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Mathematical & Mechanical Method in Mechanical Engineering Dr. Wang Xingbo Fall,2005 Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds 1. Review of Topology 2. Concepts of manifolds 3. Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds How can we describe it? Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds • Many engineering objects have a shape of complicated surface. These complicated surfaces can be described by manifolds. Theories of manifolds have exhibited their elegances and excellences in many aspects of engineering, e.g. in controlling of robots, in structural analysis of mechanical engineering. Manifold is regarded to be a powerful tool for a senior engineer or a researcher to master. Mathematical & Mechanical Method in Mechanical Engineering Review of Topology A topological space is a pair (X, T) where X is a set and T is a class of subsets of X, called topology, which satisfies the following three properties. (i) X, ∈T. (ii) If { X i}i∈I∈T, then ∪i∈I X i ∈T (iii) If X 1,…, X n ∈T, then ∩i=1,…,n X i∈T. Mathematical & Mechanical Method in Mechanical Engineering Review of Topology If (X, T) is a topological space, the elements of T are said open sets. A subset K of X is said closed if its complementary set X \K is open The closure Uof a set U X is the intersection of all the closed sets K X with U K Mathematical & Mechanical Method in Mechanical Engineering Closure of a set K1 K2 X K4 U K3 Mathematical & Mechanical Method in Mechanical Engineering Continuous function If (X,T) and (Y,U) are topological spaces, a mapping f : X→Y is said continuous if is open for each T∈U f (X,T ) T f 1 (Y,U ) Mathematical & Mechanical Method in Mechanical Engineering Homeomorphism An injective, surjective and continuous mapping f : X→Y, whose inverse mapping is also continuous, is said homeomorphism from X to Y. If there is a homeomorphism from X to Y these topological spaces are said homeomorphic. Mathematical & Mechanical Method in Mechanical Engineering Base, second countable If (X,T) is a topological space, a class B Tis said base of the topology, if each open set turns out to be union of elements of B. A topological space which admits a countable base of its topology is said second countable. If (X,T) is second countable, from any base B it is possible to extract a subbase B’ B which is countable. Mathematical & Mechanical Method in Mechanical Engineering Topology generated by set-class If A is a class of subsets of X≠; and CA is the class of topologies T on X with AT, TA := T∈CA T is said the topology generated by A. Notice that CA≠ because the set of parts of X, P(X), is a topology and includes A. Mathematical & Mechanical Method in Mechanical Engineering topology induced on a set If AX, where (X,T) is a topological space, the pair (A,TA) where, TA := {UA | UT},defines a topology on A which is said the topology induced on A by X. Mathematical & mechanical Method in Mechanical Engineering Neighborhood • If (X,T) is a topological space and pX, a neighborhood of p is an open set UX with p∈U. If X and Y are topological spaces and xX, f: X→Y is said to be continuous in X, if for every neighborhood of f(x), VY , there is a neighborhood of x, UX, such that f(U) V . It is simply proven that f : X→Y as above is continuous if and only if it is continuous in every point of X. Mathematical & mechanical Method in Mechanical Engineering Connect • A topological space (X,T) is said connected if there are no open sets A, B≠ with AB = and AB = X. It turns out that if f: X→Y is continuous and the topological space X is connected, then f(Y) is a connected topological space when equipped with the topology induced by the topological space Y. Mathematical & mechanical Method in Mechanical Engineering Hausdorff • A topological space (X,T) is said Hausdorff if each pair (p,q)XX admits a pair of neighborhoods Up, Uq with p∈Up, q∈Uq and UpUq=. If X is Hausdorff and xX is a limit of the sequence {Xn}n∈NX, this limit is unique. Up p Uq q Mathematical & mechanical Method in Mechanical Engineering Semi-distance A semi metric space is a set X endowed with a semidistance. d: XX→[0,+∞], with d ( x, y ) d ( y , x ) d ( x , y ) d ( y , z ) d ( x, z ) d ( x, y ) 0 x y Mathematical & mechanical Method in Mechanical Engineering Open Ball • The semidistance is called distance and the semi metric space is called metric space. • An open metric balls are defined as Bs ( y ) : {z R | d ( z, y ) s} n Mathematical & mechanical Method in Mechanical Engineering Connected by path • A topological space (X,T) is said connected by paths if, for each pair p, qX there is a continuous path : [0,1] →X such that(0) = p,(1) = q, q=(1) p=(0) Mathematical & mechanical Method in Mechanical Engineering Cover If X is any set, a covering of X is a class {Xi}i∈I, XiX for all iI, such that i∈I Xi = X Mathematical & mechanical Method in Mechanical Engineering Compactness-Finite Cover A topological space (X,T) is said compact if from each covering of X, {Xi}i∈I are made of open sets, it is possible to extract a covering {Xj}j∈I of X with j finite. This is also called a finite covering property Mathematical & mechanical Method in Mechanical Engineering Group Let G be a set and be a operation defined on W. If W and satisfy the following regulations: 1. There is a unit e in G such that g G, e g = g 2. . g G, g 1 G g g 1 e | g 1 g e 3. g1 G , g 2 G kg1 g 21 G where Then G is called a group over R kR Mathematical & mechanical Method in Mechanical Engineering isomorphism Let S and T be tow groups with operations , respectively. If there exists a one-to-one mapping : S T such that, for any s1 , s2 S , t1 , t2 T 1 1 t ( s s ) 1. Ift ( s ), t ( s ) it results tin 1 1 2 2 1 2. If 1 2 2 et , esare unit in S and T respectively, then et (es ) then S is said to be isomorphic to T, or vice versa; the mapping is said to be a isomorphism between S and T. Two isomorphic groups can be regarded to have the same structure algebraically Mathematical & mechanical Method in Mechanical Engineering Concepts of Manifolds A topological space (X, T) is said topological manifold of dimension n if X is Hausdorff, second countable and is locally homeomorphic to Rn, i.e., for every pX there is a neighborhood pUp and a homomorphism p: Up→Vp where VpRn is a open set. Up p p VpRn Mathematical & mechanical Method in Mechanical Engineering Chart (n-chart)Let X be topological space, U is an open subset of X. Let be a homeomorphism from U X to an open subset V Rn, namely, : p→(x1(p),…,xn(p)). Then the ordered pair (U, )= C is called an n-chart on M. where Rn is the n-dimensional Euclidean space. A chart can be thought of a mapping from some open set to an open subset of Rn Mathematical & mechanical Method in Mechanical Engineering Chart C=( U,) U VRn Mathematical & mechanical Method in Mechanical Engineering K-Compatible • Let (U, ) and (U, ) be two charts on a topological space M . If UU, let V and V be image of UU under corresponding homeomorphisms and . The two charts are said to be compatible if -1 viewed as a mapping from V Rn to V Rn, is a C function. If UU= then the charts are also said to be compatible. If -1 and -1 are all Ck (k<) functions, then and are said to Ck-compatible. If any and are said to C-compatible, then M is said to be smooth. Mathematical & mechanical Method in Mechanical Engineering k-Compatible M U U -1 V Rn V Rn Mathematical & Mechanical Method in Mechanical Engineering Atlas An atlas A on a topological space M is a collection of charts{C} on M such that 1. Any two charts in atlas are piecewise k-compatible; 2. A covers M, i.e. M C A U Mathematical & Mechanical Method in Mechanical Engineering Differential structure, Differentian Manifolds A differential structure on a topological space is an atlas with the property that any chart that is compatible with the charts of the atlas is also an element of the atlas. An n-dimensional differential manifold M is a topological space endowed with a differential structure of n-charts. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems If M is a n-dimensional differential manifold, then any point PM has such a open neighborhood U that is homeomorphism to an open set V of Rn, or we can say, that there exists at least one open subset U of M that has a n-chart (U,) such that (P)=VRn. At this time, the coordinate ((P))iof image(P) corresponding to P is called coordinate of PU and is denoted by xi(P)=((P))i. (U, xi)is called a local coordinate system. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems It can be seen that, two charts (U,), (V,) on an ndimensional differential manifold M are related two local coordinate systems. If UV,then there also exist two local coordinate systems corresponding to UV. Thus any point P UV has two coordinate representations xi(P)=((P))i and yi(P)=( (P))i and the two are dependent. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems P = - 1 (P) - 1 (P) xi (P) ( (P))i ( ( 1 ( (P)))i ( 1 )( (P))i ( 1 )( y( P))i yi (P) ( (P))i ( ( 1 ( (P)))i ( 1 )( (P))i ( 1 )( x( P))i f , g 1 1 f g 1 , g f 1 xi ( P ) f ( y i ( P )), y i ( P ) g ( xi ( P )) i 1, 2,..., n Mathematical & Mechanical Method in Mechanical Engineering Differentiable Partitions of Unity on Manifolds P = - 1 (P) - 1 (P) xi (P) ( (P))i ( ( 1 ( (P)))i ( 1 )( (P))i ( 1 )( y( P))i yi (P) ( (P))i ( ( 1 ( (P)))i ( 1 )( (P))i ( 1 )( x( P))i f , g 1 1 f g 1 , g f 1 xi ( P ) f ( y i ( P )), y i ( P ) g ( xi ( P )) i 1, 2,..., n Mathematical & Mechanical Method in Mechanical Engineering Tensor Fields in Manifolds and Associated Geometric Structures Tangent , Tangent Bundle and State Space . Local representative Let (t ) be a continuous function from R to a differential manifold M. VRn (t) p C=(U,) R - Mathematical & Mechanical Method in Mechanical Engineering Tangent , Tangent Bundle and State Space . Local representative VRn (t) p C=(U,) R - Mathematical & Mechanical Method in Mechanical Engineering Related Two curves f and g are said to be related at p if and only if 1. f(0)=g(0)=p; 2.The derivatives of the representations of f and g are equal dx if dt |t 0 dxgi dt |t 0 local Mathematical & Mechanical Method in Mechanical Engineering Related properties If f(t) and g(t) are related in chart (U, ), they are also related in chart (V, ) 1 g 1 f g f Mathematical & Mechanical Method in Mechanical Engineering Related properties 1 d d d d d 1 ( g (t )) ( ( ( g (t )))) ( g (t )) 1 dt dt d d dt 1 d d d d d 1 ( f (t )) ( ( ( f (t )))) ( f (t )) 1 dt dt d d dt f(t) and g(t) are related in chart (U, ) d d ( f (t )) |t 0 ( g (t )) |t 0 dt dt Mathematical & Mechanical Method in Mechanical Engineering Related properties 1 1 d d d d d d ( f (t )) |t 0 ( g (t )) |t 0 1 1 d d dt d d dt 1 1 D( ) d d ( 1 ) d d Mathematical & Mechanical Method in Mechanical Engineering Related Properties 1 D( ) ( g (0)) d d ( g (t )) |t 0 D ( 1 ) ( f (0)) ( f (t )) |t 0 dt dt d d ( f (t )) |t 0 ( g (t )) |t 0 dt dt Vp Up p f(t) g(t) Mathematical & Mechanical Method in Mechanical Engineering Tangent space If M is a differentiable manifold and pM, the tangent space at point p, denoted as TpM, is defined to be the set of all equivalent classes Qp at p in M. TpM has the same dimension as M Define a map is injective d d (p)([ ]) ( ) |t 0 dt Mathematical & Mechanical Method in Mechanical Engineering Tangent space For any v in Rn, choose such that for any |t|< (p) tv is a path through (p) in (U ) and 1 (p tv ) is a smooth path through p 1 d (p)([ (p tv)]) v d (p) is bijective , a linear isomorphic map from TpM to Rn Mathematical & Mechanical Method in Mechanical Engineering Basis induced by a chart Let M be a differentiable manifold, pM, and take a chart (U,) with pU. If E1,…,En is the canonical basis of Rn, then e pi p Ei (i 1, 2,..., n) define a basis in TpM which we call the basis induced in TpM by the chart (U,) (U,), (V, ) with pU,V and induced basis on p Tp M p { e {ei }i 1,2,...,n j } j 1,2,...,n t p t ieip t j e jp Tp M x t k | ( p ) t k x j j x p e k | ( p ) e j x j p k Mathematical & Mechanical Method in Mechanical Engineering Derivations Symbol D(M) indicates the real vector space of all differential functions from manifold M to R | Dp (M ) indicate the vector space spanned by k p x Let M be a differentiable manifold. A derivation in TpM is a R-linear map Dp: D(M)→R, such that, for each pair f, g D(M): Dp ( fg ) f ( p) D p g g ( p) D p f Symbol DpM is used to indicate the R-vector space of the derivations in p Mathematical & Mechanical Method in Mechanical Engineering Derivation Let M be a differential manifold. Take any TpM and any DpDpM (1) If hD(M) vanishes in a open neighborhood of p or, more strongly, h = 0 in the whole manifold M,then Dph= 0 (2) For every f, gD(M), Dpf = Dpg provide f(q) = g(q) in an open neighborhood of p. Mathematical & Mechanical Method in Mechanical Engineering Flander's Lemma If f: B→R is C∞(B) where BRn is an open 1 2 n p ( x , x ,..., x starshaped neighborhood of 0 0 0 0) , then there are n differentiable mappings gi: B→R such that, if p ( x1 , x 2 ,..., x,n ) then n f (p) f (p 0 ) gi (p)( x i x0i ) i 1 f gi (p 0 ) i |p x 0 Mathematical & Mechanical Method in Mechanical Engineering Flander’s lemma p ( x1 , x 2 ,..., x n ) y i (t ) x0i t ( x i x0i ) d f (p 0 t (p p 0 ))dt 0 dt 1 n f (p 0 t (p p 0 ) i i f (p 0 ) ( x x 0 ) dt i 0 x i 1 f (p) f (p 0 ) 1 f gi (p) i |p0 t (p p0 ) dt 0 x 1 f g i ( p 0 ) i | p0 x Mathematical & Mechanical Method in Mechanical Engineering basis of TpM Let M be a differentiable manifold and pM. There exists a R-value vector space isomorphism F: TpM DpM such p that, if {ei }i 1,2,...,n is the basis of TpM induced by any local coordinate system about p with coordinates (x1,..., xn), it holds: k F :t e t p t e Tp M i p i is a basis of DpM p k t | k p x k And in particular the set { | } k p k 1,2,..., n x Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle The tangent bundle of a manifold M, denoted by TM is defined as the union of the tangent spaces for all pM. That is: TM TpM pM Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle TM is itself a differential manifold of dimension 2n TM= {(p, v) |pM , vTpM} Tangent bundle is called a state space Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle Given two manifolds A and B and a function f:AB, there is a natural way to form a mapping, denoted by Tf, from TA to TB TA Tf TB f A B Mathematical & Mechanical Method in Mechanical Engineering Cotangent Space and Phase Space Let M be an n-dimensional manifold. For each pM, the dual space Tp*M is called the cotangent space on p and its elements are called cotangent vectors or differential 1-forms on p. If (x1,..., xn) are coordinates about p inducing the basis , the associated dual basis in Tp*M is denoted by {dxk|p}k=1,…,n. Mathematical & Mechanical Method in Mechanical Engineering Cotangent Space and Phase Space The cotangent bundle of a manifold M, denoted by T*M is defined as the union of the cotangent spaces for all pM. That is: TM TM * pM * p A cotangent space is also called a phase space that is a collection of all possible positions and momenta that cab be obtained by a configuration space. Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Let M be a differentiable manifold. An affine connection or covariant derivative r, is a map : (Y,X ) Y X where X , Y , Y X are differentiable contravariant vector fields on M, which obeys the following requirements: (1) fY +gZX = f▽Y X + gZ X, for all differentiable functions f , g and differentiable vector fields X , Y ,Z ; (2) YfX = Y(f)X+fY X for all differentiable vector field X , Y and differentiable functions f , (3) X (Y +Z) = X Y +X Z for all ,R and differentiable vector fields X, Y, Z. Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection In components referred to any local coordinate system j Y j i j i XY i Y XY X j j i j X x x x i i x x x j if i , j are fixed define a differentiable tensor i x x field which is the derivative of with respect to j x and thus x i x i k k , dx k : ij k j j x x x x x i Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection The coefficients = ( p) are differentiable functions of the considered coordinates and are called connection coefficients. k ij k ij Using these coefficients and the above expansion, in components, the covariant derivative of Y with respect to X can be written down as:. i X (Y X)i : X j ( j ijk Y k ) x Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection X i i k i i Y : X : X , j. jk j j x X is called covariant derivative of X (with respect to the affine connection ). In components we have (Y X)i = YjXi,j. k h k p q x x x x x k h ij h i j h i pq j x x x x x x Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection T : i jk i jk i kj define a tensor field is represented by j k T () ( ) i dx dx x i jk i kj This tensor field is symmetric in the covariant indices and is called torsion tensor field of the connection. Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection The assignment of an affine connection on a differentiable manifold M is completely equivalent to the assignment of coefficients ( p) in each local | , dx | x coordinate system, which differentially depend on the point p and transform as k ij x k | i p j p p k h k p q x x x x x h ijk ( p) h | p i j | p h | p | pq ( p) i j p x x x x x x under change of local coordinates. Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection If M is endowed with a metric, then the manifold is called a Riemann manifold. The connection on a Riemann manifold is called Levi-Civita's affine connection. Let M be a Riemann manifold with metric locally represented by. There is exactly one affine connection such that : (1). It is metric, i.e., = 0 (2). It is torsion free, i.e., T() = 0. Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection That is the Levi-Civita connection which is defined by the connection coefficients, called Christoffel's coefficients 1 is g ks g sj g jk i i jk { j k } : g ( j k s ). 2 x x x Consider a (pseudo) Euclidean space En. Fixing an orthogonal Cartesian coordinate system, we can define an affine connection locally given by: . Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Y X Y : X j j x x i j Mathematical & Mechanical Method in Mechanical Engineering Meaning of the Covariant Derivative X(p + hY ) X(p) l im h 0 h Unfortunately, there are two problems involved in the formula above: (1) What does it mean p+hY ? In general, we have not an affine structure on M and we cannot move points thorough M under the action of vectors as in affine spaces. (The reader should pay attention on the fact that affine connections and affine structures are different objects!). Mathematical & Mechanical Method in Mechanical Engineering Meaning of the Covariant Derivative X(p) TpM but X(p + hY ) Tp+hYM. If something like p + hY makes sense, we expect that p + hY ≠ p because derivatives in p should investigate the behavior of the function qX(q) in a “infinitesimal” neighborhood of p. So the difference X(p + hY ) - X(p) does not make sense because the vectors belong to different vector spaces! . Mathematical & Mechanical Method in Mechanical Engineering Meaning of the Covariant Derivative Let M be a differentiable manifold equipped with an affine connection . If X and Y are differentiable contravariant vector fields in M and pM Y X(p) lim h 0 P1[p, (h)]X( ( h)) X(p) h Mathematical & Mechanical Method in Mechanical Engineering where : [0 , ] → M is the unique geodesic segment referred to r starting from p with initial tangent vector Y(p) and P[(u),(v)]:T(u)T(v) is the vector-space isomorphism induced by the r parallel transport along a (sufficiently short) differentiable curve : [a , b] → M for u < v and u , v[a , b].