Action-angle variables for the purely nonlinear oscillator
AAction-angle variables for the purely nonlinear oscillator
Aritra Ghosh ∗ and Chandrasekhar Bhamidipati † School of Basic Sciences, Indian Institute of Technology Bhubaneswar,Jatni, Khurda, Odisha, 752050, India
Abstract
In this letter, we study the purely nonlinear oscillator by the method of action-anglevariables of Hamiltonian systems. The frequency of the non-isochronous system is ob-tained, which agrees well with the previously known result. Exact analytic solutions ofthe system involving generalized trigonometric functions are presented. We also presentarguments to show the adiabatic invariance of the action variable for a time-dependentpurely nonlinear oscillator.
Keywords :
Nonlinear Oscillations, Action-Angle Variables, Ateb Functions, GeneralizedTrigonometric Functions
In recent times, there has been a considerable amount of interest in strongly nonlinear oscillators[1-8]. In particular, oscillators with a restoring force proportional to sgn ( x ) | x | α with α beinga positive rational number have attracted much attention lately. These dynamical systems areinteresting both from a theoretical and an applications perspective. Such oscillators arise invarious applications in engineering and mechanics where the nonlinearity associated with thesystem need not be associated with an integer power in the restoring force.From an applications perspective ([6] and references therein), nonlinear oscillations occurin several physical systems of various length scales ranging from macroscopic scales to nanoscales. These are ubiquitous in nature and arise not only in mechanics by also in electronics,biological systems etc. Experimental investigations on aircraft materials, various alloys, wood,ceramic materials, hydrophilic polymers, composites etc reveal that in many real life situationsthe stress-strain properties of the material are strongly nonlinear i.e., the variation of force ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] J un ith displacement or deflection is quite rapid. In such a situation, the conventional polynomialapproximation of the nonlinear restoring force is often not quite useful and this indicates theimportance of the generalized model of the purely nonlinear oscillator where the restoring forceis not necessarily an integer power of the displacement.When the restoring force was an odd integer power of the displacement, one can inprinciple express solutions in terms of Jacobi elliptic functions (see for example [9,10]). Theinclusion of the signum function in this model allows for much general nonlinearity with eveninteger and other rational powers in the restoring force. For this model however, solutions canbe expressed in terms of ateb functions [11-13] which are inversions of the incomplete betafunctions defined as follows, B x ( a, b ) = (cid:90) ≤ x ≤ ( x (cid:48) ) a − (1 − x (cid:48) ) b − dx (cid:48) . (1.1)Senik [12,13] showed that the ateb functions are the solutions of the coupled system,˙ x = y α , ˙ y = − α + 1 x, namely x ( t ) = sa (1 , α, t ) and y ( t ) = ca ( α, , t ). They satisfy the identity: sa ( α, , t ) + ca α +1 (1 , α, t ) = 1and are 2Π α periodic with Π α := B (cid:18) α + 1 , (cid:19) = Γ( )Γ (cid:0) α +1 (cid:1) Γ (cid:16) α +32( α +1) (cid:17) (1.2). They clearly resemble the circular sine and cosine functions and are known as ateb sine sa and cosine ca functions in the literature (see [11-13] for details; Appendix A of [6]). For thecase α = 1 these reduce to the regular sine and cosine functions.The solution of the purely nonlinear oscillator,¨ x + sgn ( x ) | x | α = 0subject to the initial conditions x (0) = A and ˙ x (0) = 0 is written as [6], x ( t ) = Aca ( α, , ω ca t ) , with the frequency ω ca given by ω ca = (cid:114) α + 12 | A | ( α − / (1.3). The period function is given as [6,7]: T ( A ) = (cid:114) πα + 1 Γ (cid:0) α +1 (cid:1) Γ (cid:16) α +32( α +1) (cid:17) | A | (1 − α ) / (1.4)2 It can be verified that, 2Π α ω ca = T ( A ) (1.5). The frequency (and period function) depend on the amplitude A of the system and henceexhibit non-isochronous behavior. All the results for this system reduce to those of the har-monic oscillator with α = 1 wherein, the frequency becomes independent of amplitude andsolutions become the standard trigonometric functions. Motivation and results:
In this paper, we use an entirely different approach to solve thepurely nonlinear oscillator, which are known as action-angle variables. Action-angle variablesare well known in literature for periodic systems and they have also thrown enormous insights into quasi-periodic systems where the corresponding Hamiltonian system is time-dependent. Insituations, where the frequency or length scale of the problem is a time-dependent function andthe system is quasi-periodic, it is known that action-angle variables provide the best approach toobtain adiabatic invariants of the system, which remain unchanged during the motion. It is clearthat this system is a generalization of the linear harmonic oscillator to a nonlinear system withrestoring force being an arbitrary positive power of the displacement. For α = 1, the systemreduces to the harmonic oscillator and hence, the solutions are then the standard (circular)trigonometric functions. The solutions we obtain are in terms of generalized trigonometricfunctions being defined [14-18] as parametric equations to the curve, | x | m + | y | n = 1 , m, n ∈ (1 , ∞ ) (1.6)with, y = sin m,n θ and x = cos m,n θ which are the generalized sine and cosine functions re-spectively corresponding to parameters m and n . Here, m and n are two arbitrary rationalnumber parameters greater than one (see Appendix B for a more detailed discussion on thesefunctions). However for the physical model of the purely nonlinear oscillator, the solutions arein terms of generalized trigonometric functions with m = 2 and n = α + 1 (see below).The main result of this paper may be stated as follows, Theorem 1.1
For the purely nonlinear oscillator given by the equation of motion ¨ x + sgn ( x ) | x | α = 0 (1.7) subject to the initial conditions x (0) = A and ˙ x (0) = 0 , the solution may be expressed as x ( t ) = A sin ,α +1 (Ω ,α +1 t + π ,α +1 /
2) (1.8) with Ω ,α +1 = (cid:114) α + 1 | A | α − (1.9)3he structure of the paper is as follows. In the following section, we define the action-anglevariables for the system. We reproduce the previously known [6,7] result of the period functionof the system by the action-angle method. Exact solutions of the system are then presentedin terms of generalized trigonometric functions. In section 3, we relate the periodicity of thesethe two classes of functions and remark on connections between the two special functions. Weend with remarks showing the adiabatic invariance of the action variable for purely nonlinearoscillator. Brief appendices on action-angle variables and generalized trigonometric functionsare also included. We consider the dimensionless purely nonlinear oscillator described by the equation of motion,¨ q + sgn ( q ) | q | α = 0 (2.1)or the allied system, ¨ q + q | q | α − = 0subject to initial conditions, q (0) = q and ˙ q (0) = 0.The signum function ensures that the restoring force is an odd function of the displace-ment. The equation of motion can be derived from the potential, V ( q ) = (cid:90) | q | α dq = | q | α +1 α + 1The system admits a Hamiltonian structure with the Hamiltonian function taking theform, H ( q, p ) = p | q | α +1 α + 1 (2.2)This is equal to the total energy of the system, H ( q, p ) = E which is a constant of motion. Here we consider defining the action-angle variables (see Appendix A). Solving H ( q, p ) = E for p = p ( q ), p = √ E (cid:18) − | q | α +1 ( α + 1) E (cid:19) / = √ E (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) qq (cid:12)(cid:12)(cid:12)(cid:12) α +1 (cid:19) / The action can be defined as: 4 = (cid:73) pdq (2.3)Therefore one gets J = √ E (cid:73) (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) qq (cid:12)(cid:12)(cid:12)(cid:12) α +1 (cid:19) / dq = √ Eq I α +1 where, I α +1 is an integral independent of q defined as, I α +1 = (cid:73) (cid:112) − | ξ | α +1 dξ (2.4)with ξ = qq .We now introduce angular coordinate θ as, θ = (cid:90) ξ dx (cid:48) (cid:112) − | x (cid:48) | α +1 := arcsin ,α +1 ξ (2.5)where, arcsin m,n ( . ) is an inversion of the generalized sine function [14].Using the identity, | sin m,n φ | n + | cos m,n φ | m = 1 and that cos m,n φ := ddφ (sin m,n φ ), oneexpresses the integral as: I α +1 = (cid:90) π ,α +1 cos ,α +1 θdθ (2.6)where, π ,α +1 is a generalized pi defined as: π m,n := 2 (cid:90) dx (cid:48) n (cid:112) − | x (cid:48) | m This integral [eqn (2.6)] can be expressed as: I α +1 = 4 α + 3 B (cid:18) , α + 1 (cid:19) (2.7)Below, we give numerical proof that eqn. (2.7) is correct. The R.H.S. of eqn. (2.7) is wellknown from the value of Beta function. As for L.H.S, few generalized trigonometric functionscan be plotted by numerical integration of the corresponding differential equations defining the5
10 15 20 ϑ - - m , n ( ϑ ) Generalized Trigonometric Function: cos m , n ϑ Figure 1: Plots of generalized trigonometric functions cos ,α +1 for α = 1 (Black), α =3(Dashed), α = 5(Green)functions. For details, we refer the reader to ref. [17] and directly present the plots in Figure-1,and a few of the generalized π m,n in Figure-2. The values of generalized π m,n are used to performthe integral in eqn. (2.6) numerically. For instance, Figure-3 shows the plot of area under thecurve for cos ,α +1 required to find the value I α +1 using numerical integral. For example, some ofthe values I = 3 . , I = 3 . , I = 3 . , I = 3 . , I = 3 . J = (cid:114) α + 1 | q | α +32 α + 3 B (cid:18) , α + 1 (cid:19) (2.8) The period function of the system can be easily obtained from the action: T ( q ) := ∂J∂E (2.9)Simple manipulations give, T ( q ) = (cid:114) πα + 1 Γ (cid:0) α +1 (cid:1) Γ (cid:16) α +32( α +1) (cid:17) | q | (1 − α ) / (2.10)which matches with the previously known result [6,7].In expressing the solutions in form of generalized trigonometric functions, we recall thefollowing relationship from [15]: 6igure 2: Plot of generalized π m,n for the special case m = nπ m,n = 2 n B (cid:18) m − m , n (cid:19) This means that one can express π ,α +1 as, π ,α +1 = 2 α + 1 B (cid:18) , α + 1 (cid:19) Now one can obtain an expression for the frequency of solution as,Ω ,α +1 = 2 π ,α +1 T ( q )Simple manipulations give, Ω ,α +1 = (cid:114) α + 1 | q | α − (2.11) Since we have defined the action angle variables in the previous section, the solutions of theHamilton’s equations can be obtained trivially. In this formalism the angle drops out of thehamiltonian and hence Hamilton’s equations give,˙ J = 0˙ θ = Ω ,α +1 ( J )7 ϑ m , n ( ϑ ) Area under the curve for: cos m , n [ ϑ ] Figure 3: Sample plot of area under the curve for cos , θ The first equation implies that J is a first integral. The second equation gives, θ ( t ) = Ω ,α +1 t + θ (2.12)Hence, the solution of the purely nonlinear oscillator becomes, q ( t ) = q sin ,α +1 (Ω ,α +1 t + π ,α +1 /
2) (2.13)Where from the initial condition q (0) = q , we get θ = π ,α +1 / m,n ( π m,n /
2) = 1.
We conclude in this section with two important remarks and possible extensions of the currentwork. First, in this work, we have obtained exact analytic solutions to the purely nonlinearoscillator. The solutions we presented were in terms of generalized trigonometric functions.As discussed in the Introduction, the purely nonlinear oscillator system we considered, admitssolutions which have been written earlier in terms of ateb functions too. No known connectionexists between the ateb and generalized trigonometric functions in literature as yet, as theyare independent functions having different properties. Since, we observed here that both thefunctions appear as solutions of the same nonlinear oscillator system, it is then natural toexpect a relation between ateb and generalized trigonometric functions. Although, we leavefor later, a detailed investigation of the complete relation between the two functions, below wemake two comments in support of their possible connection. First, let us consider generalizedtrigonometric functons taking the values m = 2 and n = α + 1; their periodicity in this case8 .14121 3.3652 3.496 3.582 3.64292 3 4 5 6 α l α + , 4 Β , α + α + Figure 4: Comparative plot of L.H.S. (blue) and R.H.S. (red) of equation (2.7)takes the form: π ,α +1 = 2 α + 1 B (cid:18) , α + 1 (cid:19) . (3.1)Asymptotic approximations of ateb functions in terms of elementary functions have been con-sidered in [19,20]. The periodicity of ateb functions is known to be:Π α := B (cid:18) α + 1 , (cid:19) . (3.2)From [eqn(3.1)] and [eqn(3.2)], one may write: π ,α +1 = 2Π α α + 1 (3.3)where one uses the property B ( a, b ) = B ( b, a ). Thus, the 2Π α periodicity of ateb functionsand 2 π ,α +1 periodicity of generalized trigonometric functions with m = 2; n = α + 1 are same,up to a factor of 2 / (1 + α ). Second, we recall from [18] the following relation involving thegeneralized trigonometric functions:arcsin m,n x := 1 n B x n (cid:18) n , m − m (cid:19) , x ∈ [0 , , (3.4)where B x ( a, b ) is the incomplete beta function. The inversion of incomplete beta function isactually the ateb function, there by giving a way to obtain generalized trigonometric functions.These connections may be investigated further to learn whether the generalized trigonomet-ric functions and the periodical ateb functions also share similar identities, and in particular,whether a deeper mathematical connection exists. These works are in progress.9econd, we make some remarks on the importance of pursuing action-angle variables for purelynonlinear oscillators and open issues. One reason is the possibility of extending the resultsto situations where the system becomes time-dependent and quasi-periodic. For instance, ifthe length of the pendulum changes ever so slowly, it is important to know which physicalquantities of the system remain invariant. It is well known that for standard linear harmonicoscillator, the action variable of the Hamiltonian of the system remains unchanged during aperiod of oscillation and is an adiabatic invariant. There are in fact, exact invariants, such as theErmakov-Lewis invariant, which remain unchanged even under more general time-dependentsituations [24]. A proper proof of the adiabatic invariance of the classical action was constructedin [23] for a general Harmonic oscillator. Our aim is to check the whether the action variable J proposed here is such an invariant for the corresponding nonlinear oscillator. Time-dependentHamiltonian systems are ubiquitous in science and engineering and hence the question is wellmotivated. Following the general idea of action-angle variables, a new pair of coordinates ( I, φ )given as: I = (cid:18) p τ + | q | α +1 ( α + 1) τ (cid:19) ≡ Hτ, φ = arcsin ,α +1 q (cid:113) p τ + 2 | q | α +1 α +1 ≡ arcsin ,α +1 q √ Hτ , (3.5)may be proposed. Here H is given in eqn. (2.2) and I is a generalized momentum related tothe putative adiabatic invariant given by J , defined earlier as J = (cid:73) I dφ . (3.6)If q and p evolutions can be understood from the Hamilton’s equations following from H , then I and φ are expected to satisfy the new Hamilton’s equations following from the new Hamiltonian K , which is generally obtained via a canonical transformation from H as: K − H = (cid:18) ∂F ( I, q, τ ) ∂τ (cid:19) I,q = ˙ τ (cid:18) ∂F ( I, q, τ ) ∂t (cid:19) I,q (3.7)Here, F is the generating function given by: F ( I, q, τ ) = (cid:90) q p dq (cid:48) , (3.8)With the set up as above, the proof of adiabatic invariance of the action given in [23], dependson the crucial condition that the generating function F ( I, q, τ ) is independent of ˙ τ ( t ) (see thediscussion after eqn. 34 in [23]). If this condition is satisfied, then it was shown in [23] that | J − I | = O ( ˙ τ ) and also dJ/dt = O (¨ τ , ˙ τ ), signifying that the action variable J varies moreslowly than the slowly varying parameter τ ( t ). It is in this precise sense that the J is anadiabatic invariant. The generating function alluded to in eqn. (3.8) can be computed for10urely nonlinear oscillator, to be: F ( I, q, τ ) = (cid:90) q (cid:112) I/τ − q α +1 /τ dq = − q τ (cid:18) ( α + 1)I (cid:113) − q α +1 I τ F (cid:16) , α +1 ; 1 + α +1 ; q α +1 τ (cid:17) + 4I (cid:19) − q α +1 ( α + 3) τ (cid:113) q α +1 − ττ . (3.9)It can now be shown that the new Hamiltonian for purely nonlinear oscillator written in termsof the action I is K = I/τ + ˙ τ (cid:0) ∂F∂t (cid:1) I,q , where (cid:18) ∂F∂t (cid:19)
I,q = − q (cid:18) τ (cid:18) ( α − (cid:113) − q α +1 I τ F (cid:16) , α +1 ; 1 + α +1 ; q α +1 τ (cid:17) + 8I (cid:19) − q α +1 (cid:19) α + 3) τ (cid:113) q α +1 − ττ (3.10)is clearly independent of ˙ τ . As this key condition is satisfied, the proof given in [23] goes through,with the result that the action variable J for purely nonlinear oscillator is an adiabatic invariant.As a consistency check, for α = 1, it can be shown that the new generating function in eqn(3.10), reduces to the one given in [23], where for the case of standard linear oscillator, theadiabatic invariance of the action has been verified. From science and engineering applicationspoint of view, the full conversion of the new Hamiltonian K in eqn (3.7) and generating function F in eqn (3.10), in terms of the generalized trigonometric functions (using eqn 3.2) needs to bedone, possibly using the methods in [18]. We leave these issues for future work. Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments that ledto the improvement of the article.
Appendices
A Action-angle variables
Consider a one-dimensional Hamiltonian system that is independent of time, H ( q, p ) = p m + V ( q ) , (A.11)with generalized coordinates ( q, p ), whose evolution is given by the level curves of H . A typ-ical phase portrait is given in Figure-5, where the motion corresponds to traversing along theboundary of the curve (ellipse in this case). The families of curves and the path in the phaseportrait depends on a chosen value of the constant of motion. With the assumption that the11 - - - qp Figure 5: A phase portrait corresponding to a simple harmonic oscillatorpotential energy V ( q ) allows the system to perform periodic motion between turning points q ± ,one can naturally get an angle by parameterization of the curve. If the constant of motion isdenoted by G and its value by g with the path given by γ ( g ), the action then becomes the lineintegral enclosing the area inside the curve: I ( g ) = 12 π (cid:73) γ ( g ) p dq . (A.12)It is useful have an example in mind, such as the harmonic oscillator having the Hamiltonian H ( q, p ) = p / m + kq /
2. We proceed in two methods.Method-I: For the Harmonic oscillator, the solutions for generalized coordinates known to be: q ( t ) = A sin φ ( t ) , p ( t ) = mωA cos φ ( t ) , (A.13)where the phase φ ( t ) = ωt + φ giving us a natural angle variable, which keeps track of theposition on the ellipse in figure-(5). Here, ω = (cid:112) k/m , A stands for the amplitude and φ forthe initial phase. The action variable can now be found using the integral in eqn.(A.12) to be I = 12 π (cid:73) γ p dq = 12 π π/ω (cid:90) mωA cos φ ( t ) d ( A sin φ ( t )) = 12 mωA . (A.14)This suggests that a transformation to new variables is possible as: ( q, p ) → ( φ, I ), with anglebeing the phase of the oscillator and action standing for the amplitude. The Hamiltonian canbe rewritten in terms of these new action-angle variables, from the fact that Energy (constant)at a turning point is E = kA = ωI = H ( I ). The motion now becomes simple, as the angle φ drops out of the Hamiltonian.Method-II: The reason for employing action-angle variables in this manuscript is for an entirelydifferent reason, which is, that an explicit knowledge of the solutions in eqn.(A.13) is generally12ot required (irrespective of whether analytical solutions can be found out). This is so be-cause (for conservative systems where H = E is fixed), following eqn. (A.11), the generalizedmomentum can always be written as p = (cid:112) m ( E − V ( q )) . (A.15)Even without explicit knowledge of solutions for ( q, p ), the action variable can always be definedas I = 12 π (cid:73) γ p dq = 2 mπ √ E/k (cid:90) − √ E/k (cid:115) m (cid:18) E − kq (cid:19) dq = Eω . (A.16)The result in eqn. (A.16) matches the one found using Method-I above. The only informationused in Method-II is the shape of the level curve. Furthermore, as mentioned in the introduction,an important reason for the Action-Angle method is its application to situations where theHamiltonian of the dynamical system is time-dependent. In this situation, the action-anglemethod gives rise to adiabatic invariants [23], which do not change and are important in checkingthe integrability of the model in question.As in the harmonic oscillator case, one can express the frequency of a general system as, ω = ∂E∂I Defining J = 2 πI it is easy to see that the time period of oscillation, T = ∂J∂E (A.17)This is form of the action that we have used in section 2 and holds for the general case evenwhere the motion is not necessarily harmonic. In the harmonic oscillator case discussed abovewe showed that transforming to the action-angle variables makes the Hamiltonian independentof the angular coordinate. This holds true even for higher degrees of freedom making theHamilton’s equations trivial: the actions are constants of motion while the angles evolve linearlyin time. The action-angle formalism [21,22] can be readily extended to systems with higherdegrees of freedom where sufficient constants of motion exist and the Hamilton-Jacobi equationis separable. A general theorem in classical dynamics guarantees the existence action-anglevariables for an n dimensional system if there exist at least n independent constants of motionin involution (see for example [22] for a more detailed account). B Generalized Trigonometric Functions
Consider the general curve of the following form: | x | m + | y | n = 1 , m, n ∈ (1 , ∞ ) (B.18)13igure-6 shows plots of three such curves for three different pairs of m and n . For thecase m = n = 2 we get a standard circle and by analogy to the standard circle, one can defineparametric equations to such a ”generalized” circle to be the so called ”generalized trigonometricfunctions”. They are y = sin m,n θ and x = cos m,n θ . We remark that the parameters m and n are completely independent in the most general case. They are rational numbers and can takeany value greater than one. Figure-7 shows plots of sin m,n θ for two different pairs of m and n values. - - - - yx Figure 6: Plot of | x | m + | y | n = 1 with m = 2 , n = 2 (Blue, circle), m = 2 , n = 3 (Orange) and m = 3 , n = 4 (Green)Clearly, they satisfy the identity | cos m,n θ | m + | sin m,n θ | n = 1 (B.19)These functions exhibit periodic behaviour similar to the standard trigonometric func-tions. A rigourous definition for the generalized trigonometric functions may be given as follows.We define the sin m,n x as the inversion of the integral, x = (cid:90) y dy (cid:48) m (cid:112) − | y (cid:48) | n (B.20)This integral x = x ( y ) is monotonically increasing and hence can be inverted so that we get y ( x ) := sin m,n x, x ∈ [0 , π m,n /
2] (B.21)14efined initially in [0 , π m,n / π m,n periodic with, π m,n := 2 (cid:90) dx (cid:48) m (cid:112) − | x (cid:48) | n (B.22) ϑ - - m , n ( ϑ ) Generalised Trigonometric Function: sin m , n ϑ Figure 7: Plot of sin m,n θ with m = 2 , n = 3 (Black) and m = 3 , n = 4 (Dashed)We define the corresponding cosine function as,cos m,n x := ddx (sin m,n x ) (B.23)The following basic properties hold:sin m,n (0) = 0cos m,n (0) = 1sin m,n ( π m,n /
2) = 1cos m,n ( π m,n /
2) = 0These functions clearly show a resemblance with the familiar trigonometric functions. Itis very easily verified that for m = n = 2, all of these definitions and results reduce to those ofordinary trigonometric functions. It should however be remarked that it is not necessary thatall the properties of the standard trigonometric functions be carried out to their generalizedcounterparts. See refs [14-18] for detailed discussions on these functions.15 eferences [1] R.E Mickens, Truly Nonlinear Oscillators , World Scientific Publishing Co. Pvt Ltd (2009).[2] R.E. Mickens,
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