Table Of ContentGauge
Mechanics
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Gauge
Mechanics
L. Mangiarotti
University of Camerino, Italy
G. Sardanashvily
Moscow State University, Russia
World Scientific
Singapore *New Jersey London 'HongKong
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World Scientific Publishing Co. Pte. Ltd
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GAUGE MECHANICS
Copyright © 1998 by World Scientific Publishing Co Pte. Ltd
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Preface
This book presents in a unified way modern geometric methods in analytical
mechanics, based on the application of jet manifolds and connections. As is well
known, the technique of Poisson and symplectic spaces provide the adequate Hamil-
tonian formulation of conservative mechanics. This formulation, however, cannot be
extended to time-dependent mechanics subject to time-dependent transformations.
We will formulate non-relativistic time-dependent mechanics as a particular field
theory on fibre bundles over a time axis.
The geometric approach to field theory is based on the identification of classical
fields with sections of fibred manifolds. Jet manifolds provide the adequate mathe
matical language for Lagrangian field theory, while the Hamiltonian one is phrased
in terms of a polysymplectic structure. The 1-dimensional reduction of Lagrangian
field theory leads us in a straightforward manner to Lagrangian time-dependent
mechanics. At the same time, the canonical polysymplectic form on a momentum
phase space of time-dependent mechanics reduces to the canonical exterior 3-form
which plays the role similar to a symplectic form in conservative mechanics. With
this canonical 3-form, we introduce the canonical Poisson structure and formulate
Hamiltonian time-dependent mechanics in terms of Hamiltonian connections and
Hamiltonian forms.
Note that the theory of non-linear differential operators and the calculus of vari
ations are conventionally phrased in terms of jet manifolds. On the other hand, jet
formalism provides the contemporary language of differential geometry to deal with
non-linear connections, represented by sections of jet bundles. Only jet spaces enable
us to treat connections, Lagrangian and Hamiltonian dynamics simultaneously.
In fact, the concept of connection is the main link throughout the book. Con
nections on a configuration space of time-dependent mechanics are reference frames.
Holonomic connections on a velocity phase space define non-relativistic dynamic
equations which are also related to other types of connections, and can be writ
ten as non-relativistic geodesic equations. Hamiltonian time-dependent mechanics
deals with Hamiltonian connections whose geodesies are solutions of the Hamilton
v
vi PREFACE
equations.
The presence of a reference frame, expressed in terms of connections, is the main
peculiarity of time-dependent mechanics. In particular, each reference frame defines
an energy function, and quantizations with respect to different reference frames are
not equivalent.
Another important peculiarity is that a Hamiltonian fails to be a scalar function
under time-dependent transformations. As a consequence, many well-known con
structions of conservative mechanics fail to be valid for time-dependent mechanics,
and one should follow methods of field theory.
At the same time, there is the essential difference between field theory and time-
dependent mechanics. In contrast with gauge potentials in field theory, connections
on a configuration space of time-dependent mechanics fail to be dynamic variables
since their curvature vanishes identically. Following geometric methods of field the
ory, we obtain the frame-covariant formulation of time-dependent mechanics. By
analogy with gauge field theory, one may speak about gauge time-dependent me
chanics.
In comparison with non-relativistic time-dependent mechanics, a configuration
space of relativistic mechanics does not imply any preferable fibration over a time.
To construct the velocity phase space of relativistic mechanics, we therefore use
formalism of jets of submanifolds. At the same time, Hamiltonian relativistic me
chanics is seen as an autonomous Hamiltonian system on the constraint space of
relativistic hyperboloids.
With respect to mathematical prerequisites, the reader is expected to be familiar
with the basics of differential geometry of fibre bundles. For the convenience of the
reader, several mathematical facts and notions are included as an Interlude, thus
making our exposition self-contained.
Contents
Preface v
Introduction 1
1 Interlude: bundles, Jets, Connections 9
1.1 Fibre bundles 9
1.2 Multivector fields and differential forms 20
1.3 Jet manifolds 35
1.4 Connections 42
1.5 Bundles with symmetries 46
1.6 Composite fibre bundles 53
2 Geometry of Poisson Manifolds 57
2.1 Jacobi structure 57
2.2 Contact structure 61
2.3 Poisson structure 66
2.4 Symplectic structure 73
2.5 Presymplectic structure 81
2.6 Reduction of symplectic and Poisson structures 84
2.7 Appendix. Poisson homology and cohomology 89
2.8 Appendix. More brackets 96
2.9 Appendix. Multisymplectic structures 100
3 Hamiltonian Systems 105
3.1 Dynamic equations 106
3.2 Poisson Hamiltonian systems Ill
3.3 Symplectic Hamiltonian systems 114
3.4 Presymplectic Hamiltonian systems 118
3.5 Dirac Hamiltonian systems 123
3.6 Dirac constraint systems 129
vii
viii CONTENTS
3.7 Hamiltonian systems with symmetries 133
3.8 Appendix. Hamiltonian field theory 139
4 Lagrangian time-dependent mechanics 151
4.1 Fibre bundles over K 152
4.2 Dynamic equations 159
4.3 Dynamic connections 162
4.4 Non-relativistic geodesic equations 170
4.5 Reference frames 175
4.6 Free motion equations 179
4.7 Relative acceleration 182
4.8 Lagrangian systems 186
4.9 Newtonian systems 195
4.10 Holonomic constraints 206
4.11 Non-holonomic constraints 211
4.12 Lagrangian conservation laws 218
5 Hamiltonian time-dependent mechanics 227
5.1 Canonical Poisson structure 228
5.2 Hamiltonian connections and Hamiltonian forms 231
5.3 Canonical transformations 242
5.4 The evolution equation 247
5.5 Degenerate systems 248
5.6 Quadratic degenerate systems 262
5.7 Hamiltonian conservation laws 269
5.8 Time-dependent systems with symmetries 271
5.9 Systems with time-dependent parameters 274
5.10 Unified Lagrangian and Hamiltonian formalism 282
5.11 Vertical extension of Hamiltonian formalism 285
5.12 Appendix. Time-reparametrized mechanics 296
6 Relativistic mechanics 299
6.1 Jets of submanifolds 299
6.2 Relativistic velocity and momentum phase spaces 303
6.3 Relativistic dynamics 307
6.4 Relativistic geodesic equations 311
CONTENTS ix
Appendix A. Geometry of BRST mechanics 317
Appendix B. On quantum time-dependent mechanics 327
Bibliography 332
Index 347
Introduction
The present book deals with first order mechanical systems, governed by the sec
ond order differential equations in coordinates or the first order ones in coordinates
and momenta. Our goal is the description of non-conservative mechanical systems
subject to time-dependent transformations, including inertial and non-inertial frame
transformations and phase transformations.
Symplectic technique is well known to provide the adequate Hamiltonian for
mulation of conservative (i.e., time-independent) mechanics where Hamiltonians are
independent of time [2, 6, 72, 116, 126]. The familiar example is a mechanical sys
tem whose momentum phase space is the cotangent bundle T'M of a configuration
space M. This fibre bundle is provided with the canonical symplectic form
Q = dpi A dq', (1)
written with respect to the holonomic coordinates (g',p, = <?,) on T'M. A Hamil
tonian H of a conservative mechanical system is defined as a real function on the
momentum phase space T'M. Then a motion of this system is an integral curve of
the Hamiltonian vector field
■d = ^d' + d%
on T'M which fulfills the Hamilton equations
ti\Q = -dH,
& = sht, ■di = -diH.
Lagrangian conservative mechanics is usually seen as a particular Hamiltonian me
chanics on the tangent bundle TM of a configuration space M, which is endowed
with the presymplectic form defined by a Lagrangian.
The Hamiltonian formulation of conservative mechanics cannot be extended in a
straightforward manner to time-dependent mechanics because the symplectic form
1
