 on M. Therefore flow t exp tx s one-prmeter group of trnsformtions of M is ssocited with X. Choosing locl coordintes u i i 1... n on neighborhood U M the flow t is determined by the system of first-order ordinry differentil equtions ODEs.1 u i x i u where the prime denotes the differentition with respect to prmeter t nd x i re components of the ector field X t point u U. More precisely the flow t is locl pseudogroup of locl trnsformtions of M becuse the theorem of uniqueness nd existence of solutions of the system.1 hs locl chrcter. Such reltion between the locl nd the globl should be kept in mind. In the flow t points moe long own trjectories nd functions re drgged ccording to the composition lw: u u t t u f f t f t. Moreoer ech tensor field S of generl type p q on M my be drgged long by the flow t for ech lue of t to define one-prmeter fmily of tensor fields indicted by the bbreition S S t â t S. Then the deritie of this fmily with respect to t defines the Lie deritie of S by. S S t S L X S lim. t t The tensor fields S nd S re of the sme type. The Lie deritie of S with respect to X is usully denoted by the symbol L X S. But for the ske of conenience we use primes in order to denote the Lie deritie with respect to the fixed ector field X. The Lie deritie long the ector field X of function f on M is defined so tht it is the ordinry deritie long X i.e..3 f Xf f t 1 t lim t t f t f. The Lie deritie long the ector field X of nother ector field Y on M is defined by the Lie brcket.4 L X Y [X Y ]. Gien locl coordintes u i on some neighborhood U M the Lie deritie of the ector field Y u y i long X i u x i is defined by.4 in coordinte nottion: i.5 L X Y [X Y ] u i Xyi Y x i. 1 The smoothness of functions ector fields nd ny tensor fields mens tht the relent objects occurring will be ssumed to be differentible of sufficiently high clss C p or if it is necessry een C or C ω. It is ssumed tht tensor fields re sufficiently smooth so tht derities cn be tken. We use the Einstein summtion i.e. the conention tht repeted indices re implicitly summed oer. Any index tht is to be summed oer we write in the upper position.

3 98 V. Retsnoi 3 Integrtion of tensor fields In the preious section we he defined the Lie deritie of tensor field long flow t exp tx of ector field X. Anlogously one cn spek bout n integrtion of tensor fields. In prticulr we need to recoer tensor field from its known Lie deritie with respect to the ector field X. Definition 3.1. The indefinite integrl of function f t see.3 with respect to the prmeter t is defined s the set of ll ntiderities of f t long the flow t of X symbolized by 3.1 f tdt f t + f where f is n inrint of X i.e. Xf. Definition 3.. The definite integrl of f t on closed interl [ b] is defined by the Newton Leibniz formul 3. f tdt b f t fb f. If in 3.1 nd 3. f is tensor field then long with the Lie differentition one cn spek bout n integrtion of tensor fields long the flow of X. Let S nd Q be smooth tensor fields of the sme type on M. Definition 3.3. A tensor field Q is sid to be n ntideritie of S long the flow t of X if Q L X Q S. Let Q 1 nd Q be tensor fields of the sme type nd suppose one of them is n ntideritie of S. Then the second one is n ntideritie of S if nd only if Q 1 Q Q where Q is n inrint tensor field long the flow of X i.e. L X Q. Definition 3.4. The indefinite integrl of the tensor field S with respect to t is defined s the set of ll ntiderities of S long the flow t of X symbolized by 3.3 S t dt Q t + Q where Q is n ntideritie of S nd L X Q. The next Proposition reltes the integrtion nd the Lie differentition of tensor fields. Proposition 3.1. Let Q be n ntideritie of S long the flow t of X nd suppose S is continuous on closed interl [ b]. Then the definite integrl of S is defined by 3.4 S t dt Q b Q.

4 Integrtion of tensor fields 99 Proof. Let the closed interl [ b] be prtitioned by points t < t 1 <... < t i 1 < t i < t i+1 <... < t n 1 < t n b. Then the definite integrl of S is defined by tking the limit of the sum S t dt lim mx t i i1 n S ξi t i where S ξi is the lue of S t n rbitrry point ξ i t i 1 t i nd t i t i t i 1 is the length of the subinterl i 1... n. According to the men lue theorem there is one point ξ i in ech open interl t i 1 t i such tht We he which cn be rewritten s S ξi t i Q ti Q ti 1. Q b Q n Q ti Q ti 1 i1 3.5 Q b Q n S ξi t i. Then tking the limit of the sum in the right-hnd side of 3.5 s n we obtin 3.4. Let Y be differentible ector field on M. Then.4 nd 3.4 yield i1 3.6 [X Y ] t dt Y b Y. 4 Geometricl exmples Exmple 4.1. Let us consider the liner ector field X y + x y on the xy plne. The flow t of X is uniform circulr motion round the origin: t : x y x cos t y sin t y cos t + x sin t. The indefinite integrl of function long the flow t is defined by 3.1 where f f I is function of the inrint I x + y of X. From 3.3 it follows tht the indefinite integrl of ector field [X Y ] t is of the form [X Y ] t dt Y t + Y

5 1 V. Retsnoi where Y is differentible ector field on the xy plne nd Y is of the form According to the condition Y ζx y + ηx y y. [X Y ] Xζ + η + Xη ζ y the functions ζ nd η must stisfy the system of liner ODEs { ζ + ζ η + η where prime denotes the deritie with respect to X. Suppose two functions f x nd g y be gien on the xy plne. The drggings of these functions nd the function f + g x + y long the flow of X re described by f t x t g t y t nd f + g t x + y t respectiely. Let us clculte the corresponding definite integrls on the closed interl [ b] he f tdt g tdt f + g tdt x tdt x t y tdt y t x y x y x t + y t dt x t + y t [ y. ]. By 3. we Consider the ector field Y. The Lie derities of Y with respect to X re y Y [X Y ] Y [X Y ] y Y. Thus we he Y + Y nd the drgging of Y long the flow of X is described by the ector-function Y t T t Y sin t + cos t y. Then using 3.6 we obtin the definite integrl of the field Y [ interl ] : t dt Y t y where cos t t sin t y. on the closed The Figures 1 3 illustrte the mening of the definite integrl of ector field on the 1st qurter of the xy plne.

6 Integrtion of tensor fields 11 y y Y y X Y x x x Figure 1: The flow of X is the uniform circulr motion round the origin in the counterclockwise direction. The Lie deritie of Y south wind with respect to X is the y field Y west wind. X y y x y x x Figure : The field Y is rotted in moing frme ccording to the lw Y t Y cos t + Y sin t the wind chnges own direction rotting clockwise. The clculting of definite integrl [X Y ] t dt yields the field north-west wind. y 9 Y t i-1 t i -1 x i t i Y xi Y ti Y t i-1 Y xi Y ti Figure 3: The summnds for the integrl sum re defined by the men lue theorem. Tking the limit of the integrl sum we obtin the closing line to the hodogrph of Y t.

7 1 V. Retsnoi The hodogrph is the plot of the elocity s function of time. The hodogrph of the ector-function Y t hs the sme trjectory s X but with opposite direction. The integrl sum Y ξ i t i is broken line to the hodogrph nd the integrl y is stright line closing this broken line see Figure 3. Remrk 4.. Note tht the formuls in the Exmple 4.1 cn be obtined s follows. Consider the drgging of the nturl frme y nd cofrme dx dy long the flow of X y + x y. The derition formuls of the frme nd cofrme re y y 1 dx 1 dx dy 1 dy 1 where C is the Jcobi mtrix of the components of X. The mtrix C in 1 the flow of X remins constnt nd in the corresponding tngent spce the flow T t is determined by the exponentil of the mtrix Ct: cos t sin t 4. y t y sin t cos t dx cos t sin t dx. dy sin t cos t dy t The equlities 4.1 cn be considered s ODEs in mtrix nottions nd the corresponding solutions re presented by 4.. Thus the definite integrls on the closed interl [ ] of the corresponding ector fields nd 1-forms cn be clculted s follows: y dx dy dx dy y dx dy Exmple 4.3. Let three ector fields X z y y z y Y z + x z dx dy. 1 1 y 1 1 Z y x y be gien in the spce xyz. The flows of X Y nd Z re rottions bout three xes x y nd z respectiely. Let us consider the drgging of Y long the flow of X the drgging of Z long the flow of Y nd the drgging of X long the flow of Z: Y [X Y ] Z Y + Y Y t Y cos t + Z sin t Z [Y Z] X Z + Z Z t Z cos t + X sin t X [Z Y ] Y X + X X t X cos t + Y sin t.

8 Integrtion of tensor fields 13 Let us clculte the integrls of X Y nd Z on closed interl [ b]: Z t dt X t dt Y t dt [X Y ] t dt Y b Y sin b Y sin + b Z cos + b [Y Z] t dt Z b Z sin b Z sin + b X cos + b [Z X] t dt X b X sin b X sin + b Z cos + b. Tking nd b we obtin three ector fields Z t dt y + z x y x z X t dt y + x + z y y z Y t dt y x + z y + y z. The flow of the field 4.3 is x y z x t x cos t + y + z sin t y t y x sin t z t z x sin t y + z 1 cos t y + z 1 cos t. From the equlities y t z t y z nd x t + y t + z t x + y + z we obtin two inrints I 1 x + y + z I y z. It mens tht the leel { surfces of the trjectories of the field 4.3 re elliptic cylinders y + z with xis of rottion. The trjectories re ellipses on the intersections x of the cylinders I 1 c > with plnes I c perpendiculr to the xis of rottion.

9 14 V. Retsnoi The flow of the field 4.4 is x y z x t x y sin t x + z 1 cos t y t y cos t + x + z sin t z t z y sin t x + z 1 cos t nd the inrints re I 1 y + x + z I x z. The leel surfces { of the trjectories of the field 4.4 re elliptic cylinders with xis x + z of rottion. The trjectories re ellipses on the intersections of the y cylinders I 1 c > with plnes I c perpendiculr to the xis of rottion. From Y t dt X t dt it follows tht the flows nd inrints of the fields 4.4 nd 4.5 re the sme but the trjectories of these fields re opposite directed. In the Exmple 4.1 the definite integrl on the xy plne is stright line tht closing the integrl sum Y ξ i t i see Figure 3. In the xyz spce we he nlogous sitution but the closing line is prt of n ellipse. Acknowledgements. This reserch is prtilly supported by the Estonin Ministry of Eduction nd Reserch trget finnce grnt SF1839s8 nd by the Estonin Doctorl School in Mthemtics nd Sttistics. References [1] Gh. Atnsiu V. Bln N. Brinzei M. Rhul Differentil Geometricl Structures: Tngent bundles Connections in the Bundles Exponentil Lw in Jet Spce in Russin URSS Moscow 9. [] S. Kobyshi K. Nomizu Foundtions of Differentil Geometry Vol.I Interscience Publishers New York-London [3] J. A. Schouten D. J. Struik Einführung in die Neuren Methoden der Differentilgeometrie Noordhoff Groningen Russin trnsl.: Introduction to New Methods of Differentil geometry GONTI M.-L Author s ddress: Vitlij Retsnoi TTK Uniersity of Applied Sciences Centre of Rel Sciences 6 Pärnu mnt Str Tllinn Estoni. E-mil: itli@ut.ee