Mechanics and Dynamical Systems with Mathematica®

Bellomo, Nicola; Preziosi, Luigi; Romano, António

doi:10.1007/978-1-4612-1338-3

Cited by 31 publications

(19 citation statements)

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“…m, b > 0 models the behaviour of a single degree of freedom system -presented in [1] pag. 10 -made of a particle moving along a fixed straightline and under the elastic force of a wire fixed to another point not belonging to trajectory.…”

Section: Applicationsmentioning

confidence: 99%

“…introduced in [1] with m > 0, b > 1 and ω 2 = k m . Such a system consists of a particle of mass m flowing frictionless along a direction Ox, under the continuous effect of an elastic wire of constant k linking it to a…”

Section: S Foschi G Mingari Scarpello and D Ritellimentioning

confidence: 99%

“…1 Furthermore both F(p) and G(q) take the value zero at the origin where have a non-degenerate minimum. This may be ensured asking that: F (0) = G (0) = 0 and F (0),…”

Section: Introductionmentioning

confidence: 99%

“…The notation p(t, h) and q(t, h) means that to each energy level h, one and only one solution there will be to (1). Furthermore the period function is utilized T :…”

Section: Introductionmentioning

confidence: 99%

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Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

2004

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In 1985 Franz Rothe [16] found, by means of the thermodynamical equilibrium theory, an asymptotic estimate of period of solutions of Ordinary Differential Equations originated by predator -prey Volterra -Lotka model. We extend some of Rothe's ideas to more general systems:and succeed in calculating the period's asymptotic analytic expression as a function of the energy level. We finally check our result re-obtaining classical period's estimation of some popular Hamiltonian systems. We apply our technique also to a nonlinear Hamiltonian system whose period is not available in the literature.KEYWORD: Hamiltonian systems, series reversion, period, asymptotic expansion NOTE: this is the post print version of the paper

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Section: Applicationsmentioning

confidence: 99%

Section: S Foschi G Mingari Scarpello and D Ritellimentioning

confidence: 99%

“…1 Furthermore both F(p) and G(q) take the value zero at the origin where have a non-degenerate minimum. This may be ensured asking that: F (0) = G (0) = 0 and F (0),…”

Section: Introductionmentioning

confidence: 99%

“…The notation p(t, h) and q(t, h) means that to each energy level h, one and only one solution there will be to (1). Furthermore the period function is utilized T :…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

2004

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“…Most scientists believe, even today, that CAS language is a programming language; however, this is not so. CAS can be used like ''live'' mathematics for creating, proving, and evaluating formulas and expressions in numeric or symbolic form, see for example Freeman (1994); Helton and Merino (1998); Bellomo et al (2000) and Romeny (2003).…”

Section: Introductionmentioning

confidence: 99%

Computer algebra solution of the GPS N-points problem

2007

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A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss-Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the minimization of the distance norm. Gauss-Jacobi requires eight times more computation time than the direct numerical solution. It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file) for these algorithms (i.e., Sturmfel's resultant, Dixon's resultants, Groebner basis, reduced Groebner basis and Gauss-Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPS Toolbox

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Vector Space and Linear Maps

Romano

2012

Classical Mechanics With Mathematica®

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Mechanics and Dynamical Systems with Mathematica®

Cited by 31 publications

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Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

Computer algebra solution of the GPS N-points problem

Vector Space and Linear Maps

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