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In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li$acute{e}$nard type equation, $ddot {x} + f(x){dot {x}}^{2} + g(x)= 0$, where $f(x)$ and $g(x)$ are arbitrary functions of $x$. The symmetry analysis gets divided into two cases, $(i)$ the maximal (eight parameter) symmetry group and $(ii)$ non-maximal (three, two and one parameter) symmetry groups. We identify the most general form of the quadratic Li$acute{e}$nard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the nonmaximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.
This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov.
Infrared transmission and electrical resistivity measurements reveal that single crystals of LaMnPO$_{1-x}$F$_{x}$ (x $leq$ 0.28) are insulating. The optical gap obtained from transmission measurements is nearly unaffected by doping, decreasing only
We present a search for a non-Standard-Model invisible particle $X^0$ in the mass range $0.1textrm{-}1.8 ,{rm GeV}/{c^2}$ in $B^{+}to e^{+} X^{0}$ and $B^{+}to mu^{+} X^{0}$ decays. The results are obtained from a $711~{rm fb}^{-1}$ data sample that
Using data samples collected with the BESIII detector operating at the BEPCII storage ring at center-of-mass energies from 4.178 to 4.600 GeV, we study the process $e^+e^-rightarrowpi^{0}X(3872)gamma$ and search for $Z_c(4020)^{0}rightarrow X(3872)ga
We report on the emergence of robust superconducting order in single crystal alloys of 2H-TaSe$_{2-x}$S$_{x}$ (0$leq$x$leq$2) . The critical temperature of the alloy is surprisingly higher than that of the two end compounds TaSe$_{2}$ and TaS$_{2}$.