Constructing the time independent Hamiltonian from a time dependent one

• Authors: Michał Dobrski
• Languages of publication: EN

Abstracts

EN

In this paper we introduce a method for finding a time independent Hamiltonian of a given Hamiltonian dynamical system by canonoid transformation of canonical momenta. We find a condition that the system should satisfy to have an equivalent time independent formulation. We study the example of a damped harmonic oscillator and give the new time independent Hamiltonian for it, which has the property of tending to the standard Hamiltonian of the harmonic oscillator as damping goes to zero.

Keywords

EN

Hamiltonian; time independent; canonoid; damped oscillator

Discipline

• 45.20.Jj: Lagrangian and Hamiltonian mechanics

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2007
• Volume: 5
• Issue: 3
• Pages: 313-323

Dates

published

1 - 9 - 2007

online

13 - 5 - 2007

Contributors

author

Michał Dobrski

• Center of Mathematics and Physics, Technical University of Łódź, Al. Politechniki 11, 90-924, Łódź, Poland

References

• [1] F. Gantmacher: Lectures in Analytical Mechanics, Mir Publishers, Moscow, 1970.
• [2] V.I. Arnold: Mathematical methods of classical mechanics, Springer-Verlag, New York, 1978.
• [3] R.M. Santilli: Foundations of Theoretical Mechanics I, Springer-Verlag, New York, 1978.
• [4] R.M. Santilli: Foundations of Theoretical Mechanics II, Springer-Verlag, New York, 1983.
• [5] G. Morandi et al.: “The inverse problem in the calculus of variations and the geometry of the tangent bundle”, Phys. Rep., Vol. 188, (1990), pp. 147–284. http://dx.doi.org/10.1016/0370-1573(90)90137-Q[Crossref]
• [6] Y. Gelman and E.J. Saletan: “q-Equivalent particle Hamiltonians”, Nuovo Cimento B, Vol. 18, (1973), pp. 53–89.
• [7] P. Havas: “The range of application of Lagrange formalism”, Suppl. Nuovo Cimento, Vol. 5, (1957), pp. 364–388.
• [8] J.F. Plebański and H. Garcá-Compeán: “The Lagrangian for a causal curve”, Rev. Mex. Fis., Vol. 43, (1997), pp. 634–648.
• [9] G. Dito and F.J. Turrubiates: “The damped harmonic oscillator in deformation quantization”, Phys. Lett. A, Vol. 352, (2006), pp. 309–316. http://dx.doi.org/10.1016/j.physleta.2005.12.013[Crossref]
• [10] Inverse of the regularized incomplete beta function, Wolfram Research, http://functions.wolfram.com/GammaBetaErf/InverseBetaRegularized/.

Identifiers

DOI

10.2478/s11534-007-0024-7