derivation<\/a>. That is to say it obeys the following properties.<\/p>\nIt is linear<\/p>\n
\\[ D[f(z) + g(z) ] = D[f(z)] + D[g(z)] \\; ,\\]<\/p>\n
it operation on a constant results in zero<\/p>\n
\\[ D[ c f(z) ] = c D f(z) \\; , \\]<\/p>\n
and it obeys the Liebniz rule<\/p>\n
\\[ D[f(z) g(z) ] = f(z) \\cdot Dg(z) + Df(z) \\cdot g(z) \\; .\\]<\/p>\n
Repeated application of the $$D$$ operator on the product of two functions results in the usual binomial expansion<\/p>\n
\\[ 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) \\; .\\]<\/p>\n
Now the Lie series operator $$L$$ is defined as<\/p>\n
\\[ L(z,t) f(z) = e^{t D} f(z) \\; , \\]<\/p>\n
where the exponential notation is short-hand for the series<\/p>\n
\\[ 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) \\; .\\]<\/p>\n
Note that often, the arguments $$(z,t)$$ are omitted.<\/p>\n
Now an interesting and useful property of the Lie series operator for the product of two functions is<\/p>\n
\\[ L[f(z)g(z)] = L[f(z)] \\cdot L[g(z)] \\; ,\\]<\/p>\n
or in words, that the Lie series of a product is the product of the Lie series. This relationship is called the interchange relation.<\/p>\n
Since we imagine that all the functions of interest are analytic, this simple relation ‘bootstraps’ to the general relation<\/p>\n
\\[ L[ Q(z)] = Q(L[z]) \\; ,\\]<\/p>\n
for $$Q(z)$$ analytic.<\/p>\n
As an example of this bootstrap relation, consider $$D = \\frac{d}{dz}$$, and note that<\/p>\n
\\[ D^n[z] = 1 \\delta_{n0} \\]<\/p>\n
and therefore<\/p>\n
\\[ L[z] = e^{t D}z = (1+ tD)z = z + t \\; .\\]<\/p>\n
So the bootstrap relation gives first<\/p>\n
\\[ L[ Q(z)] = \\sum_{n=0}^{\\infty} \\frac{t^n}{n!} \\frac{d}{dz} Q(z) \\; ,\\]<\/p>\n
which from elementary calculus is recognized to be $$Q(z+t)$$ expanded about $$t=0$$, and second<\/p>\n
\\[ Q(L[z]) = Q(z+t) \\; . \\]<\/p>\n
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<\/p>\n
\\[ \\frac{d z_i}{dt} = F_i(z) \\]<\/p>\n
with initial conditions<\/p>\n
\\[ z_i(0) = \\zeta_i \\; .\\]<\/p>\n
Then the solution of this system of equations is<\/p>\n
\\[ z_i(t) = e^{t D} \\zeta_i \\; . \\]<\/p>\n
To see this, take the derivative of the solution to get<\/p>\n
\\[ \\frac{d z_i(t)}{d t} = \\frac{d}{dt} e^{tD} \\zeta_i = D e^{t D} \\zeta_i \\; . \\]<\/p>\n
But, by definition, $$ e^{t D} \\zeta_i = z_i(t) $$, so<\/p>\n
\\[ \\frac{d z_i(t)}{d t} = D z_i = F_i \\; . \\]<\/p>\n
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.<\/p>\n
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<\/p>\n
\\[ D = \\frac{\\partial}{\\partial z_0} + \\sum_{i=0}^{n} F_i(z) \\frac{\\partial}{\\partial z_i} \\; .\\]<\/p>\n
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.<\/p>\n
Next time, I’ll apply the Lie series formalism to the Kepler problem<\/p>\n","protected":false},"excerpt":{"rendered":"
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… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-549","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/549","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=549"}],"version-history":[{"count":6,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/549\/revisions"}],"predecessor-version":[{"id":563,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/549\/revisions\/563"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=549"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=549"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=549"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}