617 research outputs found
A geometrical analysis of the field equations in field theory
In this review paper we give a geometrical formulation of the field equations
in the Lagrangian and Hamiltonian formalisms of classical field theories (of
first order) in terms of multivector fields. This formulation enables us to
discuss the existence and non-uniqueness of solutions, as well as their
integrability.Comment: 14 pages. LaTeX file. This is a review paper based on previous works
by the same author
On the Multimomentum Bundles and the Legendre Maps in Field Theories
We study the geometrical background of the Hamiltonian formalism of
first-order Classical Field Theories. In particular, different proposals of
multimomentum bundles existing in the usual literature (including their
canonical structures) are analyzed and compared. The corresponding Legendre
maps are introduced. As a consequence, the definition of regular and
almost-regular Lagrangian systems is reviewed and extended from different but
equivalent ways.Comment: LaTeX file, 19 pages. Replaced with the published version. Minor
mistakes are correcte
Symplectic Cuts and Projection Quantization
The recently proposed projection quantization, which is a method to quantize
particular subspaces of systems with known quantum theory, is shown to yield a
genuine quantization in several cases. This may be inferred from exact results
established within symplectic cutting.Comment: 12 pages, v2: additional examples and a new reference to related wor
Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries
We state the intrinsic form of the Hamiltonian equations of first-order
Classical Field theories in three equivalent geometrical ways: using
multivector fields, jet fields and connections. Thus, these equations are given
in a form similar to that in which the Hamiltonian equations of mechanics are
usually given. Then, using multivector fields, we study several aspects of
these equations, such as the existence and non-uniqueness of solutions, and the
integrability problem. In particular, these problems are analyzed for the case
of Hamiltonian systems defined in a submanifold of the multimomentum bundle.
Furthermore, the existence of first integrals of these Hamiltonian equations is
considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl
general symmetries} of the system is discussed. Noether's theorem is also
stated in this context, both the ``classical'' version and its generalization
to include higher-order Cartan-Noether symmetries. Finally, the equivalence
between the Lagrangian and Hamiltonian formalisms is also discussed.Comment: Some minor mistakes are corrected. Bibliography is updated. To be
published in J. Phys. A: Mathematical and Genera
Stabilizing s-hole dimethyl interactions
Methyl groups bound to electronegative atoms, such as N or O, are recognized to participate in tetrel bonding as Lewis acids. On the other hand, the capability of methyl groups bound to electropositive atoms, such as B or Al, to act as Lewis bases has been recently reported. Herein, we analyze the combination of these two behaviors to establish attractive methyl···methyl interactions. We have explored the Cambridge Structural Database to find experimental examples of these dimethyl-bound systems, finding a significant degree of directionality in the relative disposition of the two methyl groups. Moreover, we have carried out a comprehensive computational analysis at the DFT level of the dimethyl interactions, including the natural bond orbital, energy decomposition analysis, and topological analysis of the electron density (QTAIM and NCI). The dimethyl interaction is characterized as weak yet attractive and based on electrostatics, with a non-negligible contribution from orbital charge transfer and polarization
On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories
The objective of this work is twofold: First, we analyze the relation between
the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms
in classical field theories. In particular, we prove the equivalence between
k-symplectic field theories and the so-called autonomous k-cosymplectic field
theories, extending in this way the description of the symplectic formalism of
autonomous systems as a particular case of the cosymplectic formalism in
non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric
character of the solutions to the Hamilton-de Donder-Weyl and the
Euler-Lagrange equations in these formalisms. Second, we study the equivalence
between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and
Lagrangian field theories (those where the configuration bundle of the theory
is trivial).Comment: 25 page
Geometric reduction in optimal control theory with symmetries
A general study of symmetries in optimal control theory is given, starting
from the presymplectic description of this kind of system. Then, Noether's
theorem, as well as the corresponding reduction procedure (based on the
application of the Marsden-Weinstein theorem adapted to the presymplectic case)
are stated both in the regular and singular cases, which are previously
described.Comment: 24 pages. LaTeX file. The paper has been reorganized. Additional
comments have been included in Section 3. The example in Section 5.2 has been
revisited. Some references have been adde
Non-standard connections in classical mechanics
In the jet-bundle description of first-order classical field theories there
are some elements, such as the lagrangian energy and the construction of the
hamiltonian formalism, which require the prior choice of a connection. Bearing
these facts in mind, we analyze the situation in the jet-bundle description of
time-dependent classical mechanics. So we prove that this connection-dependence
also occurs in this case, although it is usually hidden by the use of the
``natural'' connection given by the trivial bundle structure of the phase
spaces in consideration. However, we also prove that this dependence is
dynamically irrelevant, except where the dynamical variation of the energy is
concerned. In addition, the relationship between first integrals and
connections is shown for a large enough class of lagrangians.Comment: 17 pages, Latex fil
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