A Primer on Geometric Mechanics [5pt] Why Geometric Mechanics?isg · I Lecture 4: Symmetry and...

A Primer on Geometric Mechanics

Why GeometricMechanics?

Alex L. Castro, PUC Rio de Janeiro

Henry O. Jacobs, CMS, Caltech

Christian Lessig, CMS, Caltech

Christian Lessig (Caltech) Why Geometric Mechanics? 1 / 162

Outline

Overview

How does the Geometry get into Mechanics?

Geometric Mechanics and Computations

Manifolds and Tensor Analysis

Literature

Bibliography

Overview

Course Outline

I Five lectures.

I Lecture 1: Why geometry? Manifolds and tensors.

I Lecture 2: Lagrangian mechanics.

I Lecture 3: Hamiltonian mechanics.

I Lecture 4: Symmetry and reduction.

I Lecture 5: Discrete geometric mechanics.

I http://www.cms.caltech.edu/~lessig/primer/


http://www.cms.caltech.edu/~lessig/primer/

Overview

Course Outline

I Five lectures.









Overview

Course Outline

I Five lectures.









Overview

Course Outline

I Five lectures.









Overview

Course Outline

I Five lectures.









Overview

Course Outline

I Five lectures.









Overview

Course Outline

I Five lectures.









Outline

Overview




Literature

Bibliography


The Single Pendulum13



















1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ

G g ξ µ µ−1(ξ) T ∗Q

g∗ id

J J−1(ξ)



The Double Pendulum20
















1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)

1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)




1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)

1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)

1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)

1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)


















The Rigid Body202



The Rigid Body203



The Rigid Body204



Geometric Mechanics

32

f(x) =n∑

i=1fi ϕi(x) (1)

=n∑

i=1〈f(x), ϕi(x)〉ϕi(x) (2)

f(λ) = 〈f(x), kλ(x)〉 = δλ(f) (3)

f(x) =n∑

i=1〈f(x), kλi(x)〉 ki(x) (4)

=n∑

i=1f(λi) ki(x) (5)

fκ = f(x) + κ

n∏

i=1(x− λi)2 (6)

Q T ∗Q (7)

η∗t ` ˙ = −£XH` (8)

η∗t ` ˙ = −£XH` (9)

η∗t ` ˙ = −£XH` (10)

T ∗Q XH (11)

g g∗ ˙ = ad∗δHδ`` Diffcan(T ∗Q) (12)

1

Diffµ(Q) Diffcan(T ∗Q)

ω = £vω ˙ = £XH `

T ∗qQ q

W 0a = 1

2

[I +Q U + iVU − iV I −Q

]dq ∧ dp

` = L(q, p) dq ∧ dp

ηt η∗−t`

Q ⊆ R3

ρ(q) dq3

η−t ϕt ψs

ηt = ϕt ψs

λ kλ(x) f(λ)

3

Geometric Mechanics



Geometric Mechanics

I Geometric intuition.

I Unified description.I Lagrangian mechanics.I Hamiltonian mechanics.

I Tools of differential geometry.I Reduced descriptions.I Stability.I Existence of solutions.I . . .



Geometric Mechanics






Geometric Mechanics



I Tools of differential geometry.

I Reduced descriptions.I Stability.I Existence of solutions.I . . .



Geometric Mechanics





Outline

Overview




Literature

Bibliography


Symplectic Integrators19

1

p

θ

θ φ

qi−1 qi qi+1

qt TqtQ


g∗ id

J J−1(ξ)



Symplectic Integrators

Explicit Euler Integrator

Figure 1: Three integrators in phase space (q, p): (left) explicit, (middle) implicit, (right) symplectic. Six initial conditions are shown, with their respectivetrajectories; only the symplectic integrator captures the periodic nature of the pendulum. The bold trajectories correspond to the exact same initial condition.

increases over time, rather than being conserved. Thus, in prac-tice, the solution often “blows up” and becomes unstable as timeprogresses—not a great quality for a time integrator. Fortunately,the implicit Euler is stable: the amplitude of the pendulum’s os-cillations actually decreases over time, avoiding any chance of nu-merical divergence (see Fig. 2). However, this stability comes ata cost: the pendulum loses energy, causing the pendulum to slowdown towards a stop, even if our original equations do not includeany damping forces. Effectively, we resolved the stability issuethrough the introduction of numerical dissipation—but we inducedthe opposite problem instead. The symplectic method, on the otherhand, both is stable and oscillates with constant amplitudes. Thisis obviously a superior method for physical simulation, given thatno additional numerical operations were needed to get the correctqualitative behavior!

Figure 2: The pendulum: for the equation of motion of a pendulum of lengthL and unit mass in a gravitation field g (left), our three integrators behavevery differently: while the explicit Euler integrator exhibits amplifying oscil-lations, the implicit one dampens the motion, while the symplectic integratorperfectly captures the periodic nature of the pendulum.

Now, if we are only solving for the position of the pendulumonly at one particular time, it does not really matter which methodwe use: taking small enough time steps will guarantee arbitrarilygood accuracy. However, if we wish our time integrator to be glob-ally predictive, the least we can ask for is to get a pendulum thatactually keeps on swinging. Even a simple animation of a grandfa-ther clock or a child on a swing would look unrealistic if it seemedto gain or lose amplitude inexplicably. In other words, the behaviorof energy over time is of key importance. But how do we know thatan integrator will have these good properties ahead of time? Canwe construct them for an arbitrary physical system? The answer, aswe shall see, comes from the world of geometric mechanics and aconcept called symplecticity.

4 Geometric Mechanics

In the familiar Newtonian view of mechanics, we begin by addingup the forces F on a body and writing the equations of motion using

the famous second law,

F = ma, (4)

where a represents the acceleration of the body. With geomet-ric mechanics, however, we consider mechanics from a variationalpoint of view. In this section, we review the basic foundations ofLagrangian mechanics, one of the two main flavors of geometricmechanics (we will only point to some connections with Hamilto-nian mechanics).

4.1 Lagrangian Mechanics

Consider a finite-dimensional dynamical system parameterized bythe state variable q, i.e., the vector containing all degrees of free-dom of the system. In mechanics, a function of a position q anda velocity q called the Lagrangian function L is defined as the ki-netic energy K (usually, only function of the velocity) minus thepotential energy U of the system (usually, only function of the statevariable):

L(q, q) = K(q) U(q).

Variational Principle The action functional is then introducedas the integral of L along a path q(t) for time t 2 [0, T ]:

S(q) =

Z T

0

L(q, q) dt.

With this definition, the main result of Lagrangian dynamics,Hamilton’s principle, can be expressed quite simply: this varia-tional principle states that the correct path of motion of a dynamicalsystem is such that its action has a stationary value, i.e., the inte-gral along the correct path has the same value to within first-orderinfinitesimal perturbations. As an “integral principle” this descrip-tion encompasses the entire motion of a system between two fixedtimes (0 and T in our setup). In more ways than one, this principleis very similar to a statement on the geometry of the path q(t): theaction can be seen as the analog of a measure of “curvature”, andthe path is such that this curvature is extremized (i.e., minimized ormaximized).

Euler-Lagrange Equations How do we determine which pathoptimizes the action, then? The method is similar to optimizing anordinary function. For example, given a function f(x), we knowthat its critical points exist where the derivative rf(x) = 0. Sinceq is a path, we cannot simply take a “derivative” with respect toq; instead, we take something called a variation. A variation ofthe path q is written q, and can be thought of as an infinitesimal

from A. Stern and M. Desbrun (2006). “Discrete Geometric Mechanics for Variational Time Integrators”. In:SIGGRAPH ’06: ACM SIGGRAPH 2006 Courses. ACM




Implicit Euler Integrator








F = ma, (4)




L(q, q) = K(q) U(q).


S(q) =

Z T

0

L(q, q) dt.







Symplectic Euler Integrator








F = ma, (4)




L(q, q) = K(q) U(q).


S(q) =

Z T

0

L(q, q) dt.














F = ma, (4)




L(q, q) = K(q) U(q).


S(q) =

Z T

0

L(q, q) dt.



explicit implicit symplecticEuler Euler Euler




Electromagnetic Theory

Equations~E = ∇×

(1/µ ~B

)e = d (?µ b)

~B = ∇× ~E b = de

Geometry ~E , ~B = X(R3) e ∈ Ω1, b ∈ Ω2




Equations~E = ∇×

(1/µ ~B

)e = d (?µ b)

~B = ∇× ~E b = de





Equations~E = ∇×

(1/µ ~B

)e = d (?µ b)

~B = ∇× ~E b = de





./figures/em_energy_levels_forms.pdf

from D. N. Arnold (Nov. 2002). “Differential complexes and numerical stability”. In: Proceedings of the ICM.Beijing, pp. 137–115. arXiv:0212391 [math]. url: http://arxiv.org/abs/math/0212391


http://arxiv.org/abs/0212391

http://arxiv.org/abs/math/0212391



from D. N. Arnold (Nov. 2002). “Differential complexes and numerical stability”. In: Proceedings of the ICM.Beijing, pp. 137–115. arXiv:0212391 [math]. url: http://arxiv.org/abs/math/0212391






Equations~E = ∇×

(1/µ ~B

)e = d (?µ b)

~B = ∇× ~E b = de




Electromagnetic TheoryVariational Tetrahedral Meshing

Pierre AlliezINRIA

David Cohen-SteinerINRIA

Mariette YvinecINRIA

Mathieu DesbrunCaltech

AbstractIn this paper, a novel Delaunay-based variational approach toisotropic tetrahedral meshing is presented. To achieve both robust-ness and efficiency, we minimize a simple mesh-dependent energythrough global updates of both vertex positions and connectivity.As this energy is known to be the L1 distance between an isotropicquadratic function and its linear interpolation on the mesh, our min-imization procedure generates well-shaped tetrahedra. Mesh designis controlled through a gradation smoothness parameter and selec-tion of the desired number of vertices. We provide the foundationsof our approach by explaining both the underlying variational prin-ciple and its geometric interpretation. We demonstrate the qualityof the resulting meshes through a series of examples.Keywords: Isotropic meshing,Delaunay mesh,sizing field,slivers.

1 IntroductionThree-dimensional simplicial mesh generation aims at tiling abounded 3D domain with tetrahedra so that any two of them areeither disjoint or sharing a lower dimensional face. Such a dis-cretization of space is required for most physically-based simula-tion techniques: realistic simulation of deformable objects in com-puter graphics, as well as more general numerical solvers for par-tial differential equations in computational science, need a discretedomain to apply finite-element or finite-volume methods. Most ap-plications have specific requirements on the size and shape of sim-plices in the mesh. Isotropic meshing is desirable in the commoncase where nearly-regular tetrahedra (nearly-equal edge lengths)are preferred.

Creating high quality tetrahedral meshes is a difficult task for a vari-ety of reasons. First, the mere size of the resulting meshes requiresrobust, disciplined data structures and algorithms. There are alsobasic mathematical difficulties which make tetrahedral meshingsignificantly harder than its 2D counterpart: the most isotropic 3Dsimplex, the regular tetrahedron, does not tile 3D space (let alonespecific domains), while the equilateral triangle does tile the plane;unlike the 2D case, even well-spaced vertices can create degenerate3D elements such as slivers (see Fig. 2). Dealing with boundaries isalso fundamentally more difficult in 3D: while it always exists a 2Dtriangulation conforming to any set of non intersecting constraints,this is no longer true in 3D [Shewchuk 1998a]. All these facts ren-der both the development of algorithms and suitable error analysisfor the optimal 3D meshing problem very challenging. Given thatone can often observe in applications that the worst element in thedomain dictates accuracy and/or efficiency [Shewchuk 2002a], it isclear that great care is required to design the underlying meshes andensure that they meet the desired quality standards.

1.1 Previous Work & NomenclatureThe meshing community has extensively studied a number of tech-niques over the last 20 years. We do not aim at covering all previ-

Figure 1: Variational Tetrahedral Meshing: Given the boundary of a do-main (here, a human torso), we automatically compute the local feature sizeof this boundary as well as an interior sizing field (left, cross-section), be-fore constructing a mesh with a prescribed number of vertices (here 65K)and a smooth gradation conforming to the sizing field (right, cutaway view).The resulting tetrahedra are all well-shaped (i.e., nearly regular).ous work since comprehensive surveys are available [Carey 1997;Owen 1998; Frey and George 2000; Teng et al. 2000; Eppstein2001]. To motivate our work we briefly review both the usualnomenclature and the main difficulties involved in isotropic tetra-hedral mesh generation. Throughout tet will be the abbreviation fortetrahedron.

Proper mesh generation requires a number of successive stages,which are governed by a number of key factors:! Shape Quality Measures: Element shape/size requirements are

typically application-dependent. Consequently, an extraordinar-ily large number of quality measures has been proposed, rangingfrom minimum or maximum bounds on dihedral or solid angles,to more complex geometric ratios. We recommend [Shewchuk2002a] for a clear exposition of both the history behind these mea-sures and their relation to (1) the conditioning of finite elementstiffness matrices and (2) the accuracy of linear interpolation offunctions and their gradients. Among the most popular qualitymeasures of a tet are the radius and radius-edge ratios. The lat-ter measures the ratio between the circumsphere radius and theshortest edge length. It is not a fair measure since it does notapproach zero for a class of degenerate tets called slivers (sliv-ers result when four tet vertices are close to a great circle of asphere and spaced roughly equally along this circle, see Fig. 2).The radius ratio, which takes the quotient of inscribed and cir-cumscribed sphere radii (times three for normalization purposes),is a good measure for any kind of degeneracy.

Figure 2: Tet shapes: the regular tet (leftmost) is well shaped, unlike theother tets displayed: each represents a type of degeneracy. The rightmostone with 4 near-cocircular vertices is usually referred to as a sliver.

! Sizing requirement: Accuracy and efficiency of numericalsolvers depend on the local size of tets. Consequently, a sizingfield, prescribing the ideal local edge length as a function of space,

from P. Alliez et al. (July 2005). “Variational tetrahedral meshing”. In: ACM SIGGRAPH 2005 Papers on -SIGGRAPH ’05. Vol. 24. 3. New York, New York, USA: ACM Press, p. 617




Equations~E = ∇×

(1/µ ~B

)e = d (?µ b)

~B = ∇× ~E b = de




Ideal Fluid Mechanics

Equations v +∇v v = −∇p ω = Lvω

Geometry v ∈ Xdiv(M)ω ∈ Ω2 ∼= g∗

Q = Diffµ(M)






Q = Diffµ(M)






Q = Diffµ(M)



Ideal Fluid Dynamics

from P. Mullen et al. (2009). “Energy-Preserving Integrators for Fluid Animation”. In: ACM Transactions onGraphics (Proceedings of SIGGRAPH 2009) 28.3, pp. 1–8



























Geometric Mechanics

32

f(x) =n∑

i=1fi ϕi(x) (1)

=n∑

i=1〈f(x), ϕi(x)〉ϕi(x) (2)

f(λ) = 〈f(x), kλ(x)〉 = δλ(f) (3)

f(x) =n∑

i=1〈f(x), kλi(x)〉 ki(x) (4)

=n∑

i=1f(λi) ki(x) (5)

fκ = f(x) + κ

n∏

i=1(x− λi)2 (6)

Q T ∗Q (7)

η∗t ` ˙ = −£XH` (8)

η∗t ` ˙ = −£XH` (9)

η∗t ` ˙ = −£XH` (10)

T ∗Q XH (11)


1


ω = £vω ˙ = £XH `

T ∗qQ q

W 0a = 1

2


]dq ∧ dp

` = L(q, p) dq ∧ dp

ηt η∗−t`

Q ⊆ R3

ρ(q) dq3

η−t ϕt ψs

ηt = ϕt ψs

λ kλ(x) f(λ)

3

Geometric Mechanics



Geometric MechanicsDiscrete Geometric Mechanics

51

f(x) =n∑

i=1fi ϕi(x) (1)

=n∑

i=1〈f(x), ϕi(x)〉ϕi(x) (2)

f(λ) = 〈f(x), kλ(x)〉 = δλ(f) (3)

f(x) =n∑

i=1〈f(x), kλi(x)〉 ki(x) (4)

=n∑

i=1f(λi) ki(x) (5)

fκ = f(x) + κ

n∏

i=1(x− λi)2 (6)

Q T ∗Q (7)

η∗t ` ˙ = −£XH` (8)

η∗t ` ˙ = −£XH` (9)

η∗t ` ˙ = −£XH` (10)

T ∗Q XH (11)


1


ω = £vω ˙ = £XH `

T ∗qQ q

W 0a = 1

2


]dq ∧ dp

` = L(q, p) dq ∧ dp

ηt η∗−t`

Q ⊆ R3

ρ(q) dq3

η−t ϕt ψs

ηt = ϕt ψs

λ kλ(x) f(λ)

3



Discrete Geometric Mechanics

Effective simulations require:

1. Understanding of intrinsic geometric structure.

2. Discretization of geometric structure.I Discretizing equation leads to artifacts because

usually the geometric structure will be lost.

3. Discrete mechanical system whose structuremirrors those of the continuous system.

I Geometric structure and symmetries representessential properties of a system.






2. Discretization of geometric structure.

I Discretizing equation leads to artifacts becauseusually the geometric structure will be lost.































Outline

Overview




Literature

Bibliography

Manifolds and Tensor Analysis Preliminaries

Preliminaries

DefinitionA homomorphism is a structure preserving mapbetween two algebraic objects.

ExampleLet (H, 〈 , 〉) be an n-dimensional Hilbert space andϕin

i=1 be a basis for H. Then ϕini=1 establishes a

homomorphism between H and Rn with the innerproduct 〈 , 〉 becoming the dot product.



Preliminaries

DefinitionA homomorphism is a structure preserving mapbetween two algebraic objects.


i=1 be a basis for H. Then ϕini=1 establishes a

homomorphism between H and Rn with the innerproduct 〈 , 〉 becoming the dot product.



Preliminaries

DefinitionAn isomorphism between two algebraic structures isa homomorphism whose inverse is also ahomomorphism.


i=1 be a basis for H. Then ϕini=1 establishes

an isomorphism between H and Rn with the innerproduct 〈 , 〉 becoming the dot product. The inversemap is given by reconstruction.



Preliminaries

DefinitionAn isomorphism between two algebraic structures isa homomorphism whose inverse is also ahomomorphism.


i=1 be a basis for H. Then ϕini=1 establishes

an isomorphism between H and Rn with the innerproduct 〈 , 〉 becoming the dot product. The inversemap is given by reconstruction.



PreliminariesDefinitionA diffeomorphism ϕ : A → B between two Banachspaces A, B is a smooth map that has a smoothinverse.

Example




Example




Example

http://en.wikipedia.org/wiki/Diffeomorphism






Example





Manifolds and Tensor Analysis Linear Algebra Revisited

Linear algebra revisited






















DefinitionA linear space or a vector space V over the realnumbers R is a set together with two operations:

i) addition of elements in V: x + y : V × V → V ;

ii) scalar multiplication: ax : V × R→ V .

V forms an Abelian group under addition + so that

x + 0 = x

and every element has an inverse element −x ,

x + (−x) = x − x = 0.








x + 0 = x


x + (−x) = x − x = 0.








x + 0 = x


x + (−x) = x − x = 0.







~e1

~e2

2 ~e3




~v =3∑

i=1

v i~ei

~e1

~e2

2 ~e3




~v =3∑

i=1

(~v · ~ei )~ei

~e1

~e2

2 ~e3




~e1

~e2

2 ~e3




~e1

~e2

2 ~e3




~e1 ~e1

~e2

2 ~e3




~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3




~v =3∑

i=1

v i~ei

~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3




~v =3∑

i=1

(~v · ~e i )~ei

~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3




~e1

~e22 ~e3




~e1

~e22 ~e3

~e1

1 ~e2

2 ~e3




~v =3∑

i=1

(~v · ~e i )~ei

~v ,~ei ∈ V

~e i ∈ V ∗~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3




~v =3∑

i=1

(~v · ~e i )~ei

~v ,~ei ∈ V

~e i ∈ V ∗~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3




~v =3∑

i=1

(~v · ~e i )~ei

~v ,~ei ∈ V

~e i ∈ V ∗~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3




~v =3∑

i=1

(~v · ~e i )~ei

~v ,~ei ∈ V

~e i ∈ V ∗~e1 ~e1

~e2

2 ~e3

1 ~e2

2 ~e3


Manifolds and Tensor Analysis Manifolds

Manifolds

Idea: Represent a curved space through local regionsthat “look like” Euclidean space.

M

Ui

ϕi

i ϕj

ϕij

i Uj



Manifolds


M

Ui

ϕi

i ϕj

ϕij

i Uj



Manifolds


M

Ui

ϕi

i ϕj

ϕij

i Uj



Manifolds


I Chart map ϕi : Ui ⊂M→ ϕi (Ui ) ⊂ Rn is abijection.

I All chart maps form an atlasA = (U1, ϕ1), . . . such that M =

⋃Ui .

I Transition maps ϕij = ϕi ϕ−1

j arediffeomorphisms.

I The Cartesian coordinates in ϕi (Ui ) ⊂ Rn

induce local coordinates on M.



Manifolds




⋃Ui .







Manifolds




⋃Ui .







Manifolds




⋃Ui .







The sphere S2 as manifold

1S

2




θ

φ

ϕ−1

1S

2

2π

φ π




I Chart map:

ϕ−1 : U ⊂ R2 → S2 :

xyz

=

sin θ cosφsin θ sinφ

cos θ

I Atlas: A = (U , ϕ) (up to a point).

I Local coordinates are longitudes and latitudes(θ and φ isolines).




I Chart map:

ϕ−1 : U ⊂ R2 → S2 :

xyz

=


cos θ






I Chart map:

ϕ−1 : U ⊂ R2 → S2 :

xyz

=


cos θ





ManifoldsExample: Smooth map between manifolds.

M N

m f(m)

) f



ManifoldsExample: Smooth map between manifolds.

M N

m f(m)

) f

ϕ(m)) ψ(f(m))

)) ψ f ϕ−1

1 fψϕ



Curves on manifolds

DefinitionLet M be a manifold.A curve c(t) on Mis a smooth mappingc(t) : [a, b]→M.

M

1

c1 c2



Curves on manifolds


M

ϕ

ϕ c1

1 ϕ c2

c1 c2



Curves on manifolds


M

ϕ

ϕ c1

1 ϕ c2

c1 c2



Tangent space



Tangent space



Tangent space



Tangent space



Tangent space



Tangent space



Tangent space



Tangent space

TmM



Tangent space

DefinitionLet M be a manifold. The space of all tangentvectors to curves c(t) : [a, b]→M satisfyingc(0) = m ∈M at t = 0 is the tangent space TmMof M at m. The union

TM =⋃

TmM

is the tangent space of M.



Tangent space



Tangent space



Tangent space



Tangent space



Tangent space

TmM



Tangent Spaces

DefinitionLetM be an n-dimensional manifold and (U , ϕ) be achart for U ⊂M with ϕ : U ⊂M→ V ⊂ Rn givenby ϕ(u) = (x1(u), . . . , xn(u)) in V . Then for fixed ithe x i (u) provide maps x i : u → x i (u) and these arethe local coordinates for M defined by (U , ϕ).

TmM



Tangent Spaces

DefinitionLet M be an n-dimensional manifold and (U , ϕ) bea chart for U ⊂M with ϕ : U ⊂M→ V ⊂ Rn

given by ϕ(u) = (x1(u), . . . , xn(u)) in V . Then alocal basis for TuM is given by ∂/∂x1, . . . , ∂/∂xnwhere

∂

∂x i=

∂

∂x iϕ−1 =

(∂ϕ−1

1

∂x i, · · · , ∂ϕ

−1n

∂x i

)

and ϕ−1 = (ϕ−11 , . . . , ϕ−1

n ).



Tangent Spaces for the sphere

1S

2




1S

2

TmS2




TmS2




TmS2

S tθ

tφ

θ

θ φ

ϕ−1

2π

φ π

e2

e1




ϕ−1 : U ⊂ R2 → S2 :

xyz

=


cos θ




∂

∂θ≡ ∂

∂θϕ−1 =

(∂ϕ−1

x

∂θ,∂ϕ−1

y

∂θ,∂ϕ−1

z

∂θ

)

=

(∂ϕ−1

x

∂θ,∂ϕ−1

y

∂θ,∂ϕ−1

z

∂θ

)T

=

cos θ cosφcos θ sinφ− sin θ




∂

∂θ≡ ∂

∂θϕ−1 =

(∂ϕ−1

x

∂θ,∂ϕ−1

y

∂θ,∂ϕ−1

z

∂θ

)

=

(∂ϕ−1

x

∂θ,∂ϕ−1

y

∂θ,∂ϕ−1

z

∂θ

)T

=





∂

∂θ≡ ∂

∂θϕ−1 =

(∂ϕ−1

x

∂θ,∂ϕ−1

y

∂θ,∂ϕ−1

z

∂θ

)

=

(∂ϕ−1

x

∂θ,∂ϕ−1

y

∂θ,∂ϕ−1

z

∂θ

)T

=





∂

∂θ=


∂

∂φ=


tabular with spheretangent5 as right hand figure



Vector fields

DefinitionLet M be a manifold. A vector field X (M) on Mis a section of the tangent bundle TM. The spaceof all vector fields on M is denoted by X(M).



Vector fields




Vector fields




Vector fields




Vector fields


In local coordinates, a vector field is given by

X (m) = X 1(m)e1 + . . . + X nen.



Vector fieldsDefinitionLet X (M) be a vector field on a manifold M. Anintegral curve of X (M) at m ∈M is a curvec(t) : [a, b]→M such that dc(t)/dt = X (c(t)) forall t ∈ [a, b].



Vector fieldsDefinitionLet X (M) be a vector field on a manifold M. Anintegral curve of X (M) at m ∈M is a curvec(t) : [a, b]→M such that dc(t)/dt = X (c(t)) forall t ∈ [a, b].



Vector fields

I Integral curves and flows along vector fields arecentral to describe time evolution of mechanicalsystems.

I Observables are transported along vector fields.

I Transport along a vector field satisfies groupproperty.

I In fact Lie group since the time t is a continuousparameter.



Vector fields







Vector fields







Vector fields







Cotangent space

TM



Cotangent space

TM

T ∗M



Cotangent space

DefinitionLet TM be the tangent bundle of a manifold M.Then the cotangent space T ∗mM at m ∈M isthe linear dual space to the tangent space TmM.The union of all cotangent spaces is the cotangentbundle T ∗M.

TM

T ∗M


Manifolds and Tensor Analysis Tensors

Tensors

I Natural extension of sections of X(M) andX∗M to “higher order” objects.

I Enables to define physical quantities that arenatural, that is independent of the coordinatesystem used.



Tensors

I Natural extension of sections of X(M) andX∗M to “higher order” objects.

I Enables to define physical quantities that arenatural, that is independent of the coordinatesystem used.



Tensors

DefinitionLet V be a vector space. A tensor

t rs : V ∗ × . . .× V ∗︸︷︷︸

r times

×V × . . .× V︸︷︷︸s times

on V of type (r , s), contravariant of order r andcovariant of order s, is a linear map in

T rs (V ) = Lr+s(E ∗, . . . ,E ∗︸︷︷︸

r times

,E , . . . ,E︸︷︷︸s times

)

that is linear in each of its arguments.



Tensors

The component representation of a tensor is

t =∑

i1···is

∑

j1···jr

t j1,...jri1,...,is

ej1 ⊗ . . .⊗ ejr ⊗ e i1 ⊗ . . .⊗ e is

with the components being

t j1,...,jri1,...,is

= t(e j1, . . . , e jr , ei1, . . . , eis)



Tensors

The component representation of a tensor is

t =∑

i1···is

∑

j1···jr

t j1,...jri1,...,is

ej1 ⊗ . . .⊗ ejr ⊗ e i1 ⊗ . . .⊗ e is

with the components being

t j1,...,jri1,...,is

= t(e j1, . . . , e jr , ei1, . . . , eis)



Tensors

Example: Riemannian structure.

A Riemannian structure on a manifold M enables to“compare” two tangent vectors in TmM.

gm : TmM× TmM→ R ∼= g ∈ T 02 (M)



Tensors


A Riemannian structure on a manifold M enables to“compare” two tangent vectors in TmM.

gm : TmM× TmM→ R ∼= g ∈ T 02 (M)



Tensors


A Riemannian structure on a manifold M enables to“compare” two tangent vectors in TmM

gm : TmM× TmM→ R ∼= g ∈ T 02 (M)

or in coordinates

g(~u, ~v) =∑

ij

gij ui v j ∈ R



Tensors



gm : TmM× TmM→ R ∼= g ∈ T 02 (M)

or in coordinates

g(~u, ~v) =∑

ij

gij ui v j ∈ R



Tensors



gm : TmM× TmM→ R ∼= g ∈ T 02 (M)

or in coordinates

g(~u, ~v) = gij ui v j ∈ R


Manifolds and Tensor Analysis Differential Forms and Exterior Calculus

Vector Calculus Revisited

Differential operators:

I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)





I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)





I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)





I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)





I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)




Integral laws:

I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)




Integral laws:

I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)




Integral laws:

I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)




Integral laws:

I grad: F(R3)→ X(R3) : f 7→ ∇f ∈ X(R3)

I curl: X(R3)→ X(R3) : ~A 7→ ∇ × ~A ∈ X(R3)

I div: X(R3)→ F(R3) : ~A 7→ ∇ · ~A ∈ X(R3)



Differential Forms in R3

1-forms and Gauss law




2-forms and Gauss law




3-forms and volume integrals



Differential Forms

Definition

I Objects “ready to be integrated”.

I Central to the covariant formulation ofdifferential equations.



Differential Forms

Definition

I Objects “ready to be integrated”.

I Central to the covariant formulation ofdifferential equations.



Exterior derivative

Definition



Exterior derivative: vector calculusrevisited

I grad

I curl

I div



Exterior derivative: vector calculusrevisited

I grad

I curl

I div



Exterior derivative: coordinate expression



Exterior derivative: coordinate expression


Manifolds and Tensor Analysis The Lie Derivative

The pullback

I Start again with flow along a vector field as amap ηt :M→M.

I Pullback: Transport of tensors along the flow



The Lie Derivative

I Dynamic definition: infinitesimal transport of atensor along a vector field.



The Lie Derivative 80

T ∗Q

XH

ℓ ℓ = −£XH ℓ

η∗−tℓ


Manifolds and Tensor Analysis Integration on Manifolds

Integration on Manifolds

I Known: how to integrate k-form over a regionin Rk (e.g. Riemann integral).

I k-manifold is modelled on k-dimensionalEuclidean space.

I Charts are natural in that the result is independentof the chart (or parametrization) we use.

I =¿ Integration of forms by pulling them back tothe chart.



Integration on Manifolds: the sphere S2



Integration on Manifolds

Definition


Manifolds and Tensor Analysis Summary


I Manifolds are curved spaces that locally “look”like Euclidean space.

I Curves on manifolds induce the tangent andcotangent space.

I Tensors are “higher order tangent and cotangentvectors” that represent physical quantities.

I The Lie derivative represents the infinitesimalflow of a tensor along a vector field.

I Differential forms are anti-symmetric tensorscentral for covariant differential equations andintegration.


































Outline

Overview




Literature

Bibliography

Literature

Literature for Geometric Mechanics

Introductory

I Marsden and Ratiu, Introduction to Mechanics andSymmetry: A Basic Exposition of Classical MechanicalSystems, 2004.

I Arnold, Mathematical Methods of Classical Mechanics,1989.

I Frankel, The Geometry of Physics, 2003.

I Holm, Schmah, and Stoica, Geometric Mechanics andSymmetry: From Finite to Infinite Dimensions, 2009.

I Novikov and Taimanov, Modern Geometric Structuresand Fields, 2006.


Literature

Literature for Geometric Mechanics

Advanced / Reference

I Marsden, Ratiu, and Abraham, Manifolds, TensorAnalysis, and Applications, 2004.

I Montgomery, A Tour of Subriemannian Geometries, TheirGeodesics, and Applications, 2001.

I Cushman and Bates, Global Aspects of ClassicalIntegrable Systems, 2004.

I Cendra, Marsden, and Ratiu, Lagrangian Reduction byStages, 2001.

I Marsden et al., Hamiltonian Reduction by Stages, 2007.

I Ortega and Ratiu, Momentum Maps and HamiltonianReduction, 2004.


Outline

Overview




Literature

Bibliography

Bibliography

BibliographyAlliez, P., D. Cohen-Steiner, M. Yvinec, and M. Desbrun (July 2005).“Variational tetrahedral meshing”. In: ACM SIGGRAPH 2005 Papers on -SIGGRAPH ’05. Vol. 24. 3. New York, New York, USA: ACM Press,p. 617.

Arnold, D. N. (Nov. 2002). “Differential complexes and numericalstability”. In: Proceedings of the ICM. Beijing, pp. 137–115.arXiv:0212391 [math]. url: http://arxiv.org/abs/math/0212391.

Arnold, V. I. (Sept. 1989). Mathematical Methods of ClassicalMechanics. second. Graduate Texts in Mathematics. Springer.

Cendra, H., J. E. Marsden, and T. S. Ratiu (2001). LagrangianReduction by Stages. Memoirs of the American Mathematical Society.American Mathematical Society.

Cushman, R. H. and L. M. Bates (2004). Global Aspects of ClassicalIntegrable Systems. Birkhauser Basel, p. 456.

Frankel, T. (2003). The Geometry of Physics. Cambridge UniversityPress.




Bibliography

Holm, D. D., T. Schmah, and C. Stoica (2009). Geometric Mechanicsand Symmetry: From Finite to Infinite Dimensions. Oxford texts inapplied and engineering mathematics. Oxford University Press, 515 p.

Marsden, J. E. and T. S. Ratiu (2004). Introduction to Mechanics andSymmetry: A Basic Exposition of Classical Mechanical Systems. third.Texts in Applied Mathematics. New York: Springer-Verlag.

Marsden, J. E., T. S. Ratiu, and R. Abraham (2004). Manifolds, TensorAnalysis, and Applications. third. Applied Mathematical Sciences. NewYork: Springer-Verlag.

Marsden, J. E., G. Misiolek, J.-P. Ortega, M. Perlmutter, and T. S. Ratiu(2007). Hamiltonian Reduction by Stages. Lecture Notes in Mathematics.Springer.

Montgomery, R. (2001). A Tour of Subriemannian Geometries, TheirGeodesics, and Applications. Mathematical Surveys and Monographs.American Mathematical Society.


Bibliography

Mullen, P., K. Crane, D. Pavlov, Y. Tong, and M. Desbrun (2009).“Energy-Preserving Integrators for Fluid Animation”. In: ACMTransactions on Graphics (Proceedings of SIGGRAPH 2009) 28.3,pp. 1–8.

Novikov, S. P. and I. A. Taimanov (2006). Modern Geometric Structuresand Fields. Graduate Studies in Mathematics. American MathematicalSociety.

Ortega, J.-P. and T. S. Ratiu (2004). Momentum Maps and HamiltonianReduction. Progress in Mathematics. Boston, Basel, Berlin: Birkhauser,p. 497.

Stern, A. and M. Desbrun (2006). “Discrete Geometric Mechanics forVariational Time Integrators”. In: SIGGRAPH ’06: ACM SIGGRAPH2006 Courses. ACM.


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