Abstract
The solution of the simple harmonic oscillator problem is properly determined by means of the unilateral Fourier transform.
Keywords:
Fourier transform; Simple harmonic oscillator; Unilateral Fourier transform
The differential equation for the simple harmonic oscillator (SHO)
works as an excellent pedagogical tool for illustrating in a simple way several techniques for solving second-order differential equations such as power series expansion, and also Laplace transform (see, e.g. [1[1] E. Butkov, Mathematical Physics (Addison-Wesley, Reading, 1968).]) and Fourier series expansion [2[2] A.S. Castro, Rev. Bras. Ens. Fis. 36, 2701 (2014).] (see also [3[3] S.R. Oliveira, Rev. Bras. Ens. Fis. 39, e3701 (2017).]). Here, the differential equation for the SHO is approached by unilateral Fourier transform.
Let us begin with a brief description of the unilateral Fourier transform and a few of its properties. The direct Fourier sine and cosine transforms of are denoted by and , respectively, and are defined by the integrals (see, e.g. [1[1] E. Butkov, Mathematical Physics (Addison-Wesley, Reading, 1968).])
The original function can be recovered by the inverse unilateral Fourier transforms and expressed as
We now observe that retrieved by must satisfy the homogeneous Dirichlet boundary condition at the origin, whereas retrieved by must satisfy the homogeneous Neumann boundary condition at the origin:
Those boundary conditions are often overlooked in the literature [4[4] C. Rubio-Gonzalez and J.J. Mason, J. Appl. Mech. 66, 485 (1999)., 5[5] C. Rubio-Gonzalez and J.J. Mason, Comput. Struct. 76, 237 (2000)., 6[6] C. Rubio-Gonzalez and J.J. Mason, Int. J. Fracture 108, 317 (2001)., 7[7] E. Lira-Vergara and C. Rubio-Gonzalez, Int. J. Fract. 135, 285 (2005)., 8[8] C. Rubio-Gonzalez and E. Lira-Vergara, Int. J. Fract. 169, 145 (2011)., 9[9] M. Nazar, M. Zulqarnain, M.S. Akram and M. Asif, Commun. Nonlinear Sci. Numer. Simulat. 17, 3219 (2012)., 10[10] N. Shahid, M. Rana and I. Siddique, Bound. Value Probl. 48, 1 (2012)., 11[11] L. Debnath and D. Bhatta, Integral Transforms and Their Applications (CRC Press, New York, 2015), 3rd ed., 12[12] J.C. Araújo and R.G. Márquez, Rev. Eletr. Paul. Mat. 11, 136 (2017)., 13[13] J.C. Araújo and R.G. Márquez, TEMA (São Carlos) 20, 95 (2019).] (see [14[14] A.S. Castro, Rev. Bras. Ens. Fis. 42, e20200129 (2020).] for a merciless criticism). The unilateral Fourier transforms have the following derivative properties
where the proper boundary conditions have already been used.
We are now ready to address the SHO delineated by the homogeneous Dirichlet boundary condition () and the sine Fourier transform, or the homogeneous Neumann condition () and the cosine Fourier transform. Using (2), with denoting the unilateral Fourier transform of , one obtains
The solution is certainly valid for . The complete solution for all can be found by using the property (see, e.g. [1[1] E. Butkov, Mathematical Physics (Addison-Wesley, Reading, 1968).]), where is the Dirac delta symbol. Remembering that , one can write the solution as
where and are arbitrary constants. Using the property (see, e.g. [1[1] E. Butkov, Mathematical Physics (Addison-Wesley, Reading, 1968).])
it is not difficult to see that each inverse Fourier transform yields one of the two linearly independent solutions of our problem:
In conclusion, the complete solution of the SHO can be approached with simplicity via the unilateral Fourier transform method. To the best of author’s knowledge, the SHO was never approached in this way.
Acknowledgement
Grant 09126/2019-3, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.
References
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[1]E. Butkov, Mathematical Physics (Addison-Wesley, Reading, 1968).
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[2]A.S. Castro, Rev. Bras. Ens. Fis. 36, 2701 (2014).
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[3]S.R. Oliveira, Rev. Bras. Ens. Fis. 39, e3701 (2017).
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[4]C. Rubio-Gonzalez and J.J. Mason, J. Appl. Mech. 66, 485 (1999).
-
[5]C. Rubio-Gonzalez and J.J. Mason, Comput. Struct. 76, 237 (2000).
-
[6]C. Rubio-Gonzalez and J.J. Mason, Int. J. Fracture 108, 317 (2001).
-
[7]E. Lira-Vergara and C. Rubio-Gonzalez, Int. J. Fract. 135, 285 (2005).
-
[8]C. Rubio-Gonzalez and E. Lira-Vergara, Int. J. Fract. 169, 145 (2011).
-
[9]M. Nazar, M. Zulqarnain, M.S. Akram and M. Asif, Commun. Nonlinear Sci. Numer. Simulat. 17, 3219 (2012).
-
[10]N. Shahid, M. Rana and I. Siddique, Bound. Value Probl. 48, 1 (2012).
-
[11]L. Debnath and D. Bhatta, Integral Transforms and Their Applications (CRC Press, New York, 2015), 3rd ed.
-
[12]J.C. Araújo and R.G. Márquez, Rev. Eletr. Paul. Mat. 11, 136 (2017).
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[13]J.C. Araújo and R.G. Márquez, TEMA (São Carlos) 20, 95 (2019).
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[14]A.S. Castro, Rev. Bras. Ens. Fis. 42, e20200129 (2020).
Publication Dates
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Publication in this collection
03 Feb 2023 -
Date of issue
2023
History
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Received
28 Nov 2022 -
Accepted
07 Jan 2023