A new Lagrangian of the simple harmonic oscillator 1 revisited

Transcription

1 A new Lagrangian of the siple haronic oscillator 1 revisited Faisal Ain Yassein Abdelohssin Sudan Institute for Natural Sciences, P.O.BOX 3045, Khartou, Sudan Abstract A better and syetric new Lagrangian functional of the siple haronic oscillator has been proposed. The derived equation of otion is eactly the sae as that derived fro the first variation s Lagrangian functional. The equation of otion is derived fro Euler-Lagrange equation by perforing partial derivatives on the Lagrangian functional of the second variation of the calculus of variations. Although the better new Hailtonian functional is off that derived fro the first variation by a factor of two, it syetric than in the previous paper. PACS nubers: b, 0.30.Hq, 0.30.X 3. Keywords: General physics, Haronic oscillator, Ordinary differential Equations, Analytic echanics, Euler-Lagrange equation. Introduction The siple haronic oscillator odel is very iportant in physics (Classical and Quantu). Haronic oscillators occur widely in nature and are eploited in any anade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. Discussion (1) First variation of the Calculus of Variation It is known that the Euler-Lagrange equation resulting fro applying the first variation of the Calculus of Variations of a Lagrangian functional L( t, q( t), q of a single independent variable q( t ), its first derivative q ( t) of following action I[ q( t)] L( t, q( t), q when varied with respect to the arguents of integrand and the variation are set to zero, i.e. 1 vixra: , A new Lagrangian of the Siple Haronic Oscillator, Faisal Ain Yassein Abdelohssin, Category: Classical Physics. f.a.y.abdelohssin@gail.co 3 (1)

2 is given by 0 I[ q( t)] L( t, q( t), q [ L( t, q( t), q ] d d t q q t q q q d 0 q q Provided that the variation q vanishes at the end points of the integration and the Lagrangian function doesn t depend eplicitly on tie (i.e. 0 ). t Defining the generalized oentu p as p q Then, the Euler-Lagrange equation ay be written as p q Defining the generalized force F as F q Then, the Euler-Lagrange equation has the sae atheatical for as Newton s second law of otion: F p (i) The Lagrangian functional of siple haronic oscillator The Lagrangian functional of siple haronic oscillator in one diension is written as: 1 1 L k The first ter is the potential energy and the second ter is kinetic energy of the siple haronic oscillator. The equation of otion of the siple haronic oscillator is derived fro the Euler-Lagrange equation: d 0 To give k 0 ()

3 This is the sae as the equation of otion of the siple haronic oscillator resulted fro application of Newton's second law to a ass attached to spring of spring constant k and displaced to a position fro equilibriu position. Solving this differential equation, we find that the otion is described by the function ( t) cos( t ), 0 0 k where 0 ( t t0) and 0. T (ii) The first Hailtonian functional of siple haronic oscillator The Hailtonian functional H H( q, p) is derived fro the first Lagrangain with the use of the Legendre transfo; H pq L and defining p as the generalized oentu. Calculating the right hand q side in the equation defining the Hailtonian, we get 1 1 H k p () Second Variations of the Calculus of Variations It is known that the Euler-Lagrange equation resulting fro applying the second variations of the Calculus of Variations of a Lagrangian functional L( t, q( t), q ( t), q of a single independent variable q( t ), its first and second derivatives q ( t), q ( t) of following action I[ q( t)] L( t, q( t), q ( t), q When varied with respect to the arguents of integrand and the variation are set to zero, i.e. 0 I[ q( t)] L( t, q( t), q ( t), q [ L( t, q( t), q ( t), q ] d d d d t q q q q t q q q q q q is given by q q q L d L d L 0 Provided that the variations q and q vanish at the end points of the integration. (3)

4 The Better Model (1) The new Lagrangian functional of the siple haronic oscillator In the previous paper the new Lagrangian was given by 1 L k The better new Lagrangian functional of the siple haronic oscillator in one diension ay now be written as L k The first ter (the potential energy) is ade twice as the one in the previous paper. The equation of otion is derived fro Euler-Lagrange equation by perforing the partial derivatives on the Lagrangian functional L( ( t), ( t), : d d 0 With the ters calculated as follows k ; 0;. The equation of otion is k 0 Or, k 0 Dividing both sides by, we get the standard equation of otion of the SHO. k 0 () The new Hailtonian functional of the siple haronic oscillator The Hailtonian functional H H( q( t), q ( t), p, ) of the siple haronic q oscillator in the second variation can obtained for the Euler-Lagrange equation of the second variation as follows: First, define the generalized oentu in the second variation as d p Then, the Euler-Lagrange equation ay be written as (4)

5 d d 0 d d [ ] L d [ p ] p This yield p This has the sae atheatical for as of the Euler-Lagrange equation of the first variation and the Newton s second law of otion. The corresponding Legendre transforation in the second variations is written as: H pq q L q Substituting the corresponding variables of the siple haronic oscillator d p d 0 ( ) i.e. p in the Legendre transforation above to obtain the Hailtonian of the SHO as: H pq q L q p p( ) ( ) ( k ) p p( ) k p k which is twice the Hailtonian obtained by the ethod of the first variation. Conclusion: The second variation of the ethod of calculus of variation is rich in its applicability than the first variation. Although there was no kinetic energy ter (first derivative) in the new Lagrangian functional of the siple haronic (5)

6 oscillator we obtained the sae equation of otion siilar to those derived fro the first variation and fro the Newton s second law of otion. The Hailtonian function is off by a factor of two of the one derived fro the first variation. The second variation of the calculus of variations is proising in constructing Lagrangian of dynaical syste which were difficult to construct by following the first variation. It is possible to construct the long sought for: The Lagrangian of the daped haronic oscillator using the second variation of the calculus of variations. References [1] Feynan R, Leighton R, and Sands M. The Feynan Lectures on Physics. 3 Volues, ISBN (006) [] Goldstein. H, Poole. C, Safko. J, Addison Wesley. Classical Mechanics, Third edition, July, (000) [3] Serway, Rayond A., Jewett, John W. (003). Physics for Scientists and Engineers. Brooks / Cole. ISBN [4] Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1 (4th Edition). W. H. Freean. ISBN [5] Wylie, C. R. (1975). Advanced Engineering Matheatics (4th edition). McGraw-Hill. ISBN [6] Hayek, Sabih I. (15 Apr 003). "Mechanical Vibration and Daping". Encyclopedia of Applied Physics. WILEY-VCH Verlag GbH & Co KGaA. ISBN doi:10.100/ eap31. [7] Hazewinkel, Michiel, ed. (001) [1994], "Oscillator, haronic", Encyclopedia of Matheatics, Springer Science+Business Media B.V. / Kluwer Acadeic Publishers, ISBN [8] Cornelius Lanczos, The Variational Principles of Mechanics, University of Toronto Press, Toronto, 1970 edition. [9] F. A. Y. Abdelohssin, Equation of Motion of a Particle in a Potential Proportional to Square of Second Derivative of Position W.r.t Tie in Its Lagrangian, vixra: subitted on :34:18, Category: Classical Physics. [10] F. A. Y. Abdelohssin, A haronic oscillator in a potential energy Proportional to the square of the second Derivative of the coordinate with Respect to tie, vixra: subitted on :15:04, Category: Classical Physics [11] F. A. Y. Abdelohssin, Euler-Lagrange Equations of the Einstein-Hilbert Action, ViXra: subitted on :49:6, Category: Relativity and Cosology. [1] F. A. Y. Abdelohssin, A new Lagrangian of the Siple Haronic Oscillator, vixra: subitted on :4:55, Category: Classical Physics. (6)