A periodic functional approach to the calculus of variations and the problem of time-dependent damped harmonic oscillators

I discuss the problem of time-dependent harmonic oscillators on the basis of a periodic functional approach to the calculus of variations. Both the Lagrangian and Hamiltonian formulations are explored and discussed in some detail. Some interesting consequences are revealed.

[1]  Delfim F. M. Torres,et al.  Fractional actionlike variational problems , 2008, 0804.4500.

[2]  Some Inequalities Satisfied by Periodical Solutions of Multi-Time Hamilton Equations , 2005, math/0510557.

[3]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[4]  Jacky Cresson,et al.  Inverse problem of fractional calculus of variations for partial differential equations , 2010 .

[5]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[6]  J. Marsden,et al.  Variational principles for Lie-Poisson and Hamilton-Poincaré equations , 2003 .

[7]  E. C. Oliveira,et al.  Dirac wave equation in the de Sitter universe , 1997 .

[8]  HOMOTOPIES OF EIGENFUNCTIONS AND THE SPECTRUM OF THE LAPLACIAN ON THE SIERPINSKI CARPET , 2009, 0908.2942.

[9]  The scalar wave equation on static de Sitter and anti-de Sitter spaces , 1989 .

[10]  Jerrold E. Marsden,et al.  Nonlinear stability of fluid and plasma equilibria , 1985 .

[11]  Riewe,et al.  Nonconservative Lagrangian and Hamiltonian mechanics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Y. Ben-Aryeh,et al.  Photon statistics and entanglement in coherent-squeezed linear Mach-Zehnder and Michelson interferometers , 2008 .

[13]  E. C. Oliveira,et al.  Klein–Gordon and Dirac Equations in de Sitter Space–Time , 1999 .

[14]  J. Marsden,et al.  Semidirect products and reduction in mechanics , 1984 .

[15]  Delfim F. M. Torres,et al.  Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α, β) , 2007 .

[16]  Darryl D. Holm,et al.  The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories , 1998, chao-dyn/9801015.

[17]  Y. Ben-Aryeh,et al.  Reduction of quantum noise in the Michelson interferometer by use of squeezed vacuum states , 2002 .

[18]  Sheldon R. Smith.  Symmetries and the explanation of conservation laws in the light of the inverse problem in Lagrangian mechanics , 2008 .

[19]  M. Hazewinkel.  Encyclopaedia of mathematics , 1987 .

[20]  A. R. El-Nabulsi.  THE FRACTIONAL CALCULUS OF VARIATIONS FROM EXTENDED ERDÉLYI-KOBER OPERATOR , 2009 .

[21]  Cosmology, oscillating physics, and oscillating biology. , 1993, Physical review letters.

[22]  R. López-Peña,et al.  Noether's theorem and dynamical groups in quantum mechanics , 1990 .

[23]  Y. Ben-Aryeh,et al.  Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers , 2002 .

[24]  R. El-Nabulsi.  MODIFICATIONS AT LARGE DISTANCES FROM FRACTIONAL AND FRACTAL ARGUMENTS , 2010 .

[25]  Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations , 2009, 0903.1169.

[26]  An Alternative Form of the Helmholtz Criterion in the Inverse Problem of Calculus of Variations , 2003, math/0308179.

[27]  El-nabulsi Ahmad Rami.  Oscillating Flat FRW Dark Energy Dominated Cosmology from Periodic Functional Approach , 2010 .

[28]  Francis P. Bretherton,et al.  A note on Hamilton's principle for perfect fluids , 1970, Journal of Fluid Mechanics.

[29]  A. R. El-Nabulsi.  FRACTIONAL QUANTUM EULER–CAUCHY EQUATION IN THE SCHRÖDINGER PICTURE, COMPLEXIFIED HARMONIC OSCILLATORS AND EMERGENCE OF COMPLEXIFIED LAGRANGIAN AND HAMILTONIAN DYNAMICS , 2009 .