1,461 research outputs found
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
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
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
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
Using a mathematical model to evaluate the efficacy of TB control measures.
We evaluated the efficacy of recommended tuberculosis (TB) infection control measures by using a deterministic mathematical model for airborne contagion. We examined the percentage of purified protein derivative conversions under various exposure conditions, environmental controlstrategies, and respiratory protective devices. We conclude that environmental control cannot eliminate the risk for TB transmission during high-risk procedures; respiratory protective devices, and particularly high-efficiency particulate air masks, may provide nearly complete protection if used with air filtration or ultraviolet irradiation. Nevertheless, the efficiency of these control measures decreases as the infectivity of the source case increases. Therefore, administrative control measures (e.g., indentifying and isolating patients with infectious TB) are the most effective because they substantially reduce the rate of infection
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
A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients
We present a direct approach to the construction of Lagrangians for a large
class of one-dimensional dynamical systems with a simple dependence (monomial
or polynomial) on the velocity. We rederive and generalize some recent results
and find Lagrangian formulations which seem to be new. Some of the considered
systems (e.g., motions with the friction proportional to the velocity and to
the square of the velocity) admit infinite families of different Lagrangian
formulations.Comment: 17 page
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