1,721,056 research outputs found
The Jacobi Last Multiplier and its Applications in Mechanics
No full textWe exploit the relationships between the Lie symmetries of a mechanical system, the Jacobi Last Multiplier and the Lagrangian of the system to construct alternative Lagrangians and first integrals in the case that there is a generous supply of symmetry. A Liénard-type nonlinear oscillator is used as an example. We also exemplify the sometimes impossible connection between the general solution of a dynamical system and its first integrals
The method of Ostrogradsky, quantization, and a move toward a ghost-free future
No full textThe method of Ostrogradsky has been used to construct a first-order Lagrangian, hence Hamiltonian, for the fourth-order field-theoretical model of Pais–Uhlenbeck with unfortunate results when quantization is undertaken since states with negative norm, commonly called “ghosts,” appear. We propose an alternative route based on the preservation of symmetry and this leads to a ghost-free quantization
Point and counterpoint between Mathematical Physics and Physical Mathematics
No full textIn recent years there has been a resurgence of interest in problems dating back for over half a century. In particular we refer to the questions of the consistency of quantisation and nonlinear canonical transformations and the quantisation of higher-order field theories. We present resolutions to these questions based upon considerations of symmetry. This enables one to examine these problems within the context of existing theory without the need to introduce new and exotic theories
Gauge variant symmetries for the Schrodinger equation
No full textThe 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 Schrödinger 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 Schrödinger equations possess the same number of Lie point symmetries with the same algebra. Prom the symmetries we construct the solutions of the Schrödinger equation and find that they differ only by a phase determined by the gauge
Lie integrable cases of the simplified multistrain/two-stream model for tuberculosis and Dengue fever
No full textWe apply the techniques of Lie’s symmetry analysis to a caricature of the simplified multistrain model
of Castillo-Chavez and Feng [C. Castillo-Chavez, Z. Feng, To treat or not to treat: The case of tuberculosis,
J. Math. Biol. 35 (1997) 629–656] for the transmission of tuberculosis and the coupled two-stream vectorbased
model of Feng and Velasco-Hernández [Z. Feng, J.X. Velasco-Hernández, Competitive exclusion in
a vector-host model for the dengue fever, J. Math. Biol. 35 (1997) 523–544] to identify the combinations of
parameters which lead to the existence of nontrivial symmetries. In particular we identify those combinations
which lead to the possibility of the linearization of the system and provide the corresponding solutions.
Many instances of additional symmetry are analyzed
An old method of Jacobi to find Lagrangians
No full textIn a recent paper by Ibragimov a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables one to derive (many) Lagrangians of any second-order differential equation. The method is based on the search of the Jacobi Last Multipliers for the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as Ibragimov did. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method
Singularity Analysis and Integrability of a Simplified Multistrain Model for the Transmission of Tuberculosis and Dengue Fever
No full textWe apply singularity analysis to a caricature of the simplified multistrain model of Castillo-Chavez and Feng (J Math Biol 35 (1997) 629-656) for the transmission of tuberculosis and the coupled two-stream vector-based model of Feng and Velasco-Hernandez (J Math Biol 35 (1997) 523-544) to identify values of the parameters for which the system of nonlinear first-order ordinary differential equations describing the model are integrable. A number of combinations of parameters for which the system is integrable are identified. We compare them with the results we obtained by a symmetry analysis in an earlier paper (J Math Anal Appl 333 (2007) 430-449
Some Lagrangians for Systems without a Lagrangian
No full textWe demonstrate how to construct many different Lagrangians for two famous examples that were deemed by Douglas (1941 Trans. Am. Math. Soc. 50 71-128) not to have a Lagrangian. Following Bateman's dictum (1931 Phys. Rev. 38 815-9), we determine different sets of equations that are compatible with those of Douglas and derivable from a variational principle
An algebraic approach to laying a ghost to rest
No full textIn the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck (1950 Phys. Rev. 79 145-65) which has not had a good reception over the past half a century due to the existence of ghosts in the properties of the quantum mechanical solution. Bender and Mannheim (2008 J. Phys. A: Math. Theor. 41 304018) were successful in persuading the corresponding quantum operator to 'give up the ghost'. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physics community and in the process added further credit to the cause of advancement of the use of PT symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Dirac's theory by providing an Hamiltonian that is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order (Nucci M C 1996 J. Math. Phys. 37 1772-5) that it is possible to construct a Hamiltonian that gives rise to a satisfactory quantal description without having to abandon Dirac
Gauge variant symmetries for the Schroedinger equation
No full textThe 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 Schrödinger 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 Schrödinger equations possess the same number of Lie point symmetries with the same algebra. Prom the symmetries we construct the solutions of the Schrödinger equation and find that they differ only by a phase determined by the gauge
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