Conservative nonlinear oscillators in Abel s mechanical problem

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2 68 R. MUÑOZ AND G. FERNÁNDEZ-ANAYA 2. A family of curves We now return to Abel s mechanical problem, sketched in Sec. 1: a massive bead slides down towards the origin, in a frictionless motion, along a smooth plane curve σ. The conservation of the mechanical energy of the bead dictates that, when starting its motion from rest at a height y i, it will reach the origin in a lapse of time τ(y i ) given by (where τ(y i ) = ds(y) = y i ds (2) 2g(yi y) ( ) 2 dxσ is the differential of the arch length along σ), provided that the bead is subject to a potential: V (y) = mgy. Indeed, in a frictionless motion along a wire, the conservation of energy dictates that m 2 is constant in time, as v = ds = ( ds ) 2 + mgy = E (3) (dx ) 2 + ( ) 2 (4) is just the rapidity of the bead. By considering initial conditions y(t = ) = y i and v(t = ) = we get, from (3): ds = g(y i y) which, when solved for t, gives us (2). Equation (2) is an example of Abel s integral equation. It can be inverted, in order to obtain ds/, by using the Laplace transform method. The steps are laid out in Ref. 18, among other places. The result, which constitutes the solution of Abel s mechanical problem, may be written in the following form: ds 2g (y) = π d y τ(ν) dν (5) (y ν) 1/2 We now examine the implications of imposing on curve σ a condition of the type τ(y i ) = Ky β i (6) for a fixed β R and a real constant K >. For β =, (6) corresponds to the isochronous condition (i.e. the independence of the period from the amplitude in a periodic motion) and the resulting curve turns out to be in this case (half of) uygens s tautochrone (also known as the isochrone curve), while the solutions for β = 1/2 are inclined planes. For the general case we plug (6) in (5) to obtain: ds 2g (y) = π K d y ν β dν (7) (y ν) 1/2 The integral on the r.h.s. of (7) is divergent for β 1, and for β > 1 one can apply the Laplace transform method and the convolution theorem to get ds 2g (y) = Γ(β + 1) π Γ(β + 3/2) K d yβ+1/2 (8) where Γ stands for the gamma function (as we will be tacitly using some of the properties of this function it may be convenient to have a textbook such as [2] at hand). Equation (8) tells us that s will be finite at the origin only if β 1/2, and the case β = 1/2 itself has no geometrical (let alone physical) interpretation, as it would imply that the arch-length traveled from origin to any given point on the curve would be same, irrespective of the end point. Thus, physically acceptable bounded curves exist only for β > 1/2. With this caveat in mind, from (8) we may conclude y x σ (y) = ± where, by definition: := y = ± ( 2g π ( ds σ ) 2 1 ( ) y 2β 1 1 (9) ) 2 1 2β Γ(β + 1) Γ(β + 1/2) K (1) In order to retain the physical interpretation of Eq. (9) the integrand on the r.h.s. must remain real-valued for values of y arbitrarily close to y =, and so the expression is only valid for β 1/2. Moreover, for any acceptable value of β (i.e. 1/2 < β 1/2) we get the restriction y < (11) so that the curve ends at height (a more thorough analysis shows us that /dx diverges at height ). A curve σ is thus identified by three different parameters: β, ± and. From now on we will write σ β,,± to refer to a Rev. Mex. Fís. E 57 (1) (211) 67 72

3 CONSERVATIVE NONLINEAR OSCILLATORS IN ABEL S MECANICAL PROBLEM 69 much easier to reduce the number of variables, eliminating x through (12) in order to obtain from (15) the Lagrangian L β, (y, ẏ) = m 2 ẏ2( y ) 2β 1 mgy (16) The Euler-Lagrange equation: then gives us: ( d ẏ ) L β, (y, ẏ) = (17) y FIGURE 1. Some Σ curves. Chained curve: α =.24, dashed: α =.54, continuous thick: α = 1 (the tautochrone), thin continuous: α = 1.86, dotted: α = In all cases = 1m, g = 9.8m/s 2. specific curve. The solution (9) can be written, with a suitable change of the integration variable, in a very compact way: y/ x β,,± (y) = ± η 2β 1 1dη y [, ] (12) Notice that σ β,,, is just the reflection of σ β,,+, with the vertical axis through the origin acting as the reflection axis. By plugging (11) in (8) and applying the condition s ±,β, (y = ) = (meaning that we will measure the archlength from the origin) we find the arch-length traveled from any given point (x ±,β, (y), y) σ β,,± to the origin: y ) β+1/2 s β,,± (x, y) = s β,,± (y) = (13) β + 1/2( so that s β,, (y) = s β,,+ (y) = s β, (y), as would be expected. Finally, by plugging (13) and (11) in (6) we find: π Γ(β + 1/2) τ β, (s i ) = 2g Γ(β + 1) { β + 1/2 } β/(β+1/2)s β/(β+1/2) i (14) The meaning of this equation is the following: if a bead moves along the curve σ β,,± starting from rest at an archlength s i away from the origin, it will reach the origin in a time τ β, (s i ) given by (14). 3. A family of Lagrangians In order to obtain the equation of motion of a particle moving on a σ β,,± curve, we could start from the Lagrangian of a free-falling bead L(x, y, ẋ, ẏ) = T V = m 2 {(ẏ)2 + (ẋ) 2 } mgy (15) along with condition (12), which is that a of scleronomic holonomic constraint, so that Lagrange s method of indeterminate multipliers could be applied directly. Actually, it is d 2 y 2 + 2β 1 ( ) 2 g 1 2β + 2y y 1 2β = (18) valid for y. This seems to be quite unmanageable an equation. But by applying the point-transformation y s, with the new coordinate s defined through (13) we obtain the transformed Lagrangian L β, (s, ṡ) := m { β/(β+1/2) 2 ṡ2 β+1/2 mg s} (19) so that the Euler-Lagrange equation ( d ṡ ) L β, (s, ṡ) = (2) s now renders d 2 s 2 + g(β + 1/2)2β/(2β+1) 2β 1 2β+1 s 1 2β 2β+1 =. (21) Equation (21) can be written in the form: with the exponent d 2 s 2 = Q α,s α (s > ) (22) α := 1 2β 1 + 2β taking values in [, ), and constant Q α, through: (1 + α)(1 α)/2 Q α, := g α. > defined For α > the nonlinear Eq. (22) is that of a point-like particle of mass m under the influence of the restoring force f α, (s) = mq α, s α (we will not mention anymore the case α =, which is simply free fall). Thus, Eq. (22) would be that of a anharmonic oscillator, if it would not be for the annoying fact that s in only defined in the half-line. If we chose to stick with Eq. (22) we would have to impose suitable boundary conditions at s = in order to have a well posed problem. Instead, we will learn in Sec. 4 how to circumvent this, by joining σ β,,+ with σ β,, at the origin, as laid out in Sec. 1. Rev. Mex. Fís. E 57 (1) (211) 67 72

4 7 R. MUÑOZ AND G. FERNÁNDEZ-ANAYA FIGURE 2. Examples of U Σ potentials. Thin continuous curve: α =.24 (Q = 12.4 Jm 1.24 /kg,) dotted curve: α =.54 (Q = 15.4 Jm 1.54 /kg,) continuous thick curve: a harmonic oscillator, α = 1 (Q = 2. Jm 2 /kg,) chained: α = 1.86, (Q=28.6 Jm 2.86 /kg,) dashed: α = 2.85 (Q=38.5 Jm 3.85 /kg.) In all cases m =.1 kg. FIGURE 4. The phase-space portrait of a truly nonlinear oscillator of Eq. (31), with α = 2.86, m =.1kg, and Q=38.6 Jm 3.86 /kg. Amplitudes s i = 1,.75,.5 and.25 m. p S is graphed in kg m/s and s in meters. FIGURE 3. Some examples of F Σ. Gray curve: α =.24 (mq=1nm.24 ), black line, corresponding to a harmonic oscillator: α = 1 (mq=.7n/m), dashed: α = 2.85 (mq=1nm 2.85 ). 4. A family of nonlinear oscillators From (12) it is easy to obtain the limits: dx β,, lim = + y dx β,,+ lim = (23) y that warrant that the σ β,, path can be smoothly joined at the origin with σ β,,+. We will call the resulting path: Σ β, = Im(σ +,β, ) Im(σ,β, ) (24) ere, Im(σ) stands for the path which is the image of curve σ. Each one of the Σ β, paths has one, and only one, minimum around which a bead will describe a periodic motion. The relation between the period of motion and the amplitude in each path can be deduced from (14) so that it is clear that FIGURE 5. The phase-space portrait of a truly nonlinear oscillator of Eq. (31), with α =.24, m =.1kg, and Q=12.4 Jm 1.24 /kg. Amplitudes s i = 1,.75,.5 and.25 m. p S isgraphed in kg m/s and s in meters. (with the exception of the tautochrone, i.e. β = ) the motion in a Σ β, is anharmonic. To be more precise: T β,, the period of motion in a Σ β, curve, is related with the amplitude of motion, s i, through T β, (s i ) = 4τ β, (s i ) with τ β, as given in (14). In terms of exponent α this turns out to be: T α, (s i ) = 8π g ( Γ ( Γ 1 1+α ) 3+α 2(1+α) ) { α/2 1/2s (α + 1)} 1/2 α/2. (25) In any given Σ β, path, the generalized coordinate S β, can be defined through { s,β, (x, y) if (x, y) σ β,, S β, (x) := (26) s +,β, (x, y) if (x, y) σ β,,+ From now on we will drop, for expediency, the β, subscripts. Rev. Mex. Fís. E 57 (1) (211) 67 72

6 72 R. MUÑOZ AND G. FERNÁNDEZ-ANAYA 5. Conclusions In the preceding pages we have presented a family of conservative oscillators arising from Abel s mechanical problem. This oscillators are amenable to a lagrangian treatment, as the solutions of Abel s mechanical problem always result in holonomic scleronomic constraints. We have produced the exact period-to-amplitude relation (25), valid for all discussed cases. The oscillators turn out to be truly nonlinear, impervious to the usual perturbative treatment. The material included in this paper may complement and enrich any exposition of classical and/or analytical mechanics at university intermediate level as it: - Presents little known instances of conservative nonlinear oscillators. - Provides physically relevant, and fairly simple, examples of the use of lagrangian mechanics. - Shows us that there are still interesting things to learn in classical mechanics at a fairly elementary level. - Illustrates the use of mathematical tools such as the Laplace transform and the Gamma function in physical problems. Acknowledgments The support of SNI-CONACYT (Mexico) is duly acknowledged. 1. Z.K. Peng, Z.Q. Lang, S.A. Billings, and G.R. Tomlinson, J. of Sound and Vibration 311 (28) J.J. Collins and I.N. Stewart, J. Math. Biol. 3 (1992) R. Sisto, A. Moleti, J. Acoust. Soc. Am. 16 (1999) W. Leon-acuteski, Phys. Rev. A 55 (1997) D.P. Gregg, Solar Physics 9 (1984) T. Mausbach, T. Klinger, and A. Piel Phys. Plasmas 6 (1999) V. Marinca and N. erisanu, Chaos, Solitons and Fractals 37 (28) M. Bruschi, F. Calogero, and R. Droghei, J. Math Phys. 5 (29) A. Beléndez, A. ernández, T. Beléndez, C. Neipp, and A. Márquez, Eur. J. Phys. 28 (27) A. Beléndez, J.J. Rodes, T. Beléndez, and A. ernández Eur. J. Phys. 3 (29) L A. Beléndez, M.L. Álvarez, E. Fernández, and I. Pascual, Eur. J. Phys. 3 (29) A. Beléndez, M.L. Álvarez, E. Fernández, and I. Pascual, Eur. J. Phys. 3 (29) P. Amore and F.M. Fernández, Eur. J. Phys. 3 (29) L L. Geng, Eur. J. Phys. 28 (27) C.G. Carvalhaes and P. Suppes, Am. J. Phys. 76 (28) J. Lan and R. Sari, Eur. J. Phys. 28 (27) M.J. O Shea, Eur. J. Phys. 3 (29) G.F. Simmons and J.S. Robertson, Differential Equations with Applications and istorical Notes. (Mc Graw-ill, N.Y. 1972). 19. R.E. Mickens, Truly nonlinear oscillators (armonic Balance, Parameter Expansions, Iteration and Averaging Methods) (World Scientific, Singapore, 21). 2. G.B. Arfken, and.j. Weber Mathematical Methods for Physicists. 6th edition (Elsevier, Amsterdam, 25). Rev. Mex. Fís. E 57 (1) (211) 67 72

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