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The Relativistic Harmonic Oscillator and the Generalization of Lewis' Invariant

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The Relativistic Harmonic Oscillator and the Generalization of Lewis' Invariant University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2019 The Relativistic Harmonic Oscillator and the Generalization of Lewis' Invariant Daniel Reinhart University of Central Florida Part of the Physics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Reinhart, Daniel, "The Relativistic Harmonic Oscillator and the Generalization of Lewis' Invariant" (2019). Electronic Theses and Dissertations, 2004-2019. 6564. https://stars.library.ucf.edu/etd/6564 THE RELATIVISTIC HARMONIC OSCILLATOR AND THE GENERALIZATION OF LEWIS’ INVARIANT by DANIEL REINHART B.S. The Ohio State University, 2016 A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in the Department of Physics in the College of Sciences at the University of Central Florida Orlando, Florida Summer Term 2019 c 2019 Daniel Reinhart ii ABSTRACT In this thesis, we determine an asymptotic solution for the one dimensional relativistic harmonic oscillator using multiple scale analysis and relate the resulting invariant to Lewis’ invariant. We then generalize the equations leading to Lewis’ invariant so they are relativistically correct. Next we attempt to find an asymptotic solution for the general equations by making simplifying as- sumptions on the parameter characterizing the adiabatic nature of the system. The first term in the series for Lewis’ invariant corresponds to the adiabatic invariant for systems whose frequency varies slowly. For the relativistic case we find a new conserved quantity and seek to explore its interpretation. iii TABLE OF CONTENTS LIST OF FIGURES . vi CHAPTER 1: INTRODUCTION . 1 CHAPTER 2: LITERATURE REVIEW . 2 Landau and Lifshitz Vol. 1: The Adiabatic Invariant . 2 H.R. Lewis . 4 Eliezer and Gray . 4 Shivamoggi and Muilenburg . 5 Shivamoggi: Method of Multiple Scales . 6 Chandrasekhar: Adiabatic Invariants . 7 CHAPTER 3: ASYMPTOTIC SOLUTION FOR THE RELATIVISTIC HARMONIC OS- CILLATOR . 9 CHAPTER 4: DIRECT CALCULATION OF LEWIS’ INVARIANT . 14 Derivation of Lewis’ Invariant . 15 General Solution . 16 iv First Relativistic Corrections . 19 2 2 v0 The case of = c2 ..................................... 20 CHAPTER 5: CONCLUSION . 23 APPENDIX: IMAGES OF SELECTED SOLUTIONS . 24 LIST OF REFERENCES . 27 v LIST OF FIGURES Figure 5.1: Root of the eighth degree polynomial for r0 . 25 Figure 5.2: Twice the invariant to first order . 26 vi CHAPTER 1: INTRODUCTION For the non-relativistic harmonic oscillator with slowly varying frequency, Lewis [1] determined an invariant that was the usual adiabatic invariant to first order. Eliezer and Gray [2] then investigated an alternate derivation of Lewis’ invariant that used the two dimensional harmonic oscillator. This derivation looked at the conserved angular momentum for the system, and it was noted that Lewis’ invariant coincides with half of the square of the angular momentum when the angular momentum is taken to have unit value. Shivamoggi and Muilenburg [3] then showed that Lewis’ invariant arises in the form of the amplitudes of the various asymptotic solutions. In this thesis, we develop a generalized version of Lewis’ invariant for the relativistic harmonic oscillator in much the same way. We determine an asymptotic solution for the relativistic case, and find an invariant arises of the same form as the non-relativistic case. While a series doesn’t appear to be feasible to develop, we find that in the non-relativistic limit it coincides with Lewis’ invariant. We then generalize the procedure of Eliezer and Gray in order to develop an exact expression of the invariant for the relativistic harmonic oscillator. This expression is then used to determine a series for the invariant. This development would be applicable to any one dimensional relativistic system with a potential expandable about a minimum. In the literature review we also show that for a particle in a slowly varying magnetic field the equations of motion can be reduced to those of the harmonic oscillator (Chandrasekhar [4]). 1 CHAPTER 2: LITERATURE REVIEW Landau and Lifshitz Vol. 1: The Adiabatic Invariant For a closed Hamiltonian system undergoing periodic motion, the energy of the system is an in- tegral of motion and provides a method of reducing the number of parameters parameters in the system by one in favor of the energy. In a system whose frequency is slowly varying (as compared to the period of motion), the energy is no longer a conserved quantity (Landau and Lifshitz [5]). For the sake of argument, suppose that the slowly varying parameter is ! = !(t~) where t~= t. We then have (Landau and Lifshitz [5]): dE @H d!(t~) @H @H = = = ω0(t~) : (2.1) dt @t dt @! @! Hence, the rate at which energy decreases will be small (Landau and Lifshitz [5]). This term can be averaged, where we take !0 to be constant over the average. This gives: @H @H = ω0 : (2.2) @t @! Next note (Landau and Lifshitz [5]): Z T I dq I dq T = dt = = @H : (2.3) q_ 0 @p 2 Where we have used Hamilton’s equations of motion. Putting all of this into (2.2) and performing the time average: H @H dq dE 1 Z T @H @! @H = ω0 dt = ω0 @p ; H dq (2.4) dt T 0 @! @H @p where the path of integration is along any path where ! is constant, and therefore the Hamiltonian has the constant value of the energy (Landau and Lifshitz [5]). Using this fact: dH @H @p @H = + = 0; (2.5) d! @! @! @p or (Landau and Lifshitz [5]): @H @p @! = − : (2.6) @H @! @p Taking all of this together, and inserting into (2.4), we finally arrive at (Landau and Lifshitz [5]): dI 1 d I 1 d I @p @E @p @! = p · dq = + dq = 0; (2.7) dt 2π dt 2π dt @E @t @! @t where I is the adiabatic invariant. We therefore have that this quantity is conserved to the order of approximation we are working to. For the harmonic oscillator Hamiltonian, we obtain (Landau and Lifshitz [5]) E I = : (2.8) ! 3 H.R. Lewis This paper introduces the exact expression for Lewis’ invariant when the frequency is slowly vary- ing. The exact expression is given by (Lewis [1]): 1 x2 I = + (ρp − ρx_ )2 ; (2.9) 2 ρ2 where ρ satisfies the equation: 1 ρ¨ + !2ρ − = 0 (2.10) ρ3 We will see in the next section one way of deriving this equation by examining the two dimensional oscillator. Eliezer and Gray In this paper, Lewis’ invariant is presented as an integral of motion arising from a two dimensional oscillator. Using the Lagrangian: 1 1 L = (_ρ2 + θ_2ρ2) − !(t)2ρ2: (2.11) 2 2 We have that θ is cyclic and the conjugate momentum is conserved: h = θρ_ 2 = const: (2.12) 4 Next, the equations of motion for ρ gives (Eliezer and Gray ([2]): ρ¨ − ρθ_2 + !2ρ = 0: (2.13) Using (2.12), we can eliminate θ_ obtaining (Eliezer and Gray[2]): h2 ρ¨ + !2ρ = : (2.14) ρ3 This is the same equation, (2.10), that appears for the coordinate in Lewis’ invariant when h = 1 (Eliezer and Gray [2]). The invariant that arises from this consideration is given by: 1 h2x2 I = (ρp − ρx_ )2 + : (2.15) 2 ρ2 This implies that Lewis’ invariant equivalent to half the angular momentum of a planar oscillator when the angular momentum is taken to have unit value. Shivamoggi and Muilenburg In this paper, Shivamoggi and Muilenburg explored the asymptotic solution for the adiabatic har- monic oscillator. While determining the solution for the equation of motion, the quantity (Shiva- moggi and Muilenburg [3]): 2 2 A0! = C (2.16) was a constant of the motion (here A0 is the slowly varying time dependent amplitude for the first order solution (Shivamoggi and Muilenburg[3]). They were able to obtain a series representation for C2 by finding the higher order solutions. The series obtained was (Shivamoggi and Muilenbug 5 [3]) p2 + !2x2 !0 !0 3!02 !00 C2 = + px + 2 x2 + − (!2x2 − p2) + O(3): (2.17) ! !2 4!3 8!5 4!4 Next this series was compared to the expression obtained by Lewis ([1]) and the two were found to be proportional. Shivamoggi: Method of Multiple Scales In the next section, we rely heavily on the method of multiple scales in order to determine the asymptotic solution for our problem. This method is useful when dealing with a system that has time scales of different order (Shivamoggi [6]. For instance, Shivamoggi and Muilenburg ([3]) applied the method to the adiabatic harmonic oscillator with two time scales: a slow time scale which characterized the slowly varying nature of the frequency and a fast time scale, determined as a power series in . This method will take an ordinary differential equation and turn it into a partial differential equation (Shivamoggi [6]). Much like seperation of variables, this method typically splits our solutions into separate parts. In this case though, the components are functions of the different time scales rather than the spatial variables (as in the separation of variables). 6 Chandrasekhar: Adiabatic Invariants For a charged particle moving in a weakly time dependent magnetic field, the magnetic moment: w µ = ? (2.18) B is an adiabatic invariant for the system (Chandrasekhar [4]). Here, w? is the kinetic energy associ- ated with the velocity of the particle that is perpendicular to the magnetic field B. The equations of motion associate with this system are given by (Chandrasekhar [4]): 1 ¨r = e E + v × B : (2.19) c Fixing the direction of B we have: B = B^z: (2.20) Using Maxwell’s equations for a source free region, and taking the velocity along the magnetic field lines to be zero, (Chandrasekhar [4]): 1 @B r · E = 0; r × E = − (2.21) c @t and setting: eB(t) !(t) = (2.22) mc 7 we arrive at the pair of coupled equations (Chandrasekhar [4]) 1 x¨ − !(t)_y − y!_ = 0 (2.23) 2 1 y¨ + !(t)_x + x!_ = 0: (2.24) 2 Upon the substitution: ξ = x + iy, this can be recast(Chandrasekhar [4]): 1 ξ¨+ i!(t)ξ_ + i!ξ_ = 0: (2.25) 2 If the middle term could be eliminated, then we would have the oscillator equation.
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