Lagrangians Galore

Searching for a Lagrangian may seem either a trivial endeavour or an impossible task. In this paper we show that the Jacobi last multiplier associated with the Lie symmetries admitted by simple models of classical mechanics produces (too?) many Lagrangians in a simple way. We exemplify the method by such a classic as the simple harmonic oscillator, the harmonic oscillator in disguise [H Goldstein, {\it Classical Mechanics}, 2nd edition (Addison-Wesley, Reading, 1980)] and the damped harmonic oscillator. This is the first paper in a series dedicated to this subject.Comment: 16 page

Gauge Variant Symmetries for the Schr\"odinger Equation

The last multiplier of Jacobi provides a route for the determination of families of Lagrangians for a given system. We show that the members of a family are equivalent in that they differ by a total time derivative. We derive the Schr\"odinger equation for a one-degree-of-freedom system with a constant multiplier. In the sequel we consider the particular example of the simple harmonic oscillator. In the case of the general equation for the simple harmonic oscillator which contains an arbitrary function we show that all Schr\"odinger equations possess the same number of Lie point symmetries with the same algebra. From the symmetries we construct the solutions of the Schr\"odinger equation and find that they differ only by a phase determined by the gauge.Comment: 12 page

From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.Comment: To appear in: Proceedings of Symmetries in Science XV, Journal of Physics: Conference Series, (2012

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic

Lie point symmetries and first integrals: the Kowalevsky top

We show how the Lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations. The method of reduction/increase of order developed by Nucci (J. Math. Phys. 37, 1772-1775 (1996)) is essential. Noether's theorem is neither necessary nor considered. The most striking example we present is the relationship between Lie group analysis and the famous first integral of the Kowalevski top.Comment: 23 page

Ermakov's Superintegrable Toy and Nonlocal Symmetries

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2,R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

The reaction to nailing or cementing of the femur in rats. A microangiographic and fluorescence study.

Bone reaction to cement and to a cementless stem was studied in the rat femur with histological fluorescence and microangiographic techniques. Periosteal and endosteal apposition, and consequent remodelling, appeared as a reaction to reaming rather than caused by cement or a cementless stem. Every change in bone began with proliferation, progression and orientation of the vessels. Endosteal apposition was absent in cemented femurs because the entire medulla was occupied by the acrylic cement, but remodelling of the subendosteal cortex followed medullary revascularisation which was far advanced after 90 days. In cementless stems, endosteal apposition of primary woven bone and remodelling was the basis for bony ingrowth and anchorage through bony bridges. Our results suggest that the pattern of blood supply is relevant to the structural organisation of mature lamellar bone around the implant. Cemented stems have maximum anchorage and stability as soon as they are inserted, but this decreases with time as revascularisation occurs. Cementless stems can reach maximum integration later after insertion, and revascularisation is less critical because they usually do not fill the canal completely

CLU "in and out": looking for a link

Cancer cells need to interact synergistically with their surrounding microenvironment to form a neoplasm and to progress further to colonize distant organs. The microenvironment can exert profound epigenetic effects on cells through cell-derived interactions between cells, or through cell-derived factors deposited into the microenvironment. Tumor progression implies immune-escaping and triggers several processes that synergistically induce a cooperation among transformed and stromal cells, that compete for space and resources such as oxygen and nutrients. Therefore, the extra cellular milieu and tissue microenvironment heterotypic interactions cooperate to promote tumor growth, angiogenesis, and cancer cell motility, through elevated secretion of pleiotropic cytokines and soluble factors. Clusterin (CLU), widely viewed as an enigmatic protein represents one of the numerous cellular factors sharing the intracellular information with the microenvironment and it has also a systemic diffusion, tightly joining the "In and the Out" of the cell with a still debated variety of antagonistic functions. The multiplicity of names for CLU is an indication of the complexity of the problem and could reflect, on one hand its multifunctionality, or alternatively could mask a commonality of function. The posited role for CLU, further supported as a cytoprotective prosurvival chaperone-like molecule, seems compelling, in contrast its tumor suppressor function, as a guide of the guardians of the genome (DNA-repair proteins Ku70/80, Bax cell death inducer), could really reflect the balanced expression of its different forms, most certainly depending on the intra- and extracellular microenvironment cross talk. The complicated balance of cytokines network and the regulation of CLU forms production in cancer and stromal cells undoubtedly represent a potential link among adaptative responses, genomic stability, and bystander effect after oxidative stresses and damage. This review focuses on the tumor-microenvironment interactions strictly involved in controlling local cancer growth, invasion, and distant metastases that play a decisive role in the regulation of CLU different forms expression and release. In addition, we focus on the pleiotropic action of the extracellular form of this protein, sCLU, that may play a crucial role in redirecting stromal changes, altering intercellular communications binding cell surface receptors and contributing to influence the secretion of chemokines in paracrine and autocrine fashion. Further elucidation of CLU functions inside and outside ("in and out") of cancer cell are warranted for a deeper understanding of the interplay between tumor and stroma, suggesting new therapeutic cotargeting strategies

The effects of mechanical forces on bones and joints. Experimental study on the rat tail.

We have used an experimental model employing the bent tail of rats to investigate the effects of mechanical forces on bones and joints. Mechanical strain could be applied to the bones and joints of the tail without direct surgical exposure or the application of pins and wires. The intervertebral disc showed stretched annular lamellae on the convex side, while the annulus fibrosus on the concave side was pinched between the inner corners of the vertebral epiphysis. In young rats with an active growth plate, a transverse fissure appeared at the level of the hypertrophic cell layer or the primary metaphyseal trabecular zone. Metaphyseal and epiphyseal trabeculae on the compressed side were thicker and more dense than those of the distracted part of the vertebra. In growing animals, morphometric analysis of hemiepiphyseal and hemimetaphyseal areas, and the corresponding trabecular bone density, showed significant differences between the compressed and distracted sides. No differences were observed in adult rats. We found no significant differences in osteoclast number between compressed and distracted sides in either age group. Our results provide quantitative evidence of the working of 'Wolff's law'. The differences in trabecular density are examples of remodelling by osteoclasts and osteoblasts; our finding of no significant difference in osteoclast numbers between the hemiepiphyses in the experimental and control groups suggests that the response of living bone to altered strain is mediated by osteoblasts