How does Lie group help to solve ode's?

[ayan849]

Being not an expert, my question might sound naive to students of mahematics. My question is how on Earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?

 

[Alesak]

Suposedly if the ODE is somehow symmetric, the solutions are limited to representations of corresponding Lie group. I´ve seen this in Hall´s book on Lie groups, section "Why study representations", but I don´t understand it either.

Also I find it weird no one here understands it. (or why else there is no response?)

 

[ayan849]

Many thanks Alesak for your reply and the cited reference. It would be nice if someone could post a geometrical answer...

 

[Stephen Tashi]

My question is how on Earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?

I've often wondered about it myself. I even have a modern book on the subject: "Solution of Ordinary Differential Equations by Continuous Groups", by Geore Emanuel, Champion and Hall/CRC. I've glanced at it from time to time, but never studied it throroughly.

I'd be glad to discuss it here and that might motivate me to study it more. I think you need at least a facility with the calculus of several variables in order to understand the material. After introducing continouous gropus, Emanuel discusses "The Method Characteristics", so understanding the geometric interpretation of that technique is probably necessary.

How big a non-expert are you? How serious are you about findng an answer?

 

[lavinia]

I'd be glad to discuss it here and that might motivate me to study it more. I think you need at least a facility with the calculus of several variables in order to understand the material. After introducing continouous gropus, Emanuel discusses "The Method Characteristics", so understanding the geometric interpretation of that technique is probably necessary.

How big a non-expert are you? How serious are you about findng an answer?

I would love to have a discussion of this topic - which I know zero about.

 

[Stephen Tashi]

Do you already know something about Lie goups?

And I should have written "The Method Of Characteristics".

 

[lavinia]

Do you already know something about Lie goups?

And I should have written "The Method Of Characteristics".

I have read most Chevalley's book.

 

[Stephen Tashi]

I have read most Chevalley's book.

Then you know more than I do about Lie groups.

How shall we go about this? It might be boring to go through Emanuel's book in order. I'll jump to a crucial point and we can go back and pick up the background. Perhaps you can explain what a "once extended group" is.

Summarizing the first section of Chaper 5 with some excerpts:

The general form for a first-order ODE is:
$f(x,y,y') = 0 \$ (5.1)

(I don't know how to make LaTex leave more spaces between the equations and the numbers, which follow them.)

"We assume this ODE is invariant under a [one parameter] group whose symbol is $U f$.

That sounds like old-fashioned jargon. What would the "symbol" of the group be? Is it the so-called "infinitesimal transformation"? I'll explain what he says about it below.



In a previous chapter he discusses "function invariance" under groups. Of "function invariance", he begins by saying:

We start with a function $f(x,y)$. The condition under which it is invariant with respect to the the group is determined next. By this we mean that

$f_1 = f(x_1, y_1) = f(x,y)$ (4.1)

I have to go back another chapter to make sense of that notation. In the previous chapters he has defined the group element corresponding to the value of the parameter $\alpha$ as the transformation that sends $(x,y)$ to $(x_1,y_1)$ by the formulae:

$x_1 = \phi(x,y,\alpha)$ (2.3)
$y_1 = \psi(x,y,\alpha)$

Where $\phi$ and $\psi$ are known functions.

In terms of those functions, he defines:

$\zeta = \frac{\partial \phi}{\partial \alpha}|_0$
$\eta = \frac{\partial \psi}{\partial \alpha}|_0$

The differential operator $U$ is defined by
$U f = \zeta f_x + \eta f_y$, which is (I think)
$U f = \zeta \frac{\partial f}{\partial x} + \eta \frac{\partial f}{\partial y}$.

Returning to his treatment of function invariance, he says

$$f(x_1,y_1) = f(x,y) + \alpha U f + \frac{\alpha^2}{2} U^2 f + ...$$
we observe that a necessary a sufficient condition for $f$ to be an invariant function of the group, is for

$Uf = 0$ (4.2)

I'll end this post with that note. To formulate the definition of invariance for $f(x,y,y')$ he has to introduce the notion of the "once extended" group.

Anyone have any geometric insight about the above concepts?

 

[Sina]

One idea is that if the ODE admits enough 1-parameter group of symmetries (that are transformations that take solutions to solutions, where solutions are the curves in the phase space) which are Lie groups, one by one you find change of coordinates that takes each of these transformations to their normal form (i.e they become translations along a coordinate [call this coordinate s], if you are in R^n with sufficient smoothness conditions such transformations always exist where basically 1-parameter curves become one of your coordinates) and this change of coordinates reduces the order of your equation by 1. Ofcourse the order in which you choose these symmetries is an important detail. Moreover for an nth order ODE you have to have atleast n symmetries. To avoid locality issues I assume this was done on R^n. I don't how this is generalized to ODE on manifolds. Geometrically this means that solutions in these new coordinates do not essentially depend on the redundant coordinate s but only on its derivatives. So in the new form your ODE lacks the variable s but depends on its derivatives, reducing the equation by one order.

An accesible introduction (to a physics student) is Stephani ODE and Symmetries. However there are an immense amount of geometrical ideas hidden in that book (which are not explained for simplicity and mostly everything is introduced as a metodology) so it is very instructive to read or return back to that book after you had a proper course on differential geometry covering Lie groups and algebras.

 

[Sina]

So really what you care about is the Lie group of transformations on the phase space which take solution curves to solution curves. Intiutively, existence of a symmetry may mean redundancy which can be used to lower the degree of the equation.

 

[Stephen Tashi]

So really what you care about is the Lie group of transformations on the phase space which take solution curves to solution curves.

It will be interesting to square this with Emmauel's statements. The invariance I gave in my post is what he calls "function invariance". He defines another type of invariance (which he says is less important) called "curve invariance".

Another type of invariance is now discusses. Our goal will be to find the conditions wherein one curve of a one-parameter family of curves transforms into itself or into another curve of the family.

Let
$f = f(x,y) = c$ (4.8)
be a one-parameter family of curves. We select one of these curves, which is written as
$f_1 = f(x_1,y_1) = c.$
Under the Transformation (2.3) this becomes
$f_1 = f(\phi(x,y,\alpha),\psi(x,y,\alpha)) = \omega(x,y,\alpha)$ (4.9)
or
$\omega(x,y,\alpha) = c_1$.
By letting $\alpha$ vary with $c_1$ fixed this relation becomes the other members of the family of curves.

 

[Alesak]

Hi guys,

I found this awesome book: http://www.physics.drexel.edu/~bob/LieGroups.html.

It´s spiritual succesor of his earlier book (Lie Groups, Lie Algebras, and Some of Their Applications) and it seems to have the best motivation I´ve seen in a book about lie groups. Definitely worth checking out.

 

[Sina]

A word of caution, Gilmore is not for the mathematically twitchy :p by meaning which you should be able to withstand some amount of informal arguments or should enjoy trying to make them rigorous. The contents of the book looks like a miracle which should also tell you that not everything is trated in as much detail as Knapp's Lie groups.

It is also a nice reference book for lie groups and algebras but many properties are given without proof.

 

[Alesak]

I´m trying to go through Knapp for a school course, and I think he has the highest "proof/other text" ratio I´ve seen in a long time. The proofs seem form a dense set in his book even. It is really helpful to read about lie groups from other perspectives as well.

One good companion seems Quantum Mechanics. Symmetries, too bad I don´t know any quantum mechanics!

 

[Sina]

yea I meant gilmore's book when I said many properties are given without proof

 

[Stephen Tashi]

My own interest in this is simply to obtain a geometric intuition for what's going on. I think that is in line with ayan849's original post.

Let's start with a simple question. What is the geometric interpretation of the operator U?.

As I recall, the convention in Lie groups is that when the parameter(s) are zero then the particular transformation they define is the identity. So, for a 1 parameter group, is U f(x,y) something like a directional derivative of f(x,y) taken along the direction that the group is moving the point (x,y)?

I say "something like" since there have been other threads about whether a "direction derivative" must be taken with respect to a direction defined by a unit vector. It was mentioned that Apostol, in a book, defines a derivative of a function "with respect to a vector" without requiring that the vector be a unit vector. However, he does not call this a "directional derivative". Is it correct to say that U is a "derivative with respect to a vector" that may not be a unit vector?

 

[ayan849]

Can anybody care to give a geometrical interpretation? I can't understand a bit of what is going on here :(

 

[Stephen Tashi]

While we wait to see if anybody cares, could you explain whether your mathematical background includes directional derivatives and differential operators?

Currently, I can't answer the question ("In geometric terms, how does Lie theory helps solve differential equations?") and I only get a hazy idea from the post that said that it amounts to changing coordinates so every transformation amounts to a translation. I'll flatter myself by thinking that I am capable of getting geometric picture if I wade through some of the books that I have. (I like the Gilmore text.) However, I probably don't have the self discipline to do this without some motivation ( being a very busy person - retired, you know. It takes up all your time.) We might figure this out if you want to have a long discussion about it, which will keep me interested.

If you were just asking a casual question and wanted a quick answer without getting into a lot of posts, I can't help you.

 

[jojo12345]

Here is a geometric picture of how symmetries can aid in the solution of certain ODEs. I'll begin with some technical discussion and conclude with some qualitative remarks.

Let $M$ denote the phase space. This is the manifold where solution curves to the ODE are contained. Let $X:M\rightarrow TM$ denote the vector field that defines the ODE. A geometric way of saying that this ODE possesses symmetry is $M$ admits a (left) Lie group action $\Phi:G\times M\rightarrow M$ that commutes with the flow map of $X$. Recall that the flow map $F_\lambda$ of $X$ takes a point $m\in M$ and returns the point $F_\lambda(m)$ equal to where the initial condition $m$ ends up after moving $\lambda$ seconds according to the dynamics defined by the ODE. So, restated in symbols, the condition that the ODE possesses a symmetry is
$$\forall g\in G,\lambda\in\mathbb{R},~~\Phi_g\circ F_\lambda=F_\lambda\circ\Phi_g.$$
Provided the action $\Phi$ is sufficiently nice (e.g., free and proper), then it is possible to form the quotient manifold $M/G$, which is just the collection of group orbits (see http://en.wikipedia.org/wiki/Group_action for more discussion on group actions, particularly the section on continuous group actions). On this smaller manifold $M/G$, the original ODE induces a new ODE $x:M/G\rightarrow T(M/G)$ via the following construction.

Let $\pi:M\rightarrow M/G$ denote the map that assigns to every point $m\in M$ the group orbit $m$ lives on. For each $\lambda\in\mathbb{R}$, define the mapping $f_\lambda:M/G\rightarrow M/G$ by $$f_\lambda(\pi(m))=\pi(F_\lambda(m)).$$
This is a well-defined mapping precisely because the symmetry condition holds. That is, if $m^\prime=\Phi_g(m)$ lives on the same group orbit as $m$, then
$$f_\lambda(\pi(m^\prime))=\pi(F_\lambda(\Phi_g(m)))=\pi(F_\lambda(m))=f_\lambda(\pi(m)).$$
The key property this map possesses is that it defines a one-parameter subgroup of $M/G$, which in symbols means $f_{a+b}=f_a\circ f_b$ for any real numbers $a,b$. Thus, $f_\lambda$ is the flow map of some vector field $x$ on $M/G$.

The vector field $x:M/G\rightarrow T(M/G)$ contains all of the "essential" information contained in the original vector field $X:M\rightarrow TM$. In fact, if it turns out to be possible to solve the ODE defined by $x$, then the solution to the original ODE can be explicitly reconstructed in terms of integrals of the solution to $x$ (I won't explain how this works here). Note that because $x$ is defined on a space with smaller dimension than $M$ (in fact $\text{dim}G$ dimensions smaller), it is reasonable to expect that $x$ is easier to solve than $X$.

Finally, I'd like to point out that symmetry is even more useful when your ODE is Hamiltonian. In the Hamiltonian case, it is often possible to reduce the dimensionality of the system by TWICE the dimension of $G$! This is because Hamiltonian symmetries also give rise to conserved momentum maps. An authoritative source for learning about Hamiltonian symmetries is Abraham and Marsden's Foundations of Mechanics.

 

[algebrat]

I believe symmetry can be use to explain many of the tricks from the chapter in Boyce on differential equations.

$(x,y)\mapsto(x,y/x)$, homogeneous change of variables.

So we're to notice the lie group of symmetries along any y=mx. If we travel from $(x,y)$ to $(x',y')$, but stay on the line, ie y'=mx', then if y'=F(y/x), a certain symmetry arises.

Since $dy/dx(x,y)=f(x,y=F(y/x)=F(m)=F(y'/x')=f(x',y')=dy/dx(x',y')$

So the theory of lie groups says the differential equation involving two variables splits into two of fewer variables if we use the new coordinates, instead of $(x,y)$, we sort of use $(x,\tan(\theta))$, so we move only radially and in $x$, and our shape simplifies.

Other changes in path due to some type of symmetry,

$(x,y)\mapsto(x,\mu(x)y)$, the integrating factor. Splits the differential equation for a certain $\mu(x)$.

$(x,y)\mapsto (x,x+y)$, if $y'=f(x,y)$ has symmetry $y'=F(x+y)$

It would be an exercise to see what corresponds with what from an advanced text. Singer has a couple good books on symmetry and mechanics that are hiding some intuitions too. I haven't visited this topic in a while, so I just remember the parts that were relevant to me more often. Picking symmetries in calculus of variations is another related topic, giving nice "first integrals", which I think relates to energy conservation. Singer's correspondence between symmetries and conserved quantities in her book on Hydrogen makes it seem really simple, but it takes some mathematical buildup. The main idea was something like if an operator commutes with a transformation.

I think these are all good things to try to take into account to create a nice picture of the topic.

 

[D H]

Can anybody care to give a geometrical interpretation? I can't understand a bit of what is going on here :(

This is obviously a late response, but I'll take a swing at answering this given that there's another thread on Lie groups and ODEs. What I have to say fits this somewhat old thread much better than the new one.

Let's start simple. The problem is to the numerical initial value problem of integrating the radius vector of a unit circle in ℝ² with respect to time given the angular velocity ω as a function of time. The obvious solution is to integrate angular velocity to yield angular displacement and use this to compute the unit vector.

Let's not do that. What happens if we instead numerically integrate the radius vector directly? There's a simple relation between the angular velocity ω and the time derivative of this unit vector. Keeping it simple, let's use good old Euler integration: $x(t+\Delta t) = x(t) + \Delta t\dot{x}(t)$. Yes, Euler is bad, but it's also the basis for many of the more advanced techniques. Bad as Euler integration is in general, it's even worse here. Our vector is no longer a unit vector. It can't be. The time derivative of any constant length vector is orthogonal to the vector being differentiated. The very first integration step takes us off the unit circle, and it goes downhill from there.

One way around this problem is to project the vector back onto the unit circle. This way we do have a unit vector, but there's still a big problem: It's not pointing in the right direction. One way around this new problem is to introduce a fudge factor in the step so that the projection takes us to exactly to that point where integrating angular velocity would have taken us. It's a fudge, it's ugly, it doesn't generalize.

So far I haven't addressed Lie groups at all. Let's add a second radius vector that's orthogonal to the first one to the things we want to integrate. Stack these together and you have a transformation matrix. Assuming the determinant is 1, this is SO(2). The Lie algebra of this group is the 2x2 skew symmetric matrices, with our angular velocity determining the off-diagonal elements. Now we can take advantage of the fact that integrating angular velocity is the right thing to do, but do so using the exponential map. The Lie group equivalent of $x(t+\Delta t) = x(t) + \Delta t\dot{x}(t)$ is $T(t+\Delta t) = \Delta t \exp(Sk(\omega))T(t)$. As opposed to the ugly kludge mentioned above, this is elegant, it stays true to the underlying mathematics, and it generalizes. It generalizes in many, many ways. It generalizes to any Lie group, and it generalizes beyond Euler integration.

A large body of work has been developed over the last twenty years on Lie group integrators. Their power arises from staying true to the mathematics. Here's a fairly recent and rather nice overview of that work: http://arxiv.org/abs/1207.0069. Note: Celledoni et al do not use my motivation above. My degree is in physics but I'm essentially an aerospace engineer. My example is far too "engineerish" for a mathematician. The papers on Lie group integrators tend to quickly dive deep into dense mathematical jargon. A background in Lie groups is helpful in reading them.