This week the Lie series faces a stiff challenge in the form of the Kepler problem. Unlike some of the cases previously examined in this blog (harmonic oscillator, quantum wave propagation, etc.) the Kepler problem is nonlinear when written in the usual fashion in terms of Cartesian coordinates. There is a standard ‘clever’ substitution that make everything look linear ($$u = 1/r$$) but this won’t be employed.
In terms of the Cartesian coordinates, the orbital state for the Kepler problem is
\[ \bar s = \left[ \begin{array}{c} \vec r \\ \vec v \end{array} \right] \; ,\]
where $$\vec r$$ and $$\vec v$$ are the position and velocity of a reduced mass moving in a central potential $$V(r) = 1/r$$.
The equations of motion in state space notation are
\[ \frac{d}{dt} \bar S = \left[ \begin{array}{c} \vec v \\ -\frac{\mu \vec r}{r^3} \end{array} \right] \; ,\]
with initial conditions $$\vec r(t_0) = \vec \rho$$ and $$\vec v(t_0) = \vec \nu$$.
The corresponding Lie operator, which is written in terms of the initial conditions is
\[ D = \sum_i \nu_i \frac{\partial}{\partial \rho_i} – \frac{\mu \rho_i} {\rho^3} \frac{\partial}{\partial \nu_i } \; \]
or more compactly (with some slight abuse of notation)
\[ D = \vec \nu \cdot \partial_{\vec \rho} – \epsilon \vec \rho \cdot \partial_{\vec \nu} \; ,\]
where $$\epsilon = \mu/\rho^3$$ has been defined for future convenience.
The full solution is then given by the application of the Lie series operator $$L$$ to the initial position $$\vec \rho$$
\[ \vec r(t) = L[ \vec \rho ] \; .\]
With $$D$$ being nonlinear, the expansion implicit in this solution is very difficult to evaluate to all orders so a truncation of the expansion to order $$(t-t_0)^3$$
\[ L[\vec \rho] = e^{(t-t_0) D} \vec \rho \approx \left[ 1 + (t-t_0) D + \frac{ (t-t_0)^2}{2!} D^2 + \frac{(t-t_0)^3}{3!} D^3 \right] \vec \rho \]
will be employed with an eye for what patterns emerge.
A little bit of organization usually serves well in these types of expansions and, with that in mind, we note the following evaluations
\[ D \vec \rho = \vec \nu \; ,\]
\[ D^2 \vec \rho = D \vec \nu = – \frac{ \mu }{\rho^3} \vec \rho = – \epsilon \vec \rho \; ,\]
and
\[ D^3 \vec \rho = D( -\epsilon \vec \rho) = -\vec \nu \cdot \partial_{\vec \rho} \epsilon – \epsilon \vec \nu \; . \]
The third order expansion takes the form
\[ L[\vec \rho] \approx \vec \rho + (t-t_0) \vec \nu – \frac{ (t-t_0)^2 }{2!} \epsilon \vec \rho – \frac{ (t-t_0)^3 }{3!} \left( \vec \nu \cdot \partial_{\vec \rho} \epsilon + \epsilon \vec \nu \right) \; .\]
The last remaining partial derivative evaluates to
\[ \vec \nu \cdot \partial_{\vec \rho} \epsilon = \vec \nu \cdot \partial_{\vec \rho} \frac{\mu}{\rho^3} = -\frac{3 \vec \nu \cdot \vec \rho \mu}{\rho^5} \; . \]
Defining one last ‘greek’ shorthand (the reason for this will emerge shortly)
\[ \lambda = \frac{\vec \nu \cdot \vec \rho}{\rho^2} \]
allows the expansion to take the form
\[ L[\rho] \approx \vec \rho + (t-t_0) \vec \nu – \frac{ (t-t_0)^2 }{2!} \epsilon \vec \rho – \frac{ (t-t_0)^3 }{3!} \left( 3 \epsilon \lambda \vec \rho – \epsilon \vec \nu \right) \; .\]
What to make of this expansion? Compare this against the Taylor’s series expansion of the position (see ‘An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition’ by Battin, pp 110-112)
\[ \vec r(t) = \vec r0 + (t-t0) \left . \frac{d \vec r}{dt} \right|_{t=t0} + \frac{(t-t0)^2}{2!} \left. \frac{d^2 \vec r}{d t^2} \right|_{t=t0} \\ + \frac{(t-t0)^3}{3!} \left. \frac{d^3 \vec r}{d t^3} \right|_{t=t0} + \cdots \; .\]
Apparently, Lagrange defined the following ‘invariants’ (there are three of them but I’m only using the first 2)
\[ \epsilon = \frac{\mu}{r^3} \]
and
\[ \lambda = \frac{ \vec r \cdot \vec v }{ r^2 } \; \]
In terms of these classical invariants, the first three derivatives in the Taylors series become
\[ \frac{d \vec r}{dt} = \vec v \; , \]
\[ \frac{d^2 \vec r}{dt^2} = -\epsilon \vec r \; ,\]
and
\[ \frac{d^3 \vec r}{dt^3} = 3 \epsilon \lambda \vec r – \epsilon \vec v \; .\]
The series expansion thus agrees with the Lie series method or, in other words, the Lie series has reproduced the classical Lagrange f-g series
\[ \vec r(t) = f(t) \vec \rho + g(t) \vec \nu \; .\]
While there is really no new content that the Lie series has provided it does provide a few advantages. The first is that the notation is much more compact than the traditional approach using the Lagrange invariants. Second, the Lie series is clearly extensible to other, more complex problems and in such a fashion that is seems to provide a theoretical insight.
I’ll try to flesh this idea out next week when I apply the Lie series to the pendulum problem.