Presentacion de Variedades

7/29/2019 Presentacion de Variedades

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Control &Dynamical Systems

CA L T EC H

Manifolds, Mappings, Vector Fields

Jerrold E. Marsden

Control and Dynamical Systems, Caltech

http://www.cds.caltech.edu/ marsden/


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Some History

Carl Friedrich Gauss

Born: 30 April 1777 in Brunswick (Germany)

Died: 23 Feb 1855 in Gottingen (Germany)

PhD, 1799, University of Helmstedt, advisor was Pfaff. Dissertation

was on the fundamental theorem of algebra. 1801, predicted position of asteroid Ceres; used averaging methods;

famous as an astronomer; 1807, director of the Gottingen observatory

18011830, Continued to work on algebra, and increasingly in geometry

of surfaces, motivated by non-Euclidean geometry. 1818, Fame as a geologist; used geometry to do a geodesic survey.

1831, work on potential theory (Gauss theorem), terrestrial mag-netism, etc.

1845 to 1851, worked in finance and got rich.2


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Some History

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Some History

Carl Gustav Jacob Jacobi

Born: 10 Dec 1804 in Potsdam

Died: 18 Feb 1851 in Berlin

1821, entered the University of Berlin

1825, PhD Disquisitiones Analyticae de Fractionibus Simplicibus

1825-26 Humboldt-Universitat, Berlin

18261844 University of Konigsberg

1827, Fundamental work on elliptic functions 18441851 University of Berlin

Lectures on analytical mechanics (Berlin, 1847 - 1848); gave a de-tailed and critical discussion of Lagranges mechanics.

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Some History

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Some History

Georg Friedrich Bernhard Riemann

Born: 17 Sept 1826 in Breselenz (Germany)

Died: 20 July 1866 in Selasca, Italy

PhD, 1851, Gottingen, advisor was Gauss; thesis in complex analysis

on Riemann surfaces 1854, Habilitation lecture: Uber die Hypothesen welche der Geome-

trie zu Grunde liegen (On the hypotheses that lie at the foun-dations of geometry), laid the foundations of manifold theory and

Riemannian geometry 1857, Professor at Gottingen

1859, Reported on the Riemann hypothesis as part of his inductioninto the Berlin academy

1866, Died at the age of 40 in Italy,6


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Some History

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Some History

Felix Klein

Born: 25 April 1849 in Dsseldorf, Germany

Died: 22 June 1925 in Gottingen, Germany

1871; University of Christiania (the city which became Kristiania, thenOslo in 1925)

PhD, 1868, University of Bonn under Plucker, on applications of ge-ometry to mechanics.

1872, professor at Erlangen, in Bavaria; laid foundations of geometryand how it connected to group theory (Erlangen Programm); collabo-rations with Lie.

1880 to 1886, chair of geometry at Leipzig

1886, moved to Gottingen to join Hilberts group

18901900+, More on mathematical physics, mechanics; correspondedwith Poincare8


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Some History

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Some History

Marius Sophus Lie

Born: 17 Dec 1842 in Nordfjordeide, Norway

Died: 18 Feb 1899 in Kristiania (now Oslo), Norway

1871; University of Christiania (the city which became Kristiania, then

Oslo in 1925) PhD, 1872, University of Christiania On a class of geometric trans-

formations

18861898 University of Leipzig (got the chair of Klein); where he wrote

his famous 3 volume work Theorie der Transformationsgruppen 1894, Kummer was his student at Leipzig

18981899, Returned to the University of Christiania; dies shortly afterhis return.

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Some History

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Some History

Henri Poincare

Born: 29 April 1854 in Nancy, Lorraine, France

Died: 17 July 1912 in Paris, France

1879, Ph.D. University of Paris, Advisor: Charles Hermite

1879, University of Caen 1881, Faculty of Science in Paris in 1881

1886, Sorbonne and Ecole Polytechnique until his death at age 58

1887, elected to the Acadmie des Sciences; 1906, President.

1895, Analysis situs(algebraic topology)

18921899, Les Methodes nouvelles de la mechanique celeste (3 vol-umes) and, in 1905, Lecons de mecanique celeste

His cousin, Raymond Poincare, was President of France during WorldWar I.

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Manifolds

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Manifolds

Manifolds are sets M that locally look like linear

spacessuch as Rn.

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Manifolds


spacessuch as Rn.

A chart on M is a subset U of M together with abijective map : U (U) Rn.

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Manifolds


spacessuch as Rn.

A chart on M is a subset U of M together with abijective map : U (U) Rn.

Usually, we denote (m) by (x1, . . . , xn) and call thexi the coordinates of the point m U M.

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Manifolds


spacessuch as Rn.A chart on M is a subset U of M together with a

bijective map : U (U) Rn.

Usually, we denote (m) by (x1, . . . , xn) and call thexi the coordinates of the point m U M.

Two charts (U,) and (U, ) such that U U =

are called compatible if(U U

) and

(U

U)are open subsets ofRn and the maps

1|(U U) : (U U) (U U)

and

()1|(U U) : (U U) (U U)14


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Manifolds

are C. Here, 1|(UU) denotes the restric-

tion of the map 1 to the set (U U).

(U)

(U)

U

M

x1

x1

xn

x

n

xn

U

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Manifolds

We call M a differentiable n-manifold when:

M1. The setM is covered by a collection of charts, that is, everypoint is represented in at least one chart.

M2. M has an atlas; that is, M can be written as a union ofcompatible charts.

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f


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Manifolds




Differentiable structure: Start with a given atlasand (be democratic and) include all charts compatiblewith the given ones.

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M if ld


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Manifolds




Differentiable structure: Start with a given atlasand (be democratic and) include all charts compatiblewith the given ones.

Example: Start with R3 as a manifold with simplyone (identity) chart. We then allow other charts suchas those defined by spherical coordinatesthis is then adifferentiable structure on R3. This will be understoodto have been done when we say we have a manifold.

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M if ld


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Manifolds

Topology. Every manifold has a topology obtained

by declaring open neighborhoods in charts to be openneighborhoods when mapped to M by the chart.

Tangent Vectors. Two curves t c1(t) and t

c2(t) in an n-manifold Mare called equivalent at thepoint m if

c1(0) = c2(0) = m

andddt

( c1)t=0

= ddt

( c2)t=0

in some chart .

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M if ld


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Manifolds

0

I

c1

c2

m

F

U

R

U

M

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T V


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Tangent Vectors

A tangent vector v to a manifold M at a point

m M is an equivalence class of curves at m.

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T V


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Tangent Vectors


m M is an equivalence class of curves at m. Set of tangent vectors to M at m is a vector space.

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T V


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Tangent Vectors



Notation: TmM = tangent space to M at m M.

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T t V t


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Tangent Vectors



Notation: TmM = tangent space to M at m M.

We think of v TmM as tangent to a curve in M.

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T t V t


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Tangent Vectors

m

TmM

v

M

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T t V t


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Tangent Vectors

Components of a tangent vector.

: U M Rn a chart M

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T t V t


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Tangent Vectors


: U M Rn a chart M

gives associated coordinates (x1, . . . , xn) for points in U.

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T t V t


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Tangent Vectors


: U M Rn a chart M gives associated coordinates (x1, . . . , xn) for points in U.

Let v TmM be a tangent vector to M at m;

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T t V t


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Tangent Vectors




c a curve representative of the equivalence class v.

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T t V t


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Tangent Vectors





The components ofv are the numbers v1, . . . , vn defined by tak-ing the derivatives of the components of the curve c:

vi =d

dt( c)i

t=0

,

where i = 1, . . . , n.

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T t V t


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Tangent Vectors





The components ofv are the numbers v1, . . . , vn defined by tak-ing the derivatives of the components of the curve c:

vi =d

dt( c)i

t=0

,

where i = 1, . . . , n. Components are independent of the representative curve chosen, but

they do depend on the chart chosen.

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Ta e t B dles


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Tangent Bundles

Tangent bundle ofM, denoted by TM, is the dis-

joint union of the tangent spaces to M at the pointsm M, that is,

TM = mM

TmM.

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Tangent Bundles


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Tangent Bundles



TM = mM

TmM.

Points ofTM: vectors v tangent at some m M.

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Tangent Bundles


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Tangent Bundles



TM =

mM

TmM.

Points ofTM: vectors v tangent at some m M.

IfM is an n-manifold, then TM is a 2n-manifold.

The natural projection is the map M

: TM Mthat takes a tangent vector v to the point m M atwhich the vector v is attached.

The inverse image 1M (m) of m M is the tangent

space TmMthe fiber ofTM over the point m M.22

Differentiable Maps


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Differentiable Maps

A map f : M N is differentiable (resp. Ck) if

in local coordinates on M and N, the map f is repre-sented by differentiable (resp. Ck) functions.

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Differentiable Maps


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Differentiable Maps

A map f : M N is differentiable (resp. Ck) if

in local coordinates on M and N, the map f is repre-sented by differentiable (resp. Ck) functions.

The derivative off : M N at a point m M is

a linear map Tmf : TmM Tf(m)N. Heres how: For v TmM, choose a curve c in M with c(0) = m, and velocity

vector dc/dt |t=0 = v .

Tmf v is the velocity vector at t = 0 of the curve f c : R N,

that is,Tmf v =

d

dtf(c(t))

t=0

.

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Chain Rule


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Chain Rule

The vector Tmfv does not depend on the curve c but

only on the vector v.

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Diffeomorphisms


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Diffeomorphisms

A differentiable (or of class Ck) map f : M N

is called a diffeomorphism if it is bijective and itsinverse is also differentiable (or of class Ck).

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Diffeomorphisms


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Diffeomorphisms



IfTmf : TmM Tf(m)N is an isomorphism, the in-

verse function theorem states that f is a localdiffeomorphism around m M

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Diffeomorphisms


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Diffeomorphisms



IfTmf : TmM Tf(m)N is an isomorphism, the in-

verse function theorem states that f is a localdiffeomorphism around m M

The set of all diffeomorphisms f : M M formsa group under composition, and the chain rule showsthat T(f1) = (Tf)1.

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Submanifolds


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Submanifolds

Iff : M N is a smooth map, a point m M is a

regular point ifTmf is surjective; otherwise, m is acritical point off. A value n N is regular if allpoints mapping to n are regular.

Submersion theorem: Iff : M N is a smoothmap and n is a regular value of f, then f1(n) is asmooth submanifold ofMof dimension dim Mdim Nand

Tmf1(n)

= ker Tmf.

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Vector Fields and Flows


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Vector field X on M: a map X : M TM that

assigns a vector X(m) at the point m M; that is,M X= identity.

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Vector space of vector fields on M is denoted X(M).

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An integral curve ofXwith initial condition m0 att = 0 is a (differentiable) map c : ]a, b[ M such that]a, b[ is an open interval containing 0, c(0) = m0, and

c

(t) = X(c(t))for all t ]a, b[; i.e., a solution curve of this ODE.

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An integral curve ofXwith initial condition m0 att = 0 is a (differentiable) map c : ]a, b[ M such that]a, b[ is an open interval containing 0, c(0) = m0, and

c(t) = X(c(t))

for all t ]a, b[; i.e., a solution curve of this ODE.

Flow of X: The collection of maps t : M Msuch that t t(m) is the integral curve ofX with

initial condition m.28



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Existence and uniqueness theorems from ODE guaran-

tee that exists is smooth in m and t (where defined)ifX is.

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Existence and uniqueness theorems from ODE guaran-

tee that exists is smooth in m and t (where defined)ifX is.

Uniqueness flow property

t+s = t s

along with the initial condition 0 = identity.

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An important fact is that ifX is a Cr vector field, then

its flow is a Cr map in all variables.

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An important fact is that ifX is a Cr vector field, then

its flow is a Cr map in all variables.An important estimate that is the key to proving things

like this is the Gronwall inequality: if on an inter-

val [a, b], we havef(t) A +

ta

f(s)g(s)ds

where A 0, and f, g 0 are continuous, then

f(t) A exp

t

a

g(s)ds

This is used on the integral equation defining a flow

(using a chart).30



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Strategy for proving that the flow is C1:

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Formally linearize the equations and find the equationthat Dt should satisfy. This is the first variationequation.

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Formally linearize the equations and find the equationthat Dt should satisfy. This is the first variationequation.

Use the basic existence theory to see that the first vari-ation equation has a solution.

Show that this solution actually satisfies the definition

of the derivative oft by making use of Gronwall.

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Notion ofcompleteness of vector fields ; integral

curves are defined for all time.Criteria discussed in book; an important case is that

if, for example, by making a priori estimates, one can

show that any integral curve defined on a finite opentime interval (a, b) remains in a compact set, then thevector field is complete.

For instance sometimes such an estimate can be ob-

tained as an energy estimate; eg, the flow of x +x3 = 0 is complete since it has the preserved energyH(x, x) = 12x

2 + 14x4, which confines integral curves to

balls for finite time intervals.33

Differential of a Function


75/90

Derivative of f : M R at m M: gives a map

Tmf : TmM Tf(m)R = R.

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Tmf : TmM Tf(m)R = R.Gives a linear map df(m) : TmM R.

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Tmf : TmM Tf(m)R = R.Gives a linear map df(m) : TmM R.

Thus, df(m) TmM, the dual ofTmM.

If we replace each vector space TmM with its dualTmM, we obtain a new 2n-manifold called the cotan-gent bundle and denoted by TM.

We call df the differential off. For v TmM, wecall df(m) v the directional derivative of f inthe direction v.

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In a coordinate chart or in linear spaces, this notion co-

incides with the usual notion of a directional derivativelearned in vector calculus.

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Vector Fields as Differential Operators


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Specifying the directional derivatives completely deter-

mines a vector, and so we can identify a basis ofTmMusing the operators /xi. We write

{e1, . . . , en} =

x1, . . . ,

xn

for this basis, so that v = vi/xi.

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86/90

The dual basis to /xi is denoted by dxi. Thus, rel-

ative to a choice of local coordinates we get the basicformula

df(x) =f

xidxi

for any smooth function f : M R.

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87/90

The dual basis to /xi is denoted by dxi. Thus, rel-

ative to a choice of local coordinates we get the basicformula

df(x) =f

xidxi

for any smooth function f : M R.We also have

X[f] = Xif

xi

which is why we write

X= Xi

xi

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Jacobi-Lie Bracket


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Given two vector fields X and Y, here is a unique

vector field [X, Y] such that as a differential operator,[X, Y][f] = X[Y[f]] Y[X[f]]

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Jacobi-Lie Bracket


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Given two vector fields X and Y, here is a unique

vector field [X, Y] such that as a differential operator,[X, Y][f] = X[Y[f]] Y[X[f]]

Easiest to see what is going on by computing the com-

mutator in coordinates:

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Jacobi-Lie Bracket


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We get

[X, Y] =Xi

xi, Yj

xj

=

Xi

xiYj

xj Yj

xjXi

xi

= XiYj

xi

xj Yj

Xi

xj

xi

= XiYj

xi Yi

Xj

xi

xj

Note that the second derivative terms 2/xixj can-celed out. This is the secret why the bracket is a

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