dr. wang xingbo fall ， 2005

Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

1.1. Review of TopologyReview of Topology

2.2. Concepts of manifoldsConcepts of manifolds

3.3.


An Introduction to ManifoldsAn Introduction to Manifolds

How can we describe it? How can we describe it?



• Many engineering objects have a shape of compMany engineering objects have a shape of complicated surface. These complicated surfaces can licated surface. These complicated surfaces can be described by manifolds. Theories of manifoldbe described by manifolds. Theories of manifolds have exhibited their elegances and excellences s have exhibited their elegances and excellences in many aspects of engineering, e.g. in controlliin many aspects of engineering, e.g. in controlling of robots, in structural analysis of mechanicang of robots, in structural analysis of mechanical engineering. Manifold is regarded to be a powel engineering. Manifold is regarded to be a powerful tool for a senior engineer or a researcher to rful tool for a senior engineer or a researcher to master.master.



A A topological spacetopological space is a pair (X, is a pair (X, TT) ) where X is a set and where X is a set and TT is a class of is a class of subsets of X, called topology, which subsets of X, called topology, which satisfies the following three satisfies the following three

properties.properties.

(i) X, (i) X, ∈∈TT..(ii) If { X(ii) If { X i i}}ii∈I∈I∈∈TT, then , then ∪∪ii∈I∈I X X i i ∈∈TT

(iii) If X(iii) If X 1 1,…, X,…, X n n ∈∈TT, then , then ∩∩i=1,…,ni=1,…,n X X i i∈∈TT..


Review of TopologyReview of Topology

If (X,If (X,TT) is a topological space, the elements ) is a topological space, the elements of of TT are said open sets. are said open sets.

A subset K of X is said closed if its complemeA subset K of X is said closed if its complementary set X \K is openntary set X \K is open

The closure of a set U X is the intersection oThe closure of a set U X is the intersection of all the closed sets K X with U Kf all the closed sets K X with U K


Review of TopologyReview of Topology

U


Closure of a setClosure of a set

K1

K2

X K4 U

K3

If (X,If (X,TT) and (Y,) and (Y,UU) are topological spaces, ) are topological spaces, a mapping f : Xa mapping f : X→→Y is said continuous if is Y is said continuous if is open for each Topen for each T∈∈UU


Continuous functionContinuous function

f

(X,T) T (Y,U)

1f

An injective, surjective and continuous An injective, surjective and continuous mapping f : Xmapping f : X→→Y, whose inverse mapping Y, whose inverse mapping is also continuous, is said homeomorphismis also continuous, is said homeomorphismfrom X to Y. If there is a homeomorphismfrom X to Y. If there is a homeomorphismfrom X to Y these topological spaces arefrom X to Y these topological spaces aresaid homeomorphic.said homeomorphic.


HomeomorphismHomeomorphism

If (X,If (X,TT) is a topological space, a class ) is a topological space, a class B TB T is said base of the topology, if each is said base of the topology, if each open set turns out to be union of elements ofopen set turns out to be union of elements ofBB. A topological space which admits a countable. A topological space which admits a countablebase of its topology is said second countable.base of its topology is said second countable.

If (X,If (X,TT) is second countable, from any base ) is second countable, from any base BB it is possible to extract a subbase it is possible to extract a subbase BB’ ’ BB which is countable.which is countable.


Base, second countable Base, second countable

If A is a class of subsets of X≠; and CA isthe class of topologies T on X with AT,TA := T∈CA T is said the topology generated by A. Notice that CA≠ because the setof parts of X, P(X), is a topology and includes A.


Topology generated by set-classTopology generated by set-class

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.


topology induced on a set

• 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 & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Neighborhood

• 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.


Connect Connect

• 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 li

mit is unique.


Hausdorff

Up Uq

p

q

A semi metric space is a set X endowed with a semidistance. d: XX→[0,+∞], with


Semi-distance

( , ) ( , )d x y d y x

( , ) ( , ) ( , )d x y d y z d x z

( , ) 0d x y x y

• The semidistance is called distance and the semi metric space is called metric space.

• An open metric balls are defined as


Open Ball Open Ball

( ) : { | ( , ) }nsB y z d z y s R

• 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,


Connected by path Connected by path

q=(1)

p=(0)


Cover 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


Compactness-Finite Cover 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 extra

ct a covering {Xj}j∈I of X with j finite. This is also called a fi

nite covering property


Group 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

2. .

3. where

Then G is called a group over R

1 1 1, |g g g g e g g e G G

g e g = g G,

11 2 1 2,g g kg g G G G k R


isomorphismisomorphism

Let S and T be tow groups with operations , respectively. If there exists a one-to-one mapping : S T such that, for any

1. If it results in

2. If are unit in S and T respectively, then

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

1 2 1 2, , ,s s t t S T

1 1 2 2( ), ( )t s t s 2 2

1 11 1( )t t s s

,t se e ( )t se e


Concepts of Manifolds 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

(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


Chart Chart


ChartChart

C=( U,)

U

VRn

• 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.


K-Compatible K-Compatible


k-Compatiblek-Compatible

M

U U

-1

V Rn

V Rn

An An atlasatlas AA on a topological space on a topological space MM is a co is a collection of charts{llection of charts{CC} on M such that} on M such that

1. Any two charts in atlas are piecewise k-compatible;

2. A covers M, i.e.


AtlasAtlas

C AM U

A A differential structuredifferential structure on a topological on a topological space is an atlas with the property that any space is an atlas with the property that any chart that is compatible with the charts of chart that is compatible with the charts of the atlas is also an element of the atlasthe atlas is also an element of the atlas..


Differential structure, Differentian ManifoldsDifferential structure, Differentian Manifolds

An n-dimensional differential manifold M is a topological space endowed with a differential structure of n-charts.

IfIf M M is a is a nn-dimensional differential manifold, then -dimensional differential manifold, then any point any point PPMM has such a open neighborhood has such a open neighborhood UU t that is homeomorphism to an open set hat is homeomorphism to an open set VV of of RRnn, or , or we can say, that there exists at least one open subwe can say, that there exists at least one open subset set UU of of MM that has a that has a nn-chart (-chart (UU,,) such that ) such that ((PP))==VVRRnn. At this time, the coordinate . At this time, the coordinate ((((PP))))iiof imagof imagee((PP) corresponding to) corresponding to P P is called coordinate of is called coordinate of PPUU and is denoted by and is denoted by xxii((PP)=()=(((PP))))ii. (. (UU, , xxii)is calle)is called a d a local coordinate systemlocal coordinate system..


Local Cooridnate SystemsLocal Cooridnate Systems



It can be seen that, two charts (U,), (V,) on an n-dimensional 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.



( ) ( ) -1 -1=P P P1 1 1( ) ( ( )) ( ( ( ( ))) ( )( ( )) ( )( ( ))i i i i ix y P P P P P

1 1 1( ) ( ( )) ( ( ( ( ))) ( )( ( )) ( )( ( ))i i i i iy x P P P P P

1 1,f g 1 1,f g g f

( ) ( ( )), ( ) ( ( )) 1,2,...,i i i ix f y y g x i n P P P P


Differentiable Partitions of Unity on Manifolds Differentiable Partitions of Unity on Manifolds

( ) ( ) -1 -1=P P P1 1 1( ) ( ( )) ( ( ( ( ))) ( )( ( )) ( )( ( ))i i i i ix y P P P P P

1 1 1( ) ( ( )) ( ( ( ( ))) ( )( ( )) ( )( ( ))i i i i iy x P P P P P

1 1,f g 1 1,f g g f

( ) ( ( )), ( ) ( ( )) 1,2,...,i i i ix f y y g x i n P P P P

Tangent , Tangent Bundle and State Space .Tangent , Tangent Bundle and State Space .


Tensor Fields in Manifolds and Associated GeomeTensor Fields in Manifolds and Associated Geometric Structures tric Structures

Local representativeLet be a continuous function from R to a differential manifold M.

( )t

VRn

(t) p

C=(U,)

R

-

.. Local representative


Tangent , Tangent Bundle and State SpaceTangent , Tangent Bundle and State Space

VRn

(t) p

C=(U,)

R

-

Two curves Two curves ff and and g g are said to be related at p if aare said to be related at p if and only ifnd only if

1. f1. f(0)=(0)=gg(0)=(0)=pp;; 2.2.The derivatives of the local representations oThe derivatives of the local representations o

f f ff and and gg are equal are equal


RelatedRelated

0 0| |i if g

t t

dx dx

dt dt

If If ff((tt) and ) and gg((tt) are related in chart ) are related in chart (U(U, , ), they are also related in ), they are also related in

chart (Vchart (V, , ))


Related propertiesRelated properties

1g g

1f f


Related properties Related properties

1

1

1( )) ( ( ( )))) ( ( ))( (g t g t g t

ddd d d

dt dt d d dt

1

1

1( )) ( ( ( )))) ( ( ))( (t f t f t

ddd d df

dt dt d d dt

f(t) and g(t) are related in chart (U, )

0 0( ( )) | ( ( )) |t t

d df t g t

dt dt


Related propertiesRelated properties

1 1

1 10 0( ( )) | ( ( )) |t tf t g td dd dd d

d d dt d d dt

1

1

1( ) ( )

ddD

d d


Related Properties Related Properties

1 1

( (0)) 0 ( (0)) 0( () ( ( )) | ) ( ( )) |g t f tD Dd d

g t f tdt dt

0 0( ( )) | ( ( )) |t t

d df t g t

dt dt

Vp

Up p

f(t) g(t)

If If MM is a differentiable manifold and is a differentiable manifold and ppMM, the tangent spa, the tangent space at point ce at point pp, denoted as , denoted as TTppMM, is defined to be the set o, is defined to be the set o

f all equivalent classes f all equivalent classes QQpp at at pp in in MM. .


Tangent space Tangent space

TpM has the same dimension as M

0( )([ ]) ( ) |td

ddt p Define a map

is injective

For any For any vv in in RRnn, choose , choose such that for any |t|< such that for any |t|<


Tangent space Tangent space

is a path through in and

is a smooth path through p

is bijective , a linear isomorphic map from Tp

M to Rn

( ) t p v

( ) p ( ) U1( )t

p v

1( )([ ( )])d t p p v v

( )d p

Let Let MM be a differentiable manifold, be a differentiable manifold, ppMM, and take a chart, and take a chart((UU,,) with ) with ppUU. If . If EE11,…,E,…,E

nn is the canonical basis of is the canonical basis of RRnn, ,

thenthen define a basis in define a basis in TTppMM which we call the basis induced in which we call the basis induced in

TTppM M by the chart (by the chart (UU,,))


Basis induced by a chart Basis induced by a chart

( 1,2,..., )pi p i i n Ee

1,2,...,{ }pi i ne 1,2,...,{ }p

j j ne (U,), (V, ) with pU,V and induced basis on TpM i p j p

p i jt t t pe e T M

( )|j

j kpk

xt t

x

( )|j

p pk p jk

x

x

e e

Let Let MM be a differentiable manifold. A derivation in be a differentiable manifold. A derivation in TTppMM is is

a a RR-linear map -linear map DDpp: : DD((MM))→→R, such that, for each pair R, such that, for each pair f,f,

g g DD((MM): ):


Derivations Derivations

( ) ( ) ( )p p pD fg f p D g g p D f

Symbol D(M) indicates the real vector space of all differential functions from manifold M to R

( )p MD indicate the vector space spanned by pkx|

Symbol DpM is used to indicate the R-vector space of the deri

vations in p

Let Let MM be a differential manifold. Take any be a differential manifold. Take any TTppMM and and any any DDppDDppMM

(1) If (1) If hhDD((MM) vanishes in a open neighborhood of ) vanishes in a open neighborhood of pp or, more strongly, or, more strongly, hh = 0 in the whole manifold = 0 in the whole manifold MM,,thenthen

DDpphh= 0= 0(2) For every (2) For every f, gf, gDD((MM),),

DDppff = = DDppgg provide provide ff((qq) = ) = gg((qq) in an open neighborhood of ) in an open neighborhood of pp. .


Derivation Derivation

If If ff: : BB→→RR is is CC∞∞((BB) where ) where BBRRnn is an open starshap is an open starshaped neighborhood of ed neighborhood of

, then there are , then there are nn differentiable mappings differentiable mappings ggii: : BB→→RR such that, if , thensuch that, if , then


Flander's Lemma Flander's Lemma

1 20 0 0 0( , ,..., )nx x xp

1 2( , ,..., )nx x xp

0 01

( ) ( ) ( )( )n

i ii

i

f f g x x

p p p

00( ) |i i

fg

x

pp


Flander’s lemmaFlander’s lemma1 2( , ,..., )nx x xp )()( 00

iiii xxtxty

1

0 0 00

10 0

0 001

( ) ( ) ( ( ))

( ( )( ) ( )

ni i

ii

df f f t dt

dtf t

f x x dtx

p p p p p

p p pp

0 0

1

( )0( ) |i ti

fg dt

x

p p pp

0|)( 0 pii x

fpg


basis of Tbasis of TppM M

Let M be a differentiable manifold and pM. There exists a R-value vector space isomorphism F: TpM DpM such that, if

is the basis of TpM induced by any local coordin

ate system about p with coordinates (x1,..., xn), it holds: 1,2,...,{ }p

i i ne

: |k p kk pk

t tx

F e

i pp it t pe T M And in particular the set 1,2,...,{ | }p k nkx

is a basis of DpM

The tangent bundle of a manifold The tangent bundle of a manifold MM, denoted by , denoted by TMTM is defined as the union of the tangent spac is defined as the union of the tangent spaces for all es for all ppM. That is: M. That is:


Tangent Bundle Tangent Bundle

p M pTM T M

TMTM is itself a differential manifold of dimension 2 is itself a differential manifold of dimension 2nn

TMTM= {(= {(pp, , vv) |) |ppMM , , vvTTppMM} }

Tangent bundle is called a Tangent bundle is called a state spacestate space



Given two manifolds Given two manifolds AA and and BB and a function and a function ff::AABB, , there is a natural way to form a mapping, denoted by there is a natural way to form a mapping, denoted by Tf,Tf,

from from TATA to to TBTB



TA Tf TB

f

A B

Let Let MM be an be an nn-dimensional manifold. For each -dimensional manifold. For each ppMM, the dual space is called the cotangent, the dual space is called the cotangentspace on space on pp and its elements are called co- and its elements are called co-tangent vectors or differential 1-forms on tangent vectors or differential 1-forms on pp..If (If (xx11,..., ,..., xxnn) are coordinates about ) are coordinates about ppinducing the basis , the associated dual basisinducing the basis , the associated dual basisin is denoted by {din is denoted by {dxxkk||pp}}k=1,…,n.k=1,…,n.


Cotangent Space and Phase Space Cotangent Space and Phase Space

*pT M

*pT M


Cotangent Space and Phase Space 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:

* *

pp M

T M T M

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.


Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection Let M be a differentiable manifold. An affine connection or covariant derivative r, is a map

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.

: ( ) YYX X,


Covariant Derivative and Levi-Civita's ConnectionCovariant Derivative and Levi-Civita's Connection

In components referred to any local coordinate system

if i , j are fixed define a differentiable tensor field which is the derivative of with respect to and thus

ii i

jj i j i

j j i jXx x

YY X Y X

x x x x

XY

kkijk

kj

xj

xxx

dxxx ii

:,

jx

xi

jx

ix



The coefficients = are differentiable functions of the considered coordinates and are called connection coefficients.

Using these coefficients and the above expansion, in components, the covariant derivative of Y with respect to X can be written down as:.

kij )( pk

ij

( ) : ( )i

i j i kjkj

XX Y

x

YX



X is called covariant derivative of X (with respect to the affine connection ). In components we have

(Y X)i = YjXi,j.

.,:: jii

jki

jkj

i

XXYx

X

k h k p qk hij pqh i j h i j

x x x x x

x x x x x x



define a tensor field is represented by

This tensor field is symmetric in the covariant indices and is called torsion tensor field of the connection.

ikj

ijk

ijkT :

kji

ikj

ijk dxdx

xT

)()(



The assignment of an affine connection on a differentiable manifold M is completely equivalent to the assignment of coefficients in each local coordinate system, which differentially depend on the point p and transform as

under change of local coordinates.

pk

pjx

kij dx

xp

pi

|,|)(|

( ) | | | | ( )k h k p q

k hij p p p p pqh i j h i j

x x x x xp p

x x x x x x



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.



That is the Levi-Civita connection which is defined by the connection coefficients, calledChristoffel's coefficients

Consider a (pseudo) Euclidean spaceEn. Fixing an orthogonal Cartesian coordinate system, we can define an affine connection locally given by:.

).(2

1:}{

s

jk

k

sj

jksis

ki

jijk x

g

x

g

x

gg



jj

ij

X xx

YXY

:


Meaning of the Covariant Derivative Meaning of the Covariant Derivative

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!).

0

( ) ( )h

l imh

X X pp Y+h


Meaning of the Covariant DerivativeMeaning 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! .


Meaning of the Covariant DerivativeMeaning 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

1

0

[ , ( )] ( ( )) ( )( ) lim

h

h h

h

Y

p X X pX p

P


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].

dr. wang xingbo fall ， 2005

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