Metric nonlinear connections

For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange structure. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonic nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space. The metric tensor of the Lagrange space determines the symmetric part of the nonlinear connection, while the symplectic structure of the Lagrange space determines the skew-symmetric part of the nonlinear connection

Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type

We extend the results obtained in a previous paper about a class of Lagrangian systems which admit alternative kinetic energy metrics to second-order mechanical systems with explicit time-dependence. The main results are that a time-dependent alternative metric will have constant eigenvalues, and will give rise to a time-dependent coordinate transformation which partially decouples the system

Geometric properties of Lagrangian mechanical systems

The geometry of a Lagrangian mechanical system is determined by its associated evolution semispray. We uniquely determine this semispray using the symplectic structure and the energy of the Lagrange space and the external force field. We study the variation of the energy and Lagrangian functions along the evolution and the horizontal curves and give conditions by which these variations vanish. We provide examples of mechanical systems which are dissipative and for which the evolution nonlinear connection is either metric or symplectic

From Classical Trajectories to Quantum Commutation Relations

In describing a dynamical system, the greatest part of the work for a theoretician is to translate experimental data into differential equations. It is desirable for such differential equations to admit a Lagrangian and/or an Hamiltonian description because of the Noether theorem and because they are the starting point for the quantization. As a matter of fact many ambiguities arise in each step of such a reconstruction which must be solved by the ingenuity of the theoretician. In the present work we describe geometric structures emerging in Lagrangian, Hamiltonian and Quantum description of a dynamical system underlining how many of them are not really fixed only by the trajectories observed by the experimentalist.Comment: 25 pages. Comments are welcome

Geometric aspects of nonholonomic field theories

A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations, the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.Comment: 29 pages; typos remove

Targeted screening of inflammatory mediators in spontaneous degenerative disc disease in dogs reveals an upregulation of the tumor necrosis superfamily

Background: The regulation of inflammatory mediators in the degenerating intervertebral disc (IVD) and corresponding ligamentum flavum (LF) is a topic of emerging interest. The study aimed to investigate the expression of a broad array of inflammatory mediators in the degenerated LF and IVD using a dog model of spontaneous degenerative disc disease (DDD) to determine potential treatment targets. Methods: LF and IVD tissues were collected from 22 normal dogs (Pfirrmann grades I and II) and 18 dogs affected by DDD (Pfirrmann grades III and IV). A qPCR gene array was used to investigate the expression of 80 inflammatory genes for LF and IVD tissues, whereafter targets of interest were investigated in additional tissue samples using qPCR, western blot (WB), and immunohistochemistry. Results: Tumor necrosis factor superfamily (TNFSF) signaling was identified as a regulated pathway in DDD, based on the significant regulation (n-fold ± SD) of various TNFSF members in the degenerated IVD, including nerve growth factor (NGF; -8 ± 10), CD40LG (464 ± 442), CD70 (341 ± 336), TNFSF Ligand 10 (9 ± 8), and RANKL/TNFSF Ligand 11 (85 ± 74). In contrast, TNFSF genes were not significantly affected in the degenerated LF compared to the control LF. Protein expression of NGF (WB) was significantly upregulated in both the degenerated LF (4.4 ± 0.5) and IVD (11.3 ± 5.6) compared to the control group. RANKL immunopositivity was significantly upregulated in advanced stages of degeneration (Thompson grades IV and V) in the nucleus pulposus and annulus fibrosus of the IVD, but not in the LF. Conclusions: DDD involves a significant upregulation of various TN

Symmetries in Classical Field Theory

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.Comment: 70S05; 70H33; 55R10; 58A2

The Lagrangian-Hamiltonian Formalism for Higher Order Field Theories

We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to higher order field theories on fiber bundles. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for higher order Lagrangian field theories. Namely, our formalism does only depend on the action functional and, therefore, unlike previously proposed ones, is free from any relevant ambiguity.Comment: 29 pages, revised version as accepted for publication in J. Geom. Phy