3,598 research outputs found
On the characterization of infinitesimal symmetries of the relativistic phase space
The phase space of relativistic particle mechanics is defined as the 1st jet
space of motions regarded as timelike 1-dimensional submanifolds of spacetime.
A Lorentzian metric and an electromagnetic 2-form define naturally on the
odd-dimensional phase space a generalized contact structure. In the paper
infinitesimal symmetries of the phase structures are characterized. More
precisely, it is proved that all phase infinitesimal symmetries are special
Hamiltonian lifts of distinguished conserved quantities on the phase space. It
is proved that generators of infinitesimal symmetries constitute a Lie algebra
with respect to a special bracket. A momentum map for groups of symmetries of
the geometric structures is provided.Comment: 38 page
Geometric aspects of higher order variational principles on submanifolds
The geometry of jets of submanifolds is studied, with special interest in the
relationship with the calculus of variations. A new intrinsic geometric
formulation of the variational problem on jets of submanifolds is given.
Working examples are provided.Comment: 17 page
On a class of polynomial Lagrangians
In the framework of finite order variational sequences a new class of
Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are
the horizontalization of forms on a jet space of lower order. We describe their
properties together with properties of related objects, such as
Poincar\'e--Cartan and Euler--Lagrange forms, momenta and momenta of generating
forms, a new geometric object arising in variational sequences. Finally, we
provide a simple but important example of special Lagrangian, namely the
Hilbert--Einstein Lagrangian.Comment: LaTeX2e, amsmath, diagrams, hyperref; 15 page
On the geometry of the energy operator in quantum mechanics
We analyze the different ways to define the energy operator in geometric
theories of quantum mechanics. In some formulations the operator contains the
scalar curvature as a multiplicative term. We show that such term can be
canceled or added with an arbitrary constant factor, both in the mainstream
Geometric Quantization and in the Covariant Quantum Mechanics, developed by
Jadczyk and Modugno with several contributions from many authors.Comment: 18 pages; paper in honour of the 70th birthday of Luigi Mangiarotti
and Marco Modugn
The geometry of real reducible polarizations in quantum mechanics
The formulation of Geometric Quantization contains several axioms and
assumptions. We show that for real polarizations we can generalize the standard
geometric quantization procedure by introducing an arbitrary connection on the
polarization bundle. The existence of reducible quantum structures leads to
considering the class of Liouville symplectic manifolds. Our main application
of this modified geometric quantization scheme is to Quantum Mechanics on
Riemannian manifolds. With this method we obtain an energy operator without the
scalar curvature term that appears in the standard formulation, thus agreeing
with the usual expression found in the Physics literature.Comment: 29 page
On the formalism of local variational differential operators
The calculus of local variational differential operators introduced by B. L. Voronov, I. V. Tyutin, and Sh. S. Shakhverdiev is studied in the context of jet super space geometry. In a coordinate-free way, we relate these operators to variational multivectors, for which we introduce and compute the variational Poisson and Schouten brackets by means of a unifying algebraic scheme. We give a geometric definition of the algebra of multilocal functionals and prove that local variational differential operators are well defined on this algebra. To achieve this, we obtain some analytical results on the calculus of variations in smooth vector bundles, which may be of independent interest. In addition, our results give a new a new efficient method for finding Hamiltonian structures of differential equations
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