A Short Note on Nonlocal Transformations and First Integrals for Certain Nonlinear Oscillator Equations

Abstract

This short note presents a simple and effective application of the generalized Sundman transformations and the Jacobi last multiplier approach for two nonlinear oscillator equations. We reobtain the first integrals with this method very easily. In addition, we also compute the Sundman symmetries and Lagrangians.

1. Introduction and Preliminaries

The problem of integrability of ordinary differential equations (ODEs) has been extensively studied. In [1], Guha et al. proposed the generalized Sundman transformation (GST) method for second-order nonlinear ODEs. With this method, one can obtain easily the first integrals of the given equation. Another alternative method is the Jacobi last multiplier (JLM) approach [2]. In this approach, one can construct the Lagrangian of the given equation. Then using this Lagrangian via Noether’s theorem, one finds the first integrals of the given equation. We applied both methods to the following two nonlinear oscillator equations: ̈𝑥̇𝑥−𝑘𝑥²1+𝑘𝑥²+𝛼²𝑥1+𝑘𝑥²̈𝑥̇=0,𝑥+𝑘𝑥²1+𝑘𝑥²+𝛼²𝑥1+𝑘𝑥²³=0, (1)(2) where 𝑘 and 𝛼 are arbitrary parameters. Bruzon et al. [3]. studied and obtained first integrals of the previous equations by the nonlocal symmetry method. In this work, we reobtain these first integrals without much more efforts by the methods of [1, 2].

Now, we present notation to be used and recall the theoretical backgrounds that appear in [1, 2].

The Jacobi equation is given by ̈1𝑥+2𝜙_𝑥𝑥̇²+𝜙_𝑡𝑥̇+𝐵(𝑡,𝑥)=0 (3) and may be transformed to𝑋^′′=0 under the GST 𝑋(𝑇)=𝐹(𝑡,𝑥),𝑑𝑇=𝐺(𝑡,𝑥)𝑑𝑡,𝜕𝐹𝜕𝑥≠0,𝐺≠0. (4) The important subclass of (3) is 𝜙_𝑡=0=𝐵_𝑡, namely, ̈1𝑥+2𝜙_𝑥𝑥̇²+𝐵(𝑥)=0. (5) There are a number of equations which belong to this subclass. In this case, (5) is mapped to the equation 𝑋^′′+𝑋=0, (6) where 𝑋^′=𝑑𝑋/𝑑𝑇. The form of GST which maps (5) to (6) is as follows (𝑥)=±2𝑋=𝐹𝛽²𝐵(𝑥)exp𝜙(𝑥)𝑑𝑥^1/2,𝐵(𝑥)𝜙(𝑥)𝑑𝑇=exp/2±∫𝐵(𝑥)exp𝜙(𝑥)𝑑𝑥^1/2𝑑𝑡, (7)(8) where 𝛽 is a constant. The latter is obviously a nonlocal transformation. Note that a first integral for (6) is obviously 𝑋^′2+𝑋²=𝐼₁. (9) It is well known that a Sundman symmetry for (3) is a transformation of the form 𝐹̃𝑥(𝐹()(,𝐺̃𝑥𝑑̃(𝐹()(𝑡,=𝑀𝑡,𝑥,𝐺𝑡,𝑥))𝑡,𝑡=𝑁𝑡,𝑥,𝐺𝑡,𝑥))𝑑𝑡, (10) where 𝑀 and 𝑁 are some differentiable functions such that the transformation keeps (3) invariant. In other words, (3) is transformed to ^..1𝑥+2𝜙𝑥𝑥̇²+𝜙𝑡𝑥̃̇+𝐵𝑡,̃𝑥=0. (11) Therefore, we have the following Sundman symmetry for (5): 𝐹𝑡,̃𝑥=±𝐹²()𝐺̃𝑥𝑑̃()𝐹𝑡,𝑥+𝑐,𝑡,𝑡=±𝐺𝑡,𝑥𝐹²+𝑐𝑑𝑡. (12) Now, we recall the JLM and Lagrangian for second-order ODEs. Let us consider the following class of system: 𝑥̇=𝑝₁(𝑥)̇𝑦,𝑦=𝑞₀(𝑥)+𝑞₂(𝑥)𝑦². (13) Such a system corresponds to 𝑥̈+𝑓(𝑥)𝑥̇²+𝑔(𝑥)=0, (14) with 𝑓(𝑥)𝑝=−^′1(𝑥)+𝑞₂(𝑥)𝑝₁(𝑥),𝑔(𝑥)=−𝑝₁(𝑥)𝑞₀(𝑥). (15) The JLM for (5) is given by 𝑀(𝑥)=exp^𝑥𝑓(𝑠)𝑑𝑠². (16) It is necessary to point out that for (5) the JLM is independent of 𝑥̇. It is well known that the relationship between Lagrangian and the JLM has the following identity: 𝜕𝑀=²𝐿𝜕𝑥̇². (17) Using (17) and (16), the Lagrangian of (5) has the form of 𝐿(𝑥̇)(𝑥)𝑥̇𝑥,=𝑀²2+^𝑥𝑀(𝑠)𝑞₀(𝑠)𝑝₁(𝑠)𝑑𝑠. (18)

2. Main Results

Firstly, we consider the nonlinear oscillator equation (1). Comparison with the Jacobi equation (5) reveals that 12𝜙_𝑥=−𝑘𝑥1+𝑘𝑥²,𝜙_𝑡()=𝛼=0,𝐵𝑡,𝑥²𝑥1+𝑘𝑥². (19) Hence, from (19), 𝜙(𝑥,𝑡)=ln(1+𝑘𝑥²)⁻¹. As a result from (8) taking the positive square root, we find 𝐹(𝑥)=(𝑖/𝛽)(𝛼/𝑘²𝑥²+𝑘), and it turns out that 𝐺=−(𝑖𝛼√𝑘𝑥/(1+𝑘𝑥²)). Hence, the Sundman transformation has the explicit form (𝑥)=𝑖𝑋=𝐹𝛽𝛼𝑘²𝑥²√+𝑘,𝑑𝑇=−𝑖𝛼𝑘𝑥1+𝑘𝑥²𝑑𝑡. (20) We aim to obtain the first integral of (1) using the transformations (20). Calculating the first derivative of 𝑋(𝑇), 𝑑𝑋=1𝑑𝑇𝛽1+𝑘𝑥^2−1/2𝑥̇, (21) and then substituting it to (9), we obtain the following first integral: 𝐼₁=𝛼²−𝑘𝑥̇²1+𝑘𝑥². (22) We note that (22) has exactly the same result which appeared in [3].

To explore the first integral associated with nonlinear oscillator of (2), repeating the procedure previously given, we obtain the following Sundman transformation: (𝑥)=𝑖𝑋=𝐹𝛽𝛼𝑘²𝑥²√+𝑘,𝑑𝑇=−𝑖𝛼𝑘𝑥1+𝑘𝑥²²𝑑𝑡. (23) Calculating the first derivative of 𝑋(𝑇) and then substituting it to (9), we obtain the following first integral: 𝐼₂=𝑘𝑥²+1𝑥̇²−𝛼²𝑘1+𝑘𝑥². (24) We note that (24) has exactly the same result which appeared in [3]. Now, we try to find the Sundman symmetries of (1) and (2). The Sundman symmetry for (1) being of the form (10), we use (12). Consequently with 𝐹 and 𝐺 given as in (20), one finds that 𝑥=−𝛼²𝑥²−𝑐𝛽²𝑘𝑥²+1−𝛼²+𝑐𝛽²𝑘²𝑥²,̃𝛼+𝑘𝑡=𝐴+𝑖1+𝑘−𝛼²𝑥²−𝑐𝛽²𝑘𝑥²+1−𝛼²+𝑐𝛽²𝑘²𝑥²×𝑥+𝑘1+𝑘𝑥²−𝛼²𝑥²−𝑐𝛽²𝑘𝑥²+1𝑑𝑡, (25) where 𝐴 is a constant of integration. Repeating (12) for (2) with 𝐹 and 𝐺 given as in (23), one finds that 𝑥=−𝛼²𝑥²−𝑐𝛽²𝑘𝑥²+1−𝛼²+𝑐𝛽²𝑘²𝑥²,̃+𝑘𝑡=𝐴+𝑖𝛼1+𝑘−𝛼²𝑥²−𝑐𝛽²𝑘𝑥²+1−𝛼²+𝑐𝛽²𝑘²𝑥²+𝑘²×𝑥1+𝑘𝑥²²−𝛼²𝑥²−𝑐𝛽²𝑘𝑥²+1𝑑𝑡, (26) where 𝐴 is a constant of integration. Now, we try to find the Lagrangians of (1) and (2). Equation (1) can be considered as the following system: 𝑥̇=𝑦,𝑦̇=𝑘𝑥𝑦²1+𝑘𝑥²−𝛼²𝑥1+𝑘𝑥². (27) Consequently from (14), we have 𝑓(𝑥)=−𝑘𝑥/(1+𝑘𝑥²) and𝑔(𝑥)=𝛼²𝑥/(1+𝑘𝑥²). The JLM from (16) is given by 𝑀(𝑥)=1/(1+𝑘𝑥²), and the Lagrangian from (18) is 𝐿(𝑥̇)=𝑥̇𝑥,²21+𝑘𝑥²+𝛼²2𝑘²𝑥².+𝑘 (28)

We note that (28) has exactly the same result which appeared in [4]. Repeating the procedure previously given, we find the following Lagrangian for (2): 𝐿(𝑥̇)=1𝑥,21+𝑘𝑥²𝑥̇²+𝛼²2𝑘²𝑥².+𝑘 (29)