My Library: Completely integrable and superintegrable Hamiltonian systems with noncompact invariant submanifolds

The file Library6.pdf (3Mb) contains the attached PDF files of my main works on generalization of Noether theorems to an arbitrary Lagrangian system  Generalization of the Liouville - Arnold, Nekhoroshev and Mishchenko - Fomenko theorems on action-angle variables of completely integrable, partially integrable and superintegrable Hamiltonian systems to the case of non-compact invariant submanifolds.

Contents

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Action-angle coordinates for time-dependent completely integrable Hamiltonian systems, J. Phys. A 35 (2002) L439-L445

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric quantization of completely integrable Hamiltonian systems in action-angle coordinates, Phys. Lett. A 301 (2002) 53-57

E.Fiorani, G.Giachetta and G.Sardanashvily, Geometric quantization of time-dependent completely integrable Hamiltonian systems, J. Math. Phys. 43 (2002) 5013-5025

E.Fiorani, G.Giachetta and G.Sardanashvily, The Liouville -- Arnold -- Nekhoroshev theorem for non-compact invariant manifolds, J. Phys. A 36 (2003) L101-L107

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Jacobi fields of completely integrable systems, Phys. Lett. A 309 (2003) 382-386

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Bi-Hamiltonian partially integrable systems, J. Math. Phys. 44 (2003) 1984-1987

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Nonadiabatic holonomy operators in classical and quantum completely integrable systems, J. Math. Phys. 45 (2004) 76-86

E.Fiorani and G.Sardanashvily, Noncommutative integrability on noncompact invariant manifolds, J. Phys. A 39 (2006) 14035-14042

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Quantization of noncommutative completely integrable systems, Phys. Lett. A 362 (2007) 138-142

E.Fiorani and G.Sardanashvily, Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds, J. Math. Phys. 48 (2007) 032901

G.Sardanashvily, Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1391-1420

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (World Scientific, Singapore, 2010)

Lagrangian dynamics of higher-dimensional submanifolds

Classical non-relativistic mechanics is adequately formulated as Lagrangian and Hamiltonian theory on a fibre bundle Q-> R over the time axis R, where R is provided with the Cartesian coordinate t possessing the transition functions t'=t+const.  A velocity space of non-relativistic mechanics is the first order jet manifold JQ of sections of Q-> R. Lagrangians of non-relativistic mechanics are defined as densities on JQ. This formulation is extended to time-reparametrized non-relativistic mechanics subject to time-dependent transformations which are bundle automorphisms of Q-> R.

Thus, one can think of non-relativistic mechanics as being particular classical field theory on fibre bundles over X=R. However, an essential difference between non-relativistic mechanics and field theory on fibre bundles Y->X, dim X>1, lies in the fact that connections on Q-> R always are flat. Therefore, they fail to be dynamic variables, but characterize non-relativistic reference frames.

In comparison with non-relativistic mechanics, relativistic mechanics admits transformations of the time depending on other variables, e.g., the Lorentz transformations in Special Relativity on a Minkowski space Q. Therefore, a configuration space Q of relativistic mechanics has no preferable fibration Q-> R, and its velocity space is the first order jet manifold J[1]Q of one-dimensional submanifolds of a configuration space Q. Fibres of the jet bundle J[1]Q-> Q are projective spaces, and one can think of them as being spaces of the three-velocities of a relativistic system. The four-velocities of a relativistic system are represented by elements of the tangent bundle TQ of a configuration space Q.

One can provide a generalization of the above mentioned formulation of relativistic mechanics to the case of submanifolds of arbitrary dimension n. For instance, if n=2, this is the case of classical string theory.

Reference:

G.Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics arXiv: 1112.0216

A problem of an inertial reference frame in classical mechanics

The key problem of classical mechanics is that there is no intrinsic definition of an inertial reference frame. We have different inertial reference frames which are not inertial with respect to each other.

Classical Lagrangian and Hamiltonian non-relativistic mechanics admits the adequate mathematical formulation in terns of fibre bundle Q->R over the time axis R. In this framework, a reference frame is defined as a trivialization of this fibre bundle or, equivalently, as a connection on Q->R.

A second order dynamic equation is called a free motion equation if it can be brought into the form of a zero acceleration ddq/dtdt=0 with respect to some reference frame,  and this reference frame is said to be inertial for this equation. Thus a definition of an inertial frame depends on the choice of a free motion equation. Given such an equation, different inertial reference frames for this differ from each other in constant velocities.

A problem is that, given a different free motion equation ddq’/dtdt=0, an inertial reference frame for it fails to be so the first free motion equation ddq/dtdt=0, and their relative velocity is not constant.

In view of this problem, one should write dynamic equations of non-relativistic mechanics in terms of relative velocities and accelerations with respect to an arbitrary reference frame. However, in this case the strict mathematical notions of a relative acceleration and a non-inertial force are rather sophisticated.

References:

G.Sardanashvily, Relative non-relativistic mechanics, arXiv: 0708.2998

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

My Library: General Noether theorems

The first and second Noether theorems are formulated in a very general setting of reducible degenerate Lagrangian theories of even and odd variables on fibre bundles and

graded manifolds.

The file Library5.pdf (3 Mb) contains the attached PDF files of my main works on generalization of Noether theorems to an arbitrary Lagrangian system

Contents

G.Sardanashvily, Noether identities of a differential operator. The Koszul--Tate complex, Int. J. Geom. Methods Mod. Phys. 2 (2005) 873-886

D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The antifield Koszul--Tate complex of reducible Noether identities, J. Math. Phys. 46 (2005) 103513

D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237-252

G.Giachetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys. 50 (2009) 012903

G.Sardanashvily, Gauge conservation laws in a general setting: Superpotential, Int. J. Geom. Methods Mod. Phys. 6 (2009) 1047-1056

G.Giachetta, L.Mangiarotti and G.Sardanashvily, Advanced Classical Field Theory  (World Scientific, Singapore, 2009)

On a gauge model of the fifth force

The Poincare group gauge approach which dominated gauge gravitation researches for a long time has not succeeded in providing a gravitational field with the status of a translation gauge potential. Therefore, the question on the physical meaning of this potential arises. At the same time, gauge potentials of spatial translations seem to possess the satisfactory physical utilization for description of dislocations in the theory of continuous media. Based on this result, we have suggested that gauge potentials of Poincare space-time translations can also describe sui generis dislocations of a space-time manifold.

The source of these potentials turns out to be the canonical energy-momentum tensor of matter, and they are inserted into the motion equations of a spinless matter via an effective metric. Therefore, translation gauge fields can contribute to standard gravitational effects. In particular, they may be responsible for an additional exponential (Yukawa) type term to the Newton gravitation potential, i.e., the so-called "fifth force".

This is a hypothetic fifth fundamental interaction which is weaker than gravity. Its experimental verification attracted much attention in 80^th, but nothing was found at least at laboratory distances. However, one discusses the fifth force as possible explanation of the “dark matter” phenomena on the astrophysical and cosmological level.

References