Laplace–Runge–Lenz Vector - Alternative Scalings, Symbols and Formulations

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

\mathbf{e} = \frac{1}{mk} \left(\mathbf{p} \times \mathbf{L} \right) - \mathbf{\hat{r}} = \frac{m}{k} \left(\mathbf{v} \times \left( \mathbf{r} \times \mathbf{v} \right) \right) - \mathbf{\hat{r}}

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. Other scaled versions are also possible, e.g., by dividing A by m alone

\mathbf{M} = \mathbf{v} \times \mathbf{L} - k\mathbf{\hat{r}}

or by p0

\mathbf{D} = \frac{\mathbf{A}}{p_{0}} = \frac{1}{\sqrt{2m\left| E \right|}} \left\{ \mathbf{p} \times \mathbf{L} - m k \mathbf{\hat{r}} \right\}

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.

An alternative conserved vector is the binormal vector B studied by William Rowan Hamilton

\mathbf{B} = \mathbf{p} - \left(\frac{mk}{L^{2}r} \right) \ \left( \mathbf{L} \times \mathbf{r} \right)

which is conserved and points along the minor semiaxis of the ellipse; the LRL vector A = B × L is the cross product of B and L (Figure 4). The vector B is denoted as "binormal" since it is perpendicular to both A and L. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.

The two conserved vectors, A and B can be combined to form a conserved dyadic tensor W

\mathbf{W} = \alpha \mathbf{A} \otimes \mathbf{A} + \beta \, \mathbf{B} \otimes \mathbf{B}

where α and β are arbitrary scaling constants and represents the tensor product (which is not related to the vector cross product, despite their similar symbol). Written in explicit components, this equation reads

W_{ij} = \alpha A_{i} A_{j} + \beta B_{i} B_{j} \,

Being perpendicular to each another, the vectors A and B can be viewed as the principal axes of the conserved tensor W, i.e., its scaled eigenvectors. W is perpendicular to L

\mathbf{L} \cdot \mathbf{W} = \alpha \left( \mathbf{L} \cdot \mathbf{A} \right) \mathbf{A} + \beta \left( \mathbf{L} \cdot \mathbf{B} \right) \mathbf{B} = 0

since A and B are both perpendicular to L as well, L ⋅ A = L ⋅ B = 0. For clarification, this equation reads in explicit components

\left( \mathbf{L} \cdot \mathbf{W} \right)_{j} = \alpha \left( \sum_{i=1}^{3} L_{i} A_{i} \right) A_{j} + \beta \left( \sum_{i=1}^{3} L_{i} B_{i} \right) B_{j} = 0

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