Ivancevic_Applied-Diff-Geom

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Applied Manifold Geometry 453results in the Jacobi Hamiltonian [Lanczos (1986)]. Another closely relatedgeometrization was used in [Rosquist (1997)]. Both methods involvea re–parameterization of the independent variable. Usually we will refer tothe independent variable as the time, although its physical interpretationmay vary. As a consequence of this feature, the original Lax representationis not preserved. It is known how to transform the invariants themselvesunder the time re–parameterization [Rosquist and Pucacco (1995);Rosquist (1997)]. Given that the geometrized invariants are also in involution,the existence of a Lax representation is guaranteed [Babelon andViallet (1990)]. However, to actually find such a Lax representation isnon–trivial. Another geometrization scheme which does preserve the originalLax representation is to apply a suitable canonical transformation. Thisis however only possible for Hamiltonians with a potential of a special form.One such system that we will consider in this paper is the 3–particle openToda latticeH = 1 2(¯p1 2 + ¯p 2 2 + ¯p 32 ) + e 2(¯q1 −¯q 2) + e 2(¯q2 −¯q 3) . (3.265)Below we will discuss two canonical transformations which correspond to inequivalentgeometric representations of (3.265). For an explicit integrationof the Toda lattice, see e.g. [Perelomov (1990)]. The standard symmetricLax representation is [Perelomov (1990)]⎛ ⎞ ⎛⎞¯p 1 ā 1 00 ā 1 0L = ⎝ ā 1 ¯p 2 ā 2⎠ , A = ⎝ −ā 1 0 ā 2⎠ ,0 ā 2 ¯p 3 0 −ā 2 0where ā 1 = exp(¯q 1 − ¯q 2 ), ā 2 = exp(¯q 2 − ¯q 3 ).Note that the definitions of ā 1 and ā 2 differ from the ones used in [Rosquist(1997)]. The Hamiltonian (3.265) admits the linear invariant, I 1 = L = ¯p 1 +¯p 2 + ¯p 3 , corresponding to translational invariance. The Lax representationalso gives rise to the two invariants, I 2 = 1 2 L2 = H and I 3 = 1 3 L3 . We willassume that the tensorial Lax representation is linear and homogeneous inthe momenta. A homogeneous Lax representation can be obtained fromthe standard representation by applying a canonical transformation of thephase space.3.16.2.1 Tensorial Lax RepresentationHere we straightforwardly apply a simple canonical transformation that willgive a linear and homogeneous Lax representation [Rosquist and Goliath
454 Applied Differential Geometry: A Modern Introduction(1997)]The resulting Lax pair matrices are¯q 1 = q 1 + ln p 1 , ¯p 1 = p 1 ,¯q 2 = q 2 , ¯p 2 = p 2 ,¯q 3 = q 3 − ln p 3 , ¯p 3 = p 3 .⎛⎞ ⎛⎞p 1 a 1 p 1 00 a 1 p 1 0L = ⎝ a 1 p 1 p 2 a 2 p 3⎠ , A = ⎝ −a 1 p 1 0 a 2 p 3⎠ ,0 a 2 p 3 p 3 0 −a 2 p 3 0where a 1 = exp(q 1 − q 2 ), a 2 = exp(q 2 − q 3 ).The Hamiltonian is now purely kineticH = 1 2 L2 = 1 2Using (3.262) we identify a metric[(1 + 2a12 ) p 1 2 + p 2 2 + ( 1 + 2a 22 ) p 32 ] .ds 2 = g 11 (dq 1 ) 2 + (dq 2 ) 2 + g 33 (dq 3 ) 2 , whereg 11 = ( 21 + 2a ) −11 , g33 = ( 21 + 2a ) −12 .The non–zero Levi–Civita connection coefficients, Γ α βγ = Γα (βγ), of this metricareΓ 1 11 = −2a 2 1 g 11 , Γ 2 33 = 2a 2 2 (g 33 ) 2 ,Γ 1 12 = 2a 2 1 g 11 , Γ 3 23 = −2a 2 2 g 33 ,Γ 2 11 = −2a 2 1 (g 11 ) 2 , Γ 3 33 = 2a 2 2 g 33 .Following the arguments above, the homogeneous Lax matrix should correspondto a tensor with mixed indices L α β. It is a reasonable assumptionthat the covariant Lax formulation inherits the symmetries of the standardformulation we started with. We therefore expect L αβ and B αβ to have thesymmetriesL (αβ) = L αβ and B [αβ] = B αβ .Note that the symmetry properties are not imposed on the Lax matrices,L α β and Bα β, themselves. In fact, the required symmetries are not consistentwith the representation (3.266). We can however perform a similaritytransformation (3.264) of the Lax matrix, L → ˜L in such a way that ˜L αβ
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APPLIEDDIFFERENTIALGEOMETRYA Modern
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APPLIEDDIFFERENTIALGEOMETRYA Modern
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Dedicated to:Nitya, Atma and Kali
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PrefaceApplied Differential Geometr
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ixAcknowledgmentsThe authors wish t
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Glossary of Frequently Used Symbols
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ContentsPrefaceGlossary of Frequent
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Contentsxix2.1.5.2 Forces Acting on
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Contentsxxi3.6.3.4 Stokes Theorem a
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Contentsxxiii3.10.3.1 Basis of Lagr
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Contentsxxv3.17 Applied Unorthodox
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Contentsxxvii4.9.8.2 Geometrodynami
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Contentsxxix4.14.7.5 Monopole Conde
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Contentsxxxi5.12.3 Hawking-Penrose
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Contentsxxxiii6.5.2.4 Dimensional R
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Chapter 1IntroductionIn this introd
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Introduction 3Fig. 1.1 The four cha
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Introduction 5• Riemannian manifo
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Introduction 7charts.The atlas cont
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Introduction 9every point has a nei
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Introduction 11of curves include ci
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Introduction 13the variables define
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Introduction 15U i denotes one of t
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Introduction 17which varies smoothl
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Introduction 19interactions.1.1.5.2
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Introduction 21real number. This co
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Introduction 23Such a force is inde
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Introduction 25tons. In this formul
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Introduction 27vector-field are sol
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Introduction 29or simply connected)
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Introduction 311.1.8 Application: A
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Introduction 33theory itself existe
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Introduction 35mediate the forces.
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Introduction 37theory. 50 Fig. 1.3
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Introduction 39to describe a univer
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Introduction 41which have two disti
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Introduction 43first theory can be
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Introduction 45the Casimir effect,
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Introduction 47• External coordin
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Introduction 49a connection-base de
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Chapter 2Technical Preliminaries: T
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Technical Preliminaries: Tensors, A
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Chapter 3Applied Manifold Geometry3
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Applied Manifold Geometry 139where
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Applied Manifold Geometry 141In loc
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Applied Manifold Geometry 143on the
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Applied Manifold Geometry 145exist
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Applied Manifold Geometry 147then E
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Applied Manifold Geometry 1490D man
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Applied Manifold Geometry 151Fig. 3
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Applied Manifold Geometry 153the fo
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Applied Manifold Geometry 155contin
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Applied Manifold Geometry 157In oth
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Applied Manifold Geometry 159scalar
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Applied Manifold Geometry 1633.6 Te
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Applied Manifold Geometry 165Let ϕ
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Applied Manifold Geometry 1673.6.1.
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Applied Manifold Geometry 169or, if
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Applied Manifold Geometry 171If X i
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Applied Manifold Geometry 173along
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Applied Manifold Geometry 175is dua
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Applied Manifold Geometry 179(1) α
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Applied Manifold Geometry 181diagra
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Applied Manifold Geometry 183Clearl
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Applied Manifold Geometry 187duces,
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Applied Manifold Geometry 189This i
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Applied Manifold Geometry 191Hodge
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Applied Manifold Geometry 193The in
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Applied Manifold Geometry 195since
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Applied Manifold Geometry 197If a l
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Applied Manifold Geometry 199asGive
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Applied Manifold Geometry 201The co
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Applied Manifold Geometry 203at the
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Applied Manifold Geometry 205For ex
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Applied Manifold Geometry 207axis,
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Applied Manifold Geometry 209For ex
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Applied Manifold Geometry 211The sp
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Applied Manifold Geometry 213corres
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Applied Manifold Geometry 215corres
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Applied Manifold Geometry 217Specia
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Applied Manifold Geometry 219the co
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Applied Manifold Geometry 221moment
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Applied Manifold Geometry 223produc
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Applied Manifold Geometry 225has a
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Applied Manifold Geometry 2273.8.5
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Applied Manifold Geometry 229in whi
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Applied Manifold Geometry 231with
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Applied Manifold Geometry 233itly:(
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Applied Manifold Geometry 235play t
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Applied Manifold Geometry 237Some r
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Applied Manifold Geometry 239Basic
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Applied Manifold Geometry 241orthog
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Applied Manifold Geometry 243is adm
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Applied Manifold Geometry 245Root S
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Applied Manifold Geometry 247groups
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Applied Manifold Geometry 251second
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Applied Manifold Geometry 255be a v
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Applied Manifold Geometry 257Now, a
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Applied Manifold Geometry 263other
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Applied Manifold Geometry 267result
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Applied Manifold Geometry 2713.10 R
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Applied Manifold Geometry 273More p
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Applied Manifold Geometry 275Y (t 0
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Applied Manifold Geometry 277which
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Applied Manifold Geometry 279for an
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Applied Manifold Geometry 283Thus w
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Applied Manifold Geometry 287evolut
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Applied Manifold Geometry 293‘ren
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Applied Manifold Geometry 295mute;
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Applied Manifold Geometry 297Rather
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Applied Manifold Geometry 299of the
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Applied Manifold Geometry 301clidea
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Applied Manifold Geometry 303Second
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Applied Manifold Geometry 305i.e.,
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Applied Manifold Geometry 307invari
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Applied Manifold Geometry 309The Ba
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Applied Manifold Geometry 311Let us
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Applied Manifold Geometry 313all cu
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Applied Manifold Geometry 315ders.
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Applied Manifold Geometry 317this c
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Applied Manifold Geometry 319In par
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Applied Manifold Geometry 321As I p
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Applied Manifold Geometry 323the fo
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Applied Manifold Geometry 325per (1
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Applied Manifold Geometry 327Genera
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Applied Manifold Geometry 329The ma
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Applied Manifold Geometry 331pretat
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Applied Manifold Geometry 333byMDL
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Applied Manifold Geometry 3353.12 S
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Applied Manifold Geometry 337H 2 (M
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Applied Manifold Geometry 339bracke
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Applied Manifold Geometry 341Any so
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Applied Manifold Geometry 343All on
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Applied Manifold Geometry 345The ac
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Applied Manifold Geometry 347linear
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Applied Manifold Geometry 349such t
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Applied Manifold Geometry 351Let γ
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Applied Manifold Geometry 355chart
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Applied Manifold Geometry 357phase-
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Applied Manifold Geometry 359( ) 0
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Applied Manifold Geometry 361hetero
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Applied Manifold Geometry 363thus c
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Applied Manifold Geometry 365Action
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Applied Manifold Geometry 367and we
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Applied Manifold Geometry 369action
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Applied Manifold Geometry 371and we
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Applied Manifold Geometry 373by (J(
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Applied Manifold Geometry 375fields
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Applied Manifold Geometry 377Let Tx
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Applied Manifold Geometry 381The ve
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Applied Manifold Geometry 383and gi
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Applied Manifold Geometry 385The co
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Applied Manifold Geometry 387tor Ca
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Applied Manifold Geometry 3912 −
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Applied Manifold Geometry 393comple
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Applied Manifold Geometry 395F[C]
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Applied Manifold Geometry 397become
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Applied Manifold Geometry 399of fuz
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Applied Manifold Geometry 401In our
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Page 512 and 513: Applied Manifold Geometry 477ing di
Page 514 and 515: Applied Manifold Geometry 479Note t
Page 516 and 517: Applied Manifold Geometry 481Then w
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Page 520 and 521: Chapter 4Applied Bundle Geometry4.1
Page 522 and 523: Applied Bundle Geometry 487is calle
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Page 528 and 529: Applied Bundle Geometry 493dimensio
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Page 534 and 535: Applied Bundle Geometry 499Let T Y
Page 536 and 537: Applied Bundle Geometry 501In other
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Applied Bundle Geometry 503we shall
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Applied Bundle Geometry 505Pendulum
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Applied Bundle Geometry 507Using L
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Applied Bundle Geometry 509the isom
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Applied Bundle Geometry 511theory f
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Applied Bundle Geometry 513of the c
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Applied Bundle Geometry 515where Ω
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Applied Bundle Geometry 517nals, th
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Applied Bundle Geometry 519(1985)])
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Applied Bundle Geometry 521finite-d
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Applied Bundle Geometry 523and thes
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Applied Bundle Geometry 525Now, let
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Applied Bundle Geometry 527gravity
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Applied Bundle Geometry 529a Dp−b
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Applied Bundle Geometry 531A princi
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Applied Bundle Geometry 533is the r
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Applied Bundle Geometry 535Now, let
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Applied Bundle Geometry 537By cycli
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Applied Bundle Geometry 5394.9.2 Fe
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Applied Bundle Geometry 541This lin
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Applied Bundle Geometry 543(1) L g
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Applied Bundle Geometry 545and ω 0
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Applied Bundle Geometry 547isfiesω
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Applied Bundle Geometry 549of the c
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Applied Bundle Geometry 551Controll
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Applied Bundle Geometry 553Foliatio
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Applied Bundle Geometry 555ψ : (x,
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Applied Bundle Geometry 557from res
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Applied Bundle Geometry 559Motion a
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Applied Bundle Geometry 561where Z
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Applied Bundle Geometry 563Systems
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Applied Bundle Geometry 5653. Recal
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Applied Bundle Geometry 567Therefor
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Applied Bundle Geometry 569smooth m
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Applied Bundle Geometry 571state va
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Applied Bundle Geometry 573The geom
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Applied Bundle Geometry 575Y ∋ y
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Applied Bundle Geometry 577autonomo
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Applied Bundle Geometry 579(see 3.1
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Applied Bundle Geometry 581and init
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Applied Bundle Geometry 583potentia
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Applied Bundle Geometry 585effector
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Applied Bundle Geometry 587Fig. 4.9
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Applied Bundle Geometry 589physical
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Applied Bundle Geometry 591open Lio
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Applied Bundle Geometry 593et al. (
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Applied Bundle Geometry 595(2) Acti
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Applied Bundle Geometry 597˙Φ ≤
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Applied Bundle Geometry 599number o
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Applied Bundle Geometry 601where Ψ
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Applied Bundle Geometry 603ing to c
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Applied Bundle Geometry 605where [
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Applied Bundle Geometry 607interval
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Applied Bundle Geometry 609The foll
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Applied Bundle Geometry 611In parti
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Applied Bundle Geometry 613The pull
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Applied Bundle Geometry 615(2) δ
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Applied Bundle Geometry 617of unita
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Applied Bundle Geometry 619system o
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Applied Bundle Geometry 621of H pre
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Applied Bundle Geometry 623It follo
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Applied Bundle Geometry 625a produc
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Applied Bundle Geometry 627B Ham(M)
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Applied Bundle Geometry 629b.Let us
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Applied Bundle Geometry 631If P is
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Applied Bundle Geometry 633question
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Applied Bundle Geometry 635spheres
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Applied Bundle Geometry 637be thoug
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Applied Bundle Geometry 639of the b
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Applied Bundle Geometry 641in C 1 (
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Applied Bundle Geometry 643from a t
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Applied Bundle Geometry 645Now let
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Applied Bundle Geometry 647for t <
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Applied Bundle Geometry 649is a Ham
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Applied Bundle Geometry 651The foll
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Applied Bundle Geometry 653is the u
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Applied Bundle Geometry 655every no
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Applied Bundle Geometry 657P × P w
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Applied Bundle Geometry 659that the
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Applied Bundle Geometry 661extend t
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Applied Bundle Geometry 663identity
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Applied Bundle Geometry 665The mapw
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Applied Bundle Geometry 667It is a
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Applied Bundle Geometry 669I Q is i
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Applied Bundle Geometry 671whereC i
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Applied Bundle Geometry 673form Q.
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Applied Bundle Geometry 675τ( exp(
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Applied Bundle Geometry 6774.13.2.2
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Applied Bundle Geometry 679Therefor
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Applied Bundle Geometry 683⎛⎞co
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Applied Bundle Geometry 685Usually,
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Applied Bundle Geometry 687on the s
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Applied Bundle Geometry 6914.13.3 P
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Applied Bundle Geometry 693Since T
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Applied Bundle Geometry 695On the o
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Applied Bundle Geometry 697may writ
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Applied Bundle Geometry 699determin
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Applied Bundle Geometry 701section,
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Applied Bundle Geometry 703for deri
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Applied Bundle Geometry 705dictions
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Applied Bundle Geometry 707too sing
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Applied Bundle Geometry 709In (4.18
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Applied Bundle Geometry 713sis repr
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Applied Bundle Geometry 717a (linea
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Applied Bundle Geometry 719past few
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Applied Bundle Geometry 721global q
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Applied Bundle Geometry 7239. It ha
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Applied Bundle Geometry 725needs to
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Applied Bundle Geometry 727pear at
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Applied Bundle Geometry 729X, F A i
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Applied Bundle Geometry 731ifolds,
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Applied Bundle Geometry 733and not
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Applied Bundle Geometry 735under U(
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Applied Bundle Geometry 737N = 2 br
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Applied Bundle Geometry 739thusa
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Applied Bundle Geometry 741antisymm
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Applied Bundle Geometry 743see this
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Applied Bundle Geometry 745Coupling
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Applied Bundle Geometry 747acts as
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Applied Bundle Geometry 749good coo
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Applied Bundle Geometry 751Φ. Brea
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Applied Bundle Geometry 753equating
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Applied Bundle Geometry 755differen
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Applied Bundle Geometry 7574.14.10
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Applied Bundle Geometry 759deformed
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Applied Bundle Geometry 761phenomen
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Applied Bundle Geometry 763appears
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Applied Bundle Geometry 7654.14.10.
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Applied Bundle Geometry 767For J >
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Applied Bundle Geometry 769K 1/2
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Applied Bundle Geometry 771(i) The
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Applied Bundle Geometry 773by α =
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Applied Bundle Geometry 775(4.279)
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Applied Bundle Geometry 777Λ ⊗
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Applied Bundle Geometry 779coupling
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Applied Bundle Geometry 781where we
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Applied Bundle Geometry 783The posi
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Applied Bundle Geometry 785Although
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Applied Bundle Geometry 787Let us a
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Applied Bundle Geometry 789of the t
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Applied Bundle Geometry 791The Riem
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Applied Bundle Geometry 793To compl
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Applied Bundle Geometry 795Ω 1 :Ω
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Chapter 5Applied Jet GeometryModern
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Applied Jet Geometry 799x, with coe
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Applied Jet Geometry 801s i (x), as
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Applied Jet Geometry 803It is conve
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Applied Jet Geometry 805has the 1
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Applied Jet Geometry 807which split
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Applied Jet Geometry 809gives the h
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Applied Jet Geometry 811Furthermore
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Applied Jet Geometry 813This is a c
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Applied Jet Geometry 815Due to this
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Applied Jet Geometry 817The affine
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Applied Jet Geometry 819In particul
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Applied Jet Geometry 821Building on
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Applied Jet Geometry 823on J ∞ (X
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Applied Jet Geometry 825its tangent
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Applied Jet Geometry 827Fig. 5.3 Hi
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Applied Jet Geometry 829It follows
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Applied Jet Geometry 831affine, whi
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Applied Jet Geometry 833where Γ i
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Applied Jet Geometry 8355.6.5 Jacob
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Applied Jet Geometry 837is also the
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Applied Jet Geometry 839A generic m
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Applied Jet Geometry 841where ϑ eH
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Applied Jet Geometry 843Since E L|
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Applied Jet Geometry 845Thus, every
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Applied Jet Geometry 847along u. Ev
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Applied Jet Geometry 849equations c
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Applied Jet Geometry 851A Hamiltoni
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Applied Jet Geometry 853Connections
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Applied Jet Geometry 855Lagrangian
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Applied Jet Geometry 857on V ∗ M.
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Applied Jet Geometry 859contain no
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Applied Jet Geometry 861(ii) For an
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Applied Jet Geometry 8635.6.16 Lyap
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Applied Jet Geometry 865for any t >
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Applied Jet Geometry 867vertical co
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Applied Jet Geometry 869difficulty,
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Applied Jet Geometry 871the energy
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Applied Jet Geometry 873by the tran
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Applied Jet Geometry 875The conditi
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Applied Jet Geometry 877Let Γ = (M
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Applied Jet Geometry 879geometrical
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Applied Jet Geometry 881Let (M, I)
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Applied Jet Geometry 883so referrin
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Applied Jet Geometry 885The Poincar
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Applied Jet Geometry 887where N n
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Applied Jet Geometry 889be a projec
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Applied Jet Geometry 891to differen
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Applied Jet Geometry 893for every c
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Applied Jet Geometry 895vector-fiel
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Applied Jet Geometry 897This is phr
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Applied Jet Geometry 899where a, b
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Applied Jet Geometry 901represent t
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Applied Jet Geometry 903̂L : J 1 (
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Applied Jet Geometry 905space J 1 (
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Applied Jet Geometry 907First of al
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Applied Jet Geometry 909Therefore,
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Applied Jet Geometry 911Recall that
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Applied Jet Geometry 9135.11 Applic
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Applied Jet Geometry 915the vector
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Applied Jet Geometry 9175.11.4 Gaug
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Applied Jet Geometry 9195.11.5 Lagr
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Applied Jet Geometry 921are represe
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Applied Jet Geometry 923which are t
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Applied Jet Geometry 925of this fun
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Applied Jet Geometry 927a principal
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Applied Jet Geometry 929Recall that
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Applied Jet Geometry 931Moreover, t
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Applied Jet Geometry 933other only
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Applied Jet Geometry 935In particul
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Applied Jet Geometry 937Note that,
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Applied Jet Geometry 939This equali
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Applied Jet Geometry 941Then, the s
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Applied Jet Geometry 943In the Lagr
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Applied Jet Geometry 945Let τ be a
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Applied Jet Geometry 947Using (5.42
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Applied Jet Geometry 949associated
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Applied Jet Geometry 951parts,F α
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Applied Jet Geometry 953of the Hilb
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Applied Jet Geometry 955For example
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Applied Jet Geometry 957the convent
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Applied Jet Geometry 959left ideal
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Applied Jet Geometry 961LX → Σ i
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Applied Jet Geometry 963Levi-Civita
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Applied Jet Geometry 965see this fr
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Applied Jet Geometry 967curves. And
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Applied Jet Geometry 969One sees th
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Applied Jet Geometry 971sums over a
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Applied Jet Geometry 973all this to
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Applied Jet Geometry 975universe ev
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Applied Jet Geometry 977on Σ. If M
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Applied Jet Geometry 979(1) Lorentz
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Applied Jet Geometry 981terpret thi
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Chapter 6Geometrical Path Integrals
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Geometrical Path Integrals and Thei
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BibliographyAblowitz, M.J., Clarkso
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Bibliography 1255equations. Springe
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Bibliography 1257connections, Phys.
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Bibliography 1259JHEP 0003:007.Bran
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Bibliography 1261Clementi, C. Petti
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Bibliography 1263Domany, E., Van He
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Bibliography 1265geometrical signat
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Bibliography 1267Bergmann theory of
Page 1304 and 1305:
Bibliography 1269to Complex Systems
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Bibliography 1271Action for String
Page 1308 and 1309:
Bibliography 1273Ivancevic, V., Bea
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Bibliography 1275Krener, A. (1984).
Page 1312 and 1313:
Bibliography 1277Li, M., Vitanyi, P
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Bibliography 1279Nature, 261(5560),
Page 1316 and 1317:
Bibliography 1281Lambert and R. Gol
Page 1318 and 1319:
Bibliography 1283dynamics of human
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Bibliography 1285Reshetikhin, N.Yu,
Page 1322 and 1323:
Bibliography 1287Rev. E, 58, 6333-6
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Bibliography 1289Stasheff, J.D. (19
Page 1326 and 1327:
Bibliography 1291matical Physics. S
Page 1328 and 1329:
Bibliography 1293Witten, E. (1991).
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Index1−jet bundle, 8271−jet lif
Page 1332 and 1333:
Index 1297Chan-Paton factor, 1238Ch
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Index 1299Dirac-Born-Infeld theory,
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Index 1301Gauss, 5Gauss map, 284Gau
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Index 1303isometric, 17isomorphic,
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Index 1305Maxwell Lagrangian, 34Max
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Index 1307probability manifold, 333
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Index 1309super-space, 1215supercov
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Index 1311Yang-Mills theory, 39, 29
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