Γ In particular, the conserved LRL vector component As corresponds to the variation in the coordinates where i equals 1, 2 and 3, with xi and pi being the ith components of the position and momentum vectors r and p, respectively; as usual, δis represents the Kronecker delta. The resulting first-order change in the Lagrangian is Substitution into the general formula for the conserved quantity Γ yields the conserved component As of the LRL vector, Lie transformation The Noether theorem derivation of the conservation of the LRL vector A is elegant, but has one drawback: the coordinate variation δxi involves not only the position r, but also the momentum p or, equivalently, the velocity v. This drawback may be eliminated by instead deriving the conservation of A using an approach pioneered by Sophus Lie. Specifically, one may define a Lie transformation in which the coordinates r and the time t are scaled by different powers of a parameter λ (Figure 9), This transformation changes the total angular momentum L and energy E, but preserves their product EL2. Therefore, the eccentricity e and the magnitude A are preserved, as may be seen from the equation for A2 The direction of A is preserved as well, since the semiaxes are not altered by a global scaling. This transformation also preserves Kepler's third law, namely, that the semiaxis a and the period T form a constant T2/a3. Alternative scalings, symbols and formulations Unlike the momentum and angular momentum vectors p and L, there is no universally accepted definition of the Laplace–Runge–Lenz vector; several different scaling factors and symbols are used in the scientific literature. The most common definition is given above, but another common alternative is to divide by the constant mk to obtain a dimensionless conserved eccentricity vector where v is the velocity vector. This scaled vector e has the same direction as A and its magnitude equals the eccentricity of the orbit, and thus vanishes for circular orbits. Other scaled versions are also possible, e.g., by dividing A by m alone or by p0 which has the same units as the angular momentum vector L. In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by −1. Other common symbols for the LRL vector include a, R, F, J and V. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.