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Regularization of central forces with damping in two and three dimensions

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Abstract

Regularization of damped motion under central forces in two and three dimensions is investigated and equivalent; undamped systems are obtained. The dynamics of a particle moving in \(\frac{1}{r}\) potential and subjected to a damping force is shown to be regularized using Levi-Civita transformation. We then generalize this regularization mapping to the case of damped motion in the potential \(r^{-\frac{2N}{N+1}}\). Further equation of motion of a damped Kepler motion in three dimensions is mapped to an oscillator with inverted sextic potential and couplings, in four dimensions using Kustaanheimo–Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping coefficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in two and three dimensions, we show that the regularized equation is nonlinear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin–Sundman transformation.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. Conserved angular momentum in damped Kepler problem and perturbed harmonic oscillator (see Eqs. (2.6), (2.30)) allows us to equate these numbers and this leads to the relation in Eq. (2.12).

  2. Inserting \(\gamma \) back into above equation, we find the coefficient of the last term in the Lagrangian is \(\frac{m\lambda ^2\gamma ^2}{32c^2}\).

  3. These equations follow from the BCK–Lagrangian \( L=\frac{1}{2m}\left( {\dot{q}}_{i}^2-\Omega ^2q_{i}^2\right) e^{\lambda \tau }, i=1,2\).

  4. Bohlin–Sundman transformation has been applied to derive the mapping between two-dimension harmonic oscillator to two-dimension Kepler problem. Here, angular momentum is conserved in both systems and demanding these constants of motion are proportional to each other, results in the relation between time parameters of these two systems. In the present case too, angular momentum is conserved for damped harmonic oscillator as well as (damped) Kepler problem in two dimensions.

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Acknowledgements

SKP thanks UGC, India, for support through JRF scheme (ID. 191620059604). Work by the author PG was supported by the Khalifa University of Science and Technology under Grant Number FSU-2021-014.

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Authors and Affiliations

  1. School of Physics, University of Hyderabad, Central University P.O, Hyderabad, Telangana, 500046, India

    E. Harikumar & Suman Kumar Panja

  2. Department of Mathematics, Khalifa University of Science and Technology, P.O. Box 127788, Abu Dhabi, UAE

    Partha Guha

Authors
  1. E. Harikumar
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  2. Suman Kumar Panja
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  3. Partha Guha
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Correspondence to E. Harikumar.

Appendix: A motion in \(r^2\) potential with damping in two dimensions: Bohlin–Sundman map

Appendix: A motion in \(r^2\) potential with damping in two dimensions: Bohlin–Sundman map

In this Appendix, we study the application of Bohlin–Sundman mapping to the equations of motion describing a damped harmonic oscillator. After mapping these equations to that of a shifted harmonic oscillator by a time-dependent point transformation, we re-express the equations in terms of complex coordinates. Then by applying a re-parametrization of time followed by a coordinate change, we map these equations describing a dynamics on a constant energy surface to that of a particle moving in \(\frac{1}{r}\) potential.

We start from the equations of motions

$$\begin{aligned} { q}_{i}^{\prime \prime }+\lambda q_{i}^\prime +\Omega ^2q_i,~i=1,2 \end{aligned}$$
(A.1)

describing damped harmonic motion in two dimensions.Footnote 3 Here \(q_{i}^\prime =\frac{{\hbox {d}}q_i}{{\hbox {d}}\tau }\) and \( q_{i}^{\prime \prime }=\frac{{\hbox {d}}^2q_i}{{\hbox {d}}\tau ^2}\). We now apply the time-dependent coordinate transformation

$$\begin{aligned} x_i=q_{i}e^{\frac{\lambda \tau }{2}}, i=1,2 \end{aligned}$$
(A.2)

and rewrite the above equations of motion as

$$\begin{aligned} x_{1}^{\prime \prime }+{\tilde{\Omega }}^2x_1= & {} 0,\end{aligned}$$
(A.3)
$$\begin{aligned} x_{2}^{\prime \prime }+{\tilde{\Omega }}^2x_2= & {} 0, \end{aligned}$$
(A.4)

where \({\tilde{\Omega }}^2=\Omega ^2-\frac{\lambda ^2}{4}\). These are Euler–Lagrange equations following from the Lagrangian

$$\begin{aligned} L=\frac{m}{2}\left( x_{1}^{\prime 2}+ x_{2}^{\prime 2}\right) -\frac{m}{2}{\tilde{\Omega }}^2(x_{1}^2+x_{2}^2)-\frac{m\lambda }{2}(x_1 x_{1}^\prime +x_2 x_{2}^\prime ). \end{aligned}$$
(A.5)

We re-express these equations using complex coordinate

$$\begin{aligned} \omega =x_1+ix_2 \end{aligned}$$
(A.6)

as

$$\begin{aligned} {\omega ^{\prime \prime }}+{\tilde{\Omega }}^2 \omega =0. \end{aligned}$$
(A.7)

We now apply Bohlin–Sundman transformation

$$\begin{aligned} \omega \rightarrow Z=\omega ^2. \end{aligned}$$
(A.8)

and also implement re-parametrization of timeFootnote 4 using

$$\begin{aligned} {{\bar{Z}}}\frac{{\hbox {d}}Z}{{\hbox {d}}t}=\frac{\bar{\omega }}{2}\frac{{\hbox {d}}\omega }{{\hbox {d}}\tau } \end{aligned}$$
(A.9)

Using Eq. (A.8) in Eq. (A.9), we get

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}=\frac{1}{4{{\bar{\omega }}}{\omega }}\frac{\hbox {d}}{\hbox {d}\tau } \end{aligned}$$
(A.10)

and using this we find

$$\begin{aligned} {\dot{Z}}= & {} \frac{{\hbox {d}}Z}{{\hbox {d}}t}=\frac{1}{2{\bar{\omega }}}\frac{{\hbox {d}}\omega }{{\hbox {d}}\tau } \end{aligned}$$
(A.11)
$$\begin{aligned} {\ddot{Z}}= & {} \frac{{\hbox {d}}^2Z}{{\hbox {d}}t^2}=\frac{1}{8{{\bar{\omega }}\omega }} \left[ \frac{1}{{\bar{\omega }}}\frac{{\hbox {d}}^2\omega }{{\hbox {d}}\tau ^2} -\frac{1}{{{\bar{\omega }}}^2} \left( \frac{{\hbox {d}}{{{\bar{\omega }}}}}{{\hbox {d}}\tau }\right) \left( \frac{{\hbox {d}}\omega }{{\hbox {d}}\tau }\right) \right] \end{aligned}$$
(A.12)

From Eq. (A.7), we have \(\omega ^{\prime \prime }=-{\tilde{\Omega }}^2\omega \) and using this, we rewrite the second equation in the above as

$$\begin{aligned} {{\ddot{Z}}}= & {} -\frac{1}{8{{\bar{\omega }}\omega }} \left[ \frac{1}{{\bar{\omega }}}{{\tilde{\Omega }}}^2 \omega +\frac{1}{{{\bar{\omega }}}^2}\left( \frac{{\hbox {d}}{{{\bar{\omega }}}}}{{\hbox {d}}\tau } \right) \left( \frac{{\hbox {d}}\omega }{{\hbox {d}}\tau }\right) \right] \end{aligned}$$
(A.13)
$$\begin{aligned}= & {} -\frac{Z}{8|Z|^3} \left[ \left( \frac{{\hbox {d}}{{{\bar{\omega }}}}}{{\hbox {d}}\tau }\right) \left( \frac{{\hbox {d}}\omega }{{\hbox {d}}\tau }\right) + {{\tilde{\Omega }}}^2{{{\bar{\omega }}}}\omega \right] \end{aligned}$$
(A.14)

We now derive the conserved “energy” associated with the Lagrangian in Eq. (A.5) and re-express the terms in the [ ] appearing in the above equation. To this end, we first obtain the conjugate momenta corresponding to \(x_i\) as

$$\begin{aligned} p_i=m x_{i}^\prime -\frac{m\lambda }{2} x_i, i=1,2 \end{aligned}$$
(A.15)

and construct the Hamiltonian in terms of velocities as

$$\begin{aligned} H= & {} \frac{m}{2}\left( x_{1}^{\prime 2}+ x_{2}^{\prime 2}\right) +\frac{m{{\tilde{\Omega }}}^2}{2}\left( x_{1}^2+x_{2}^2\right) \end{aligned}$$
(A.16)
$$\begin{aligned}= & {} \frac{m}{2}\left[ {{\bar{\omega }}}^\prime \omega ^\prime +{{\tilde{\Omega }}}^2{{{\bar{\omega }}}}{\omega }\right] \equiv E. \end{aligned}$$
(A.17)

Since E above is a constant, we use it to re-express Eq. (A.14) as

$$\begin{aligned} {{\ddot{Z}}}=-\frac{E}{4m}\frac{Z}{|Z|^3}. \end{aligned}$$
(A.18)

We note that with the identification of conserved E with \(4k=m{{\tilde{\Omega }}}^2\), the strength of Kepler potential,

$$\begin{aligned} \frac{E}{4}=k, \end{aligned}$$
(A.19)

Eq. (A.18) is the Kepler’s equation in two dimensions, written in the complex coordinate \(Z=X_{1}+iX_{2}\).

Thus, we have mapped the equation of “damped” harmonic oscillator in two dimensions, to the equation of motion corresponding to the (undamped) Kepler problem in two dimensions.

We now start with the expression for energy of 2-dim Kepler system, described in terms of \({{\bar{Z}}}\) and Z and re-express it in terms of \({{{\bar{\omega }}}}\) and \(\omega \), i.e.,

$$\begin{aligned} E_\mathrm{Kepler}= & {} \frac{m}{2}\frac{{\hbox {d}}{{\bar{Z}}}}{{\hbox {d}}t}\frac{{\hbox {d}}Z}{{\hbox {d}}t}-\frac{k}{|Z|}\end{aligned}$$
(A.20)
$$\begin{aligned}= & {} \frac{m}{8}\left[ \frac{1}{{{{\bar{\omega }}}}\omega } \frac{{\hbox {d}}{{{\bar{\omega }}}}}{{\hbox {d}}\tau }\frac{{\hbox {d}}\omega }{{\hbox {d}}\tau }\right] -\frac{k}{{{{\bar{\omega }}}}\omega }. \end{aligned}$$
(A.21)

Using above equation and Eq. (A.19), we get

$$\begin{aligned} -E_\mathrm{Kepler}=\frac{m}{8}\left( \Omega ^2-\frac{\lambda ^2}{4}\right) . \end{aligned}$$
(A.22)

Thus, we find

  1. 1.

    The equations of motion of two-dimensional damped Harmonic oscillator in Eq. (A.1) are first mapped to equations of a shifted harmonic oscillator given in Eq. (A.4) which are then mapped to that of (undamped) Kepler problem in two dimensions.

  2. 2.

    The strength of the Kepler potential is related to the conserved energy of the shifted harmonic oscillator.

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Harikumar, E., Panja, S.K. & Guha, P. Regularization of central forces with damping in two and three dimensions. Eur. Phys. J. Plus 136, 904 (2021). https://doi.org/10.1140/epjp/s13360-021-01903-0

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