THE GEOMETRY OF HAMILTON AND LAGRANGE SPACES

By BARBARA LEE

CONTENTS 1 THE GEOMETRY OF TANGENT BUNDLE THE MANIFOLD TM HOMOGENEITY SEMISPRAYS ON THE MANIFOLD NONLINEAR CONNECTIONS THE STRUCTURES D-TENSOR ALGEBRA N-LINEAR CONNECTIONS TORSION AND CURVATURE PARALLELISM STRUCTURE EQUATIONS 2 FINSLER SPACES FINSLER METRICS GEOMETRIC OBJECTS OF THE SPACE GEODESICS CANONICAL SPRAY CARTAN NONLINEAR CONNECTION METRICAL CARTAN CONNECTION PARALLELISM STRUCTURE EQUATIONS REMARKABLE CONNECTIONS OF FINSLER SPACES SPECIAL FINSLER MANIFOLDS ALMOST KÄHLERIAN MODEL OF A FINSLER MANIFOLD THE NOTION OF LAGRANGE SPACE VARIATIONAL PROBLEM EULER–LAGRANGE EQUATIONS CANONICAL SEMISPRAY NONLINEAR CONNECTION HAMILTON–JACOBI EQUATIONS THE STRUCTURES AND OF THE LAGRANGE SPACE THE ALMOST KÄHLERIAN MODEL OF THE SPACE METRICAL N–LINEAR CONNECTIONS GRAVITATIONAL AND ELECTROMAGNETIC FIELDS THE LAGRANGE SPACE OF ELECTRODYNAMICS GENERALIZED LAGRANGE SPACES 4 THE GEOMETRY OF COTANGENT BUNDLE THE BUNDLE THE POISSON BRACKETS THE HAMILTONIAN SYSTEMS HOMOGENEITY NONLINEAR CONNECTIONS DISTINGUISHED VECTOR AND COVECTOR FIELDS THE ALMOST PRODUCT STRUCTURE THE METRICAL STRUCTURE THE ALMOST COMPLEX STRUCTURE D-TENSOR ALGEBRA N-LINEAR CONNECTIONS TORSION AND CURVATURE THE COEFFICIENTS OF AN N-LINEAR CONNECTION THE LOCAL EXPRESSIONS OF D-TENSORS OF TORSION AND CURVATURE PARALLELISM HORIZONTAL AND VERTICAL PATHS STRUCTURE EQUATIONS OF AN N-LINEAR CONNECTION BIANCHI IDENTITIES 5 HAMILTON SPACES THE SPACES N–METRICAL CONNECTIONS IN THE N–LIFT OF HAMILTON SPACES CANONICAL NONLINEAR CONNECTION OF THE SPACE THE CANONICAL METRICAL CONNECTION OF HAMILTON SPACE STRUCTURE EQUATIONS OF BIANCHI IDENTITIES PARALLELISM HORIZONTAL AND VERTICAL PATHS THE HAMILTON SPACES OF ELECTRODYNAMICS THE ALMOST KÄHLERIAN MODEL OF AN HAMILTON SPACE 6 CARTAN SPACES THE NOTION OF CARTAN SPACE PROPERTIES OF THE FUNDAMENTAL FUNCTION K OF CARTAN SPACE CANONICAL NONLINEAR CONNECTION OF A CARTAN SPACE THE CANONICAL METRICAL CONNECTION STRUCTURE EQUATIONS BIANCHI IDENTITIES SPECIAL N-LINEAR CONNECTIONS OF CARTAN SPACE SOME SPECIAL CARTAN SPACES PARALLELISM IN CARTAN SPACE HORIZONTAL AND VERTICAL PATHS THE ALMOST KÄHLERIAN MODEL OF A CARTAN SPACE 7 THE DUALITY BETWEEN LAGRANGE AND HAMILTON SPACES THE LAGRANGE-HAMILTON DUALITY – DUAL NONLINEAR CONNECTIONS – D U A L D–CONNECTIONS THE FINSLER–CARTAN –DUALITY BERWALD CONNECTION FOR CARTAN SPACES LANDSBERG AND BERWALD SPACES LOCALLY MINKOWSKI SPACES APPLICATIONS OF THE -DUALITY 8 SYMPLECTIC TRANSFORMATIONS OF THE DIFFERENTIAL GEOMETRY OF CONNECTION-PAIRS ON COTANGENT BUNDLE SPECIAL LINEAR CONNECTIONS ON THE HOMOGENEOUS CASE F -RELATED CONNECTION-PAIRS F-RELATED CONNECTIONS THE GEOMETRY OF A HOMOGENEOUS CONTACT TRANSFORMATION EXAMPLES 9 THE DUAL BUNDLE OF A K-OSCULATOR BUNDLE THE BUNDLE THE DUAL OF THE 2–OSCULATOR BUNDLE DUAL SEMISPRAYS ON HOMOGENEITY NONLINEAR CONNECTIONS DISTINGUISHED VECTOR AND COVECTOR FIELDS LIE BRACKETS EXTERIOR DIFFERENTIALS THE ALMOST PRODUCT STRUCTURE THE ALMOST CONTACT STRUCTURE THE RIEMANNIAN STRUCTURES ON THE D–TENSOR ALGEBRA N-LINEAR CONNECTIONS TORSION AND CURVATURE THE COEFFICIENTS OF AN N-LINEAR CONNECTION THE H-, COVARIANT DERIVATIVES IN LOCAL ADAPTED BASIS RICCI IDENTITIES THE LOCAL EXPRESSIONS OF CURVATURE AND TORSION PARALLELISM OF THE VECTOR FIELDS ON THE MANIFOLD STRUCTURE EQUATIONS OF AN N–LINEAR CONNECTION 11 GENERALIZED HAMILTON SPACES OF ORDER THE SPACES METRICAL CONNECTIONS IN –SPACES THE LIFT OF A GH–METRIC EXAMPLES OF SPACES 12 HAMILTON SPACES OF ORDER THE SPACES CANONICAL PRESYMPLECTIC STRUCTURES AND CANONICAL POISSON STRUCTURES LAGRANGE SPACES OF ORDER TWO VARIATIONAL PROBLEM IN THE SPACES LEGENDRE MAPPING DETERMINED BY A SPACE LEGENDRE MAPPING DETERMINED BY CANONICAL NONLINEAR CONNECTION OF THE SPACE CANONICAL METRICAL N CONNECTION OF SPACE THE HAMILTON SPACES OF ELECTRODYNAMICS 13 CARTAN SPACES OF ORDER –SPACES CANONICAL PRESYMPLECTIC STRUCTURE OF SPACE CANONICAL NONLINEAR CONNECTION OF CANONICAL METRICAL CONNECTION OF SPACE PARALLELISM OF VECTOR FIELDS STRUCTURE EQUATIONS OF RIEMANNIAN ALMOST CONTACT STRUCTURE OF A SPACE CHAPTER THE MANIFOLD TM LET M BE A REAL DIFFERENTIABLE MANIFOLD OF DIMENSION N A POINT OF M WILL BE DENOTED BY X AND ITS LOCAL COORDINATE SYSTEM BY THE INDICES I, J, RUN OVER SET {1, , N} AND EINSTEIN CONVENTION OF SUMMARIZING IS ADOPTED ALL OVER THIS BOOK THE TANGENT BUNDLE OF THE MANIFOLD M CAN BE IDENTIFIED WITH THE 1-OSCULATOR BUNDLE SEE THE DEFINITION BELOW INDEED, LET US CONSIDER TWO CURVES HAVING IMAGES IN A DOMAIN OF LOCAL CHART WE SAY THAT AND HAVE A ”CONTACT OF ORDER 1” OR THE ”SAME TANGENT LINE” IN THE POINT IF: AND FOR ANY FUNCTION THE GEOMETRY OF HAMILTON & LAGRANGE SPACES THE RELATION ”CONTACT OF ORDER 1” IS AN EQUIVALENCE ON THE SET OF SMOOTH CURVES IN M, WHICH PASS THROUGH THE POINT LET BE A CLASS OF EQUIVALENCE IT WILL BE CALLED A ”1–OSCULATOR SPACE” IN THE POINT THE SET OF 1–OSCULATOR SPACES IN THE POINT WILL BE DENOTED BY AND WE PUT ONE CONSIDERS THE MAPPING DEFINED BY CLEARLY, IS A SURJECTION THE SET IS ENDOWED WITH A NATURAL DIFFERENTIABLE STRUCTURE, INDUCED BY THAT OF THE MANIFOLD M, SO THAT IS A DIFFERENTIABLE MAPPING IT WILL BE DESCRIBED BELOW THE CURVE IS ANALYTICALY REPRESENTED IN THE LOCAL CHART BY TAKING THE FUNCTION F FROM (1 1), SUCCESIVELY EQUAL TO THE COORDINATE FUNCTIONS THEN A REPRESENTATIVE OF THE CLASS IS GIVEN BY THE PREVIOUS POLYNOMIALS ARE DETERMINED BY THE COEFFICIENTS HENCE, THE PAIR WITH IS A LOCAL CHART ON M THUS A DIFFERENTIABLE ATLAS OF THE DIFFERENTIABLE STRUCTURE ON THE MANIFOLD M DETERMINES A DIFFERENTIABLE ATLAS ON AND THEREFORE THE TRIPLE ( M) IS A DIFFERENTIABLE BUNDLE BASED ON THE EQUATIONS (1 2) WE CAN IDENTIFY THE POINT WITH THE TANGENT VECTOR CONSEQUENTLY, WE CAN INDEED IDENTIFY THE 1–OSCULATOR BUNDLE WITH THE TANGENT BUNDLE (TM, M) BY (1 2) A TRANSFORMATION OF LOCAL COORDINATES ON THE MANIFOLD TM IS GIVEN BY CH 1 THE GEOMETRY OF TANGENT BUNDLE ONE CAN SEE THAT TM IS OF DIMENSION 2N AND IS ORIENTABLE MOREOVER, IF M IS A PARACOMPACT MANIFOLD, THEN TM IS PARACOMPACT, TOO LET US PRESENT HERE SOME NOTATIONS A POINT WHOSE PROJECTION BY IS X, I E WILL BE DENOTED BY (X,Y), ITS LOCAL COORDINATES BEING WE PUT WHERE {0} MEANS THE NULL SECTION OF THE COORDINATE TRANSFORMATION (1 3) DETERMINES THE TRANSFORMATION OF THE NATURAL BASIS OF THE TANGENT SPACE TM AT THE POINT THE FOLLOWING: BY MEANS OF (1 3) WE OBTAIN LOOKING AT THE FORMULA (1 4) WE REMARK THE EXISTENCE OF SOME NATURAL OBJECT FIELDS ON E FIRST OF ALL, THE TANGENT SPACE TO THE FIBRE IN THE POINT IS LOCALLY SPANNED BY THEREFORE, THE MAPPING V : PROVIDES A REGULAR DISTRIBUTION WHICH IS GENERATED BY THE ADAPTED BASIS (I = 1, , N) CONSEQUENTLY, V IS AN INTEGRABLE DISTRIBUTION ON TM V IS CALLED THE VERTICAL DISTRIBUTION ON TM TAKING INTO ACCOUNT (1 3), (1 4), IT FOLLOWS THAT IS A VERTICAL VECTOR FIELD ON TM, WHICH DOES NOT VANISH ON THE MANIFOLD IT IS CALLED THE LIOUVILLE VECTOR FIELD THE EXISTENCE OF THE LIOUVILLE VECTOR FIELD IS VERY IMPORTANT IN THE STUDY OF THE GEOMETRY OF THE MANIFOLD TM LET US CONSIDER THE –LINEAR MAPPING THEOREM 1 1 1 THE FOLLOWING PROPERTIES HOLD: 1° J IS GLOBALLY DEFINED ON TM 2° THE GEOMETRY OF HAMILTON & LAGRANGE SPACES 3° J IS AN INTEGRABLE STRUCTURE ON E 4° THE PROOF CAN BE FOUND IN [113] WE SAY THAT J IS THE TANGENT STRUCTURE ON E THE PREVIOUS GEOMETRICAL NOTIONS ARE USEFUL IN THE NEXT SECTIONS OF THIS BOOK 1 2 HOMOGENEITY THE NOTION OF HOMOGENEITY OF FUNCTIONS WITH RESPECT TO THE VARIABLES IS NECESSARY IN OUR CONSIDERATIONS BECAUSE SOME FUNDAMENTAL OBJECT FIELDS ON E HAVE THE HOMOGENEOUS COMPONENTS IN THE OSCULATOR MANIFOLD A POINT HAS A GEOMETRICAL MEANING, I E CHANGING OF PARAMETRIZATION OF THE CURVE DOES NOT CHANGE THE SPACE TAKING INTO ACCOUNT THE AFFINE TRANSFORMATIONS OF PARAMETER WE OBTAIN THE TRANSFORMATION OF COORDINATES OF IN THE FORM THEREFORE, THE TRANSFORMATIONS OF COORDINATES (1 3) ON THE MANIFOLD E PRESERVE THE TRANSFORMATIONS (2 2) LET US CONSIDER THE GROUP OF HOMOTETIES OF REAL NUMBERS FIELD R H ACTS AS AN UNIPARAMETER GROUP OF TRANSFORMATIONS ON E AS FOLLOWS WHERE IS THE POINT CONSEQUENTLY, H ACTS AS A GROUP OF TRANSFORMATIONS ON TM, WITH THE PRESERVING OF THE FIBRES THE ORBIT OF A POINT IS GIVEN BY THE TANGENT VECTOR TO ORBIT IN THE POINT IS GIVEN BY CH 1 THE GEOMETRY OF TANGENT BUNDLE THIS IS THE LIOUVILLE VECTOR FIELD IN THE POINT U0 NOW WE CAN FORMULATE: DEFINITION 1 2 1 A FUNCTION DIFFERENTIABLE ON AND CONTINUOUS ON THE NULL SECTION IS CALLED HOMOGENEOUS OF DEGREE R, ON THE FIBRES OF TM, (BRIEFLY: R–HOMOGENEOUS WITH RESPECT TO ) IF: THE FOLLOWING EULER THEOREM HOLDS [90], [106]: THEOREM 1 2 1 A FUNCTION DIFFERENTIABLE ON AND CONTINUOUS ON THE NULL SECTIONS IS HOMOGENEOUS OF DEGREE R ON THE FIBRES OF TM IF AND ONLY IF WE HAVE BEING THE LIE DERIVATIVE WITH RESPECT TO THE LIOUVILLE VECTOR FIELD REMARK IF WE PRESERVE DEFINITION 1 2 1 AND ASK FOR TO BE DIFFERENTIA- BLE ON TM (INCLUSIVE ON THE NULL SECTION), THEN THE PROPERTY OF 1–HOMOGENEITY OF F IMPLIES THAT F IS A LINEAR FUNCTION IN VARIABLES THE EQUALITY (2 4) IS EQUIVALENT TO THE FOLLOWING PROPERTIES HOLD: 1° 2° 3° DEFINITION 1 2 2 A VECTOR FIELD IS R-HOMOGENEOUS IF IT FOLLOWS: THEOREM 1 2 2 A VECTOR FIELD IS R-HOMOGENEOUS IF AND ONLY IF WE HAVE (2 5) OF COURSE, IS THE LIE DERIVATIVE OF X WITH RESPECT TO CONSEQUENTLY, WE CAN PROVE THE GEOMETRY OF HAMILTON & LAGRANGE SPACES 1° THE VECTOR FIELDS ARE 1 AND 0-HOMOGENEOUS, RESPECTIVELY 2° IF IS S-HOMOGENEOUS AND IS R-HOMOGENEOUS THEN F X IS S + R-HOMOGENEOUS 3° A VECTOR FIELD ON IS R–HOMOGENEOUS IF AND ONLY IF ARE FUNCTIONS ( R – L)-HOMOGENEOUS AND ARE FUNCTIONS R-HOMOGENEOUS 4° IS R-HOMOGENEOUS AND IS S-HOMOGENEOUS, THEN IS A (R + S – 1)-HOMOGENEOUS FUNCTION 5° THE LIOUVILLE VECTOR FIELD IS 1-HOMOGENEOUS 6° IF IS AN ARBITRARY S-HOMOGENEOUS FUNCTION, THEN IS A (S – 1)- HOMOGENEOUS FUNCTION AND IS (S – 2)-HOMOGENEOUS FUNCTION IN THE CASE OF Q-FORM WE CAN GIVE: DEFINITION 1 2 3 A Q-FORM IS S-HOMOGENEOUS IF IT FOLLOWS [106]: THEOREM 1 2 3 A Q-FORM IS S-HOMOGENEOUS IF AND ONLY IF (2 6) COROLLARY 1 2 1 WE HAVE, [106]: 1° S-HOMOGENEOUS AND IS 2° 3° ARE 0-HOMOGENEOUS 1-FORMS ARE 1-HOMOGENEOUS 1-FORMS THE APPLICATIONS OF THOSE PROPERTIES IN THE GEOMETRY OF FINSLER SPACE ARE NUM- BERLESS CH 1 THE GEOMETRY OF TANGENT BUNDLE