This column, which is a relatively short one as I am still grappling with the details, is the introductory piece for what I intend to be a through exploration of the use of Lie Series as a method for solving differential equations. This particular entry is heavily influenced by the NASA Technical Note TN D-4460 entitled ‘On the Application of Lie-Series to the Problems of Celestial Mechanics’ by Karl Stumpff, dated June 1968.
Stumpff starts his note with the following interesting passage:
That passage is interesting mostly due what the meaning of ‘Recently’ is. In his references, Stumpff lists only one work by Grobner, entitled “Die Lie-Reihen und ihre Anwendungen” dated 1960. Sophus Lie, from whom the Lie series derives it’s name, died in 1899 and it seems that it took 61 years for his work to really penetrate celestial mechanics.
The central idea of the Lie series is the production of an operator that makes the solution of certain differential equations manifest and their expansion in terms of a power series ‘automatic’. In presenting the prescription, I follow Stumpff’s ordering closely but with modifications of the notation to suit my tastes and needs.
Assume that there are $$n$$ functions
\[ F_i(z) = F_i(z_1, z_2, \dots, z_n) \]
in $$n$$ complex variables $$z_i$$ that are analytic in the neighborhood of
\[ \zeta = (\zeta_1, \zeta_2, \dots, \zeta_n) \; . \]
The concept of a Lie series depends on first defining the operator
\[ D = F_1(z) \frac{\partial}{\partial z_1} + F_2(z) \frac{\partial}{\partial z_2} + \dots + F_n(z) \frac{\partial}{\partial z_n} \; .\]
The operator $$D$$ is a derivation. That is to say it obeys the following properties.
It is linear
\[ D[f(z) + g(z) ] = D[f(z)] + D[g(z)] \; ,\]
it operation on a constant results in zero
\[ D[ c f(z) ] = c D f(z) \; , \]
and it obeys the Liebniz rule
\[ D[f(z) g(z) ] = f(z) \cdot Dg(z) + Df(z) \cdot g(z) \; .\]
Repeated application of the $$D$$ operator on the product of two functions results in the usual binomial expansion
\[ D^n [ f(z) g(z) ] = \sum_{\nu = 0}^{n} \left( \begin{array}{c} n \\ \nu \end{array} \right) D^{\nu} f(z) D^{n-\nu} g(z) \; .\]
Now the Lie series operator $$L$$ is defined as
\[ L(z,t) f(z) = e^{t D} f(z) \; , \]
where the exponential notation is short-hand for the series
\[ L(z,t) f(z) = \sum_{n=0}^{\infty} \frac{t^n}{n!} D^n f(z) = \left[1 + t D + \frac{t^2}{2!} D^2 + \dots \right] f(z) \; .\]
Note that often, the arguments $$(z,t)$$ are omitted.
Now an interesting and useful property of the Lie series operator for the product of two functions is
\[ L[f(z)g(z)] = L[f(z)] \cdot L[g(z)] \; ,\]
or in words, that the Lie series of a product is the product of the Lie series. This relationship is called the interchange relation.
Since we imagine that all the functions of interest are analytic, this simple relation ‘bootstraps’ to the general relation
\[ L[ Q(z)] = Q(L[z]) \; ,\]
for $$Q(z)$$ analytic.
As an example of this bootstrap relation, consider $$D = \frac{d}{dz}$$, and note that
\[ D^n[z] = 1 \delta_{n0} \]
and therefore
\[ L[z] = e^{t D}z = (1+ tD)z = z + t \; .\]
So the bootstrap relation gives first
\[ L[ Q(z)] = \sum_{n=0}^{\infty} \frac{t^n}{n!} \frac{d}{dz} Q(z) \; ,\]
which from elementary calculus is recognized to be $$Q(z+t)$$ expanded about $$t=0$$, and second
\[ Q(L[z]) = Q(z+t) \; . \]
Now the utility of the Lie series is not that it provides a compact way of representing a Taylor’s series (as convenient as that is) but rather in the fact that when the $$F_i$$ are not trivial functions it encodes solutions to a coupled set of differential equations. To see how this works, assume a system of differential equations given by
\[ \frac{d z_i}{dt} = F_i(z) \]
with initial conditions
\[ z_i(0) = \zeta_i \; .\]
Then the solution of this system of equations is
\[ z_i(t) = e^{t D} \zeta_i \; . \]
To see this, take the derivative of the solution to get
\[ \frac{d z_i(t)}{d t} = \frac{d}{dt} e^{tD} \zeta_i = D e^{t D} \zeta_i \; . \]
But, by definition, $$ e^{t D} \zeta_i = z_i(t) $$, so
\[ \frac{d z_i(t)}{d t} = D z_i = F_i \; . \]
In some sense, this is our old friend the propagator or state transition matrix written in a new form. However, this new encoding works for non-linear systems as well, a point that makes it an improvement on those approaches.
One last note. So far, the system of differential equations was assumed to be autonomous. In the event the system isn’t, a simple ‘trick’ can be used to make it look autonomous. Define a new variable $$z_0 = t$$ and the augment the definition of $$D$$ to be
\[ D = \frac{\partial}{\partial z_0} + \sum_{i=0}^{n} F_i(z) \frac{\partial}{\partial z_i} \; .\]
This manipulation has the advantage of making an non-autonomous system look formally autonomous. The only disadvantage is that all notion of equilibrium points are lost since the right-hand side of the equations can never have well-defined critical points.
Next time, I’ll apply the Lie series formalism to the Kepler problem