Understanding Nonlinear Exponential Maps for Vector Fields

[geoduck]

I'm having trouble understanding the exponential map for nonlinear vector fields.

If dσ/dt=X(σ)

for vector field X, then how does one interpret the solution:

σ(t)=exp[tX]σ(0) ?

If X is nonlinear, then X is not a matrix, so this expression wouldn't make sense.

If X is a matrix that maps:

X: point on manifold → vector (direction of flow) on manifold

then this expression makes sense.

 

[quasar987]

what's a nonlinear vector field? Plz define it.

 

[geoduck]

what's a nonlinear vector field? Plz define it.

A linear vector field has the property that X(p1+p2)=X(p1) +X(p2), where p are points in your space. In 2 dim, they look like

X=(ax+by,cx+dy)

for constant a b c d.

A nonlinear field is

X=X(f(x,y),g(x,y))

 

[quasar987]

I think I know what is confusing you. You are basically asking "what does exp(tX) means when X is not a matrix ?!?". The answer is that in the context of flows, exp(tX) is just a notation for the solution of the ODE dσ/dt=X(σ). The reason for this strange notation is tha the stolution of this equation has the "exponential property": exp([s+t]X) = exp(sX)exp(tX).

 

[geoduck]

I think I know what is confusing you. You are basically asking "what does exp(tX) means when X is not a matrix ?!?". The answer is that in the context of flows, exp(tX) is just a notation for the solution of the ODE dσ/dt=X(σ). The reason for this strange notation is tha the stolution of this equation has the "exponential property": exp([s+t]X) = exp(sX)exp(tX).

I was looking at some online notes, and they explained it in terms of linear vector fields so that it's not just notationally true, but literally true (there are a few typos, but the first two pages has it):

http://mysite.science.uottawa.ca/rossmann//Lie_book_files/Section 1-1.pdf

But in textbooks the exponential map is applied to any flow, not just linear ones.

So it seems you can define an exponential map for a lot of things...things that obey the additive group for example, or just a Lie group in general if you expand the "exponential property" via Baker-Campbell Hausdorff.

 

[quasar987]

In Lie groups, the exponential map appears also just as a flow, but we are only looking at exp(tX) for X a so-called "left-invariant" vector fields on G.

There is also a notion of exponential map in riemannian geometry which is similar in a way to the exponential map in Lie group theory but is not the flow of any vector field on M.