PDM damped-driven oscillators: exact solvability, classical states crossings, and self-crossings

Within the standard Lagrangian and Hamiltonian setting, we consider a position-dependent mass (PDM) classical particle performing a damped driven oscillatory (DDO) motion under the influence of a conservative harmonic oscillator force field $V\left( x\right) =\frac{1}{2}\omega ^{2}Q\left( x\right) x^{2}$ and subjected to a Rayleigh dissipative force field $\mathcal{R}\left( x,\dot{x}\right) =\frac{1}{2}b\,m\left( x\right) \dot{x}^{2}$ in the presence of an external periodic (non-autonomous) force $F\left( t\right) =F_{\circ }\,\cos \left( \Omega t\right) $. Where, the correlation between the coordinate deformation $\sqrt{Q(x)}$ and the velocity deformation $\sqrt{m(x)}$ is governed by a point canonical transformation $q\left( x\right) =\int \sqrt{m\left( x\right) }dx=\sqrt{% Q\left( x\right) }x$. Two illustrative examples are used: a non-singular PDM-DDO, and a power-law PDM-DDO models. Classical-states $\{x(t),p(t)\}$ crossings are analysed and reported. Yet, we observed/reported that as a classical state $\{x_{i}(t),p_{i}(t)\}$ evolves in time it may cross itself at an earlier and/or a latter time/s.

This is just one example among so many discussed in more details by Chandrasekar et al. [2] and some other nonlinear oscillators (including position-dependent mass (PDM) ones) discussed in the sample of references [3][4][5][6][7][8][9][10][11][12][13][14] . Oscillators (damped or undamped) find their applicability in many fields of interest like the search for gravitational waves, laser cooling of atoms, medical physics studies, etc (for more details on this issue, the reader may refer to Marmolejo et al. [15], and references cited therein). Moreover, a classical particle subjected to a conservative harmonic oscillator potential force field in the vicinity of damping and driving forces (referred to as a damped driven oscillator (DDO) [1]) allows the resonance phenomenon to emerge (i.e., the frequency Ω of the applied driving force matches the frequency ω of the simple harmonic oscillator, as in equation (4) below). The resonance phenomena are used in the study of classical mechanics, electromagnetism, optics, acoustics, etc (e.g., [15][16][17][18][19][20] and references cited therein).
In the current methodical proposal, we recycle/recollect (in section 2) the mathematical preliminaries for a constant mass, m • = 1, classical particle performing a damped driven oscillatory motion under the influence of a conservative harmonic oscillator force field V (q) = 1 2 ω 2 q 2 , a Rayleigh dissipative force field R (q) = 1 2 bq 2 , and in the presence of an external periodic non-autonomous force F (t) = F • cos (Ωt). This would, in effect, make our proposal self-contained.
In section 3, we use a point canonical transformation, (12) below, and report the corresponding PDM-Lagrangian as well as the corresponding PDM dynamical equation for the DDO. In the same section, we report our results for two PDM illustrative examples: a non-singular Mathews-Lakshmanan type [10] PDM and a power-law one. Our concluding remarks are given in section 4.

II. DAMPED DRIVEN OSCILLATOR: PRELIMINARIES RECOLLECTED
Consider a classical particle moving under the influence of a conservative oscillator force field V (q) = 1 2 ω 2 q 2 , a Rayleigh dissipative force field R (q) = 1 2 bq 2 , and in the presence of an external periodic non-autonomous force F (t) = F • cos (Ωt). Then the standard Lagrangian describing this particle is given by Under such dissipative and periodic forces settings, the Euler-Lagrange equation of motion reads where η = b/2ω is the damping ratio. This dynamical equation is identified as the damped-driven oscillator (DDO) equation of motion. The solution of its homogeneous part (usually called the complementary or transient solution when F • = 0) is given by Where the values of η identify the nature of damping so that one uses η < 1 for under-damping, η = 1 for criticaldamping, and η > 1 for over-damping. Moreover, under the assumption that q (0) = q • = 0 andq (0) =q • = 0 the solution ( 6) reduces to where B t = 0 to avoid imaginary settings forq (0) =q • = 0 ∈ R (i.e., β = iω 1 − η 2 ; for the under-damping case η < 1). As such, one may suggest that the general solution for (5) is of the form is the so called steady state solution [1]. When this assumption is plugged in (5), it yields Then the general solution for the DDO dynamical equation 5 is given by where Moreover, with cos(δ) = C s (ω 2 − Ω 2 )/F • and sin(δ) = 2C s ηωΩ/F • ) one obtains The phase shift δ measures the phase lag between the external force F (t) and the system's response (with a time lag τ = δ/Ω). Moreover, one may find the amplitude resonance frequency by requiring that dC (Ω) /dΩ = 0 to yield

III. DAMPED DRIVEN OSCILLATOR: PDM-COUNTERPART
A point canonical transformation of the form would yieldq Under such settings, the DDO-Lagrangian (4) would transform into a PDM-DDO one so that Then, the corresponding dynamical equation (5) transforms into the PDM-DDO dynamical equation and inherits its exact solution from (9) through the point transformation (12). Obviously, such a point transformation secures invariance between the two dynamical systems, the constant mass system (i.e., the dynamical equation on the left of (15) and the PDM one (i.e., the dynamical equation on the right of (15)). This methodical proposal is clarified through the following illustrative example.

A. A non-singular Mathews-Lakshmanan PDM-DDO model
Consider a Mathews-Lakshmanan type [10] PDM-particle with In this case, our point canonical transformation (12) would result in Using (9) and (12), one would write to imply as the exact solution for the PDM-DDO dynamical equation At this point, one should notice that its solution (19) converges to q (t) as λ −→ 0. That is, This PDM-DDO equation of motion (20) describes our PDM-particle m (x) of (16) moving in the vicinity of a conservative potential force field and feels a non-conservative Rayleigh dissipative force field along with a driving force F (t) = F • cos (Ωt). Moreover, the corresponding PDM-Lagrangian and PDM-Hamiltonian are given, respectively, by and where is the PDM canonical momentum.
would imply that In this case, our point canonical transformation (12) would result in This is the exact solution for the power-law PDM-DDO dynamical equation Which describes a power-law-type PDM (27) moving in a conservative potential force field and feels a non-conservative Rayleigh dissipative force field The corresponding PDM Lagrangian and PDM Hamiltonian are, respectively, given by and where is the PDM canonical momentum.  For both PDM illustrative examples above, Such a behavior trend is also observed for classical particles with constant mass settings. This is due to the first exponentially decay term in the argument of sinh function of (19), where its contribution dies out and the second term in (19) dominants over at latter time, as in Figure 9.

IV. CONCLUDING REMARKS
To make our methodical proposal self-contained, we have recollected the mathematical preliminaries for a constant mass m • = 1 classical particle performing a DDO-motion under the influence of a conservative harmonic oscillator force field V (q) = 1 2 ω 2 q 2 , a Rayleigh dissipative force field R (q) = 1 2 bq 2 , and in the presence of an external periodic non-autonomous force F (t) = F • cos (Ωt). We have used the point canonical transformation (12) and reported the corresponding PDM-Lagrangian (14) as well as the corresponding PDM dynamical equation for the DDO (15) . We have used two illustrative examples: a non-singular PDM (16) and a power-law PDM one (27). To the best of our knowledge, such PDM-DDO proposal has never been reported elsewhere.
Early on, we have reported the phase-space trajectories crossing (i.e., classical states {x(t), p(t)} crossings) as a phenomenon associated with dissipative forces (c.f., e.g., [8,9,39]) which may as well emerge as a result of the point canonical transformation (12) into PDM-settings. In the current study, we witness yet another type of classical states crossing. That is, as a classical state {x i (t), p i (t)} evolves in time it may cross itself at an earlier and/or latter time/s following the recipe {x i (t 1 ), p i (t 1 )} = {x i (t 2 ), p i (t 2 )} = · · · = {x i (t j ), p i (t j )}; t 1 < t 2 < · · · < t j , as documented in figures 1(b), 2(b), 3(b), 4(b), 5(b), 6(b), 7(b), 8(b), 9(a), and 9(b). This is an indication that a classical state may experience instants of its past and/or future as it evolves in time..