The complete Kepler group can be derived by Lie group analysis

Abstract

It is shown that the complete symmetry group for the Kepler problem, as introduced by Krause, can be derived by Lie group analysis. The same result is true for any autonomous system.

Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

“…Then the admitted Lie symmetry algebra is no longer infinite-dimensional and nontrivial symmetries of the original system can be retrieved [43]. This idea has been successfully applied in several instances ( [43], [57], [47], [44], [49], [35], [45], [15], [53]). Also in [39] we have shown that first integrals can be obtained by Lie group analysis even if the system under study does not come from a variational problem, i.e.…”

“…Here we show how to obtain Lie symmetries by using just the reduction method [43]. If we derive w 2 from equation (3.43c), i.e., 44) then (3.43) transforms into a system of one second-order and one first-order equations in the unknowns w 1 , w 3 .…”

“…Eliminate in turn each of the variables w i -if feasible -in order to obtain an equivalent nth-order system which contains a single equation of second order and n − 2 equations of first order. The admitted Lie symmetry algebra is no longer infinite-dimensional and the Lie group analysis can be deterministically applied ( [43], [57], [49], [15], [53]). A secondary outcome of this strategy is that first integrals can also be determined ( [39], [44], [35]).…”

“…However, in [43] we have remarked that any system of n first-order equations could be transformed into an equivalent system where at least one of the equations is of secondorder. Then the admitted Lie symmetry algebra is no longer infinite-dimensional and nontrivial symmetries of the original system can be retrieved [43]. This idea has been successfully applied in several instances ( [43], [57], [47], [44], [49], [35], [45], [15], [53]).…”

Jacobi's Three-Body System Moves like a Free Particle

“…The Kepler problem and MICZ-Kepler problem were also shown to be equivalent to an isotropic two-dimensional system of linear harmonic oscillators in [26] thanks to Lie symmetries. In [33] Lie group analysis -when applied to Euler-Poisson equations as obtained from the reduction method [37] -unveiled the Kowalevski top [25] and its peculiar integral without making use of either Noether's theorem [34] or the Painlevé method [25]. In [38] Lie group analysis related the famous Lorenz system [32] to the Euler equations of a rigid body moving about a fixed point and subjected to a torsion depending on time and angular velocity, namely Lie group analysis transformed the "butterfly" into a "tornado".…”