26.02.2018

Geodesic flows: notes 1

Geodesic flows, notes 1: symplectic geometry 

Our first topic is to understand geodesics, first defined as curves γ(t)γ(t) , in terms of a Hamilton flow in the cotangent bundle. Hamilton flows belong to the realm of symplectic geometry. This is an area of mathematics which comes up in a variety of contexts, such as:

1. Historically, symplectic geometry arose from the study of classical mechanics. Consider the NN -body problem, where NN celestial objects (planets, stars etc) interact with each other gravitationally. If the objects have mass mimi , position qi(t)R3qi(t)∈R3 , and momentum (or velocity) pi(t)=q˙i(t)R3pi(t)=q˙i(t)∈R3 , by Newton's laws one has the equations miq¨i(t)=jiGmimj|qiqj|2miq¨i(t)=∑j≠iGmimj|qi−qj|2 where GG is the gravitational constant. These can be rewritten as the Hamilton equations

q˙(t=pH(q(t),p(t)),
p˙(t=qH(q(t),p(t))


where HH is the Hamilton function H=T+UH=T+U , with TT the kinetic energy and UU the self-potential energy (source: Wikipedia). The underlying symplectic structure makes it possible to consider these equations in many different (symplectic) coordinate systems, while still retaining the basic properties.

2. The geodesic flow on a Riemannian manifold (M,g)(M,g) can be understood as a Hamilton flow on the cotangent space TMT∗M . This explains the facts that geodesics have constant speed, that geodesic flow preserves volume, and that the geodesic vector field on SMSM is divergence free and formally skew-adjoint.

3. If A(x,D)A(x,D) is a linear differential operator in an open set ΩRnΩ⊂Rn , so A(x,D)u=|α|maα(x)DαuA(x,D)u=∑|α|≤maα(x)Dαu where aαC(Ω)aα∈C∞(Ω) , the principal symbol of A(x,D)A(x,D) is

a(x,ξ)=|α|=maα(x)ξα,(x,ξ)Ω×Rn.a(x,ξ)=∑|α|=maα(x)ξα,(x,ξ)∈Ω×Rn.

It turns out that under a change of coordinates, the principal symbol of AA transforms like an invariant function on the cotangent bundle TΩT∗Ω . Moreover, if AA and BB are two differential operators, their commutator [A,B]=ABBA[A,B]=AB−BA has principal symbol {a,b}{a,b} where {,}{⋅,⋅} is the Poisson bracket on TΩT∗Ω :

{a,b}(x,ξ)=ξaxbxaξb.{a,b}(x,ξ)=∇ξa⋅∇xb−∇xa⋅∇ξb.

These basic facts indicate that symplectic geometry has a fundamental role in the study of linear partial differential equations.

4. The billiard map in the context of the broken geodesic ray transform preserves the related symplectic structure and hence volumes.

Symplectic manifolds

We proceed to explain some basic ideas in the setting of a general 2n2n -dimensional symplectic manifold.

Definition. A symplectic manifold is a pair (N,σ)(N,σ) where NN is an 2n2n -dimensional smooth manifold, and σσ is symplectic form, that is, a closed 22 -form on NN which is nondegenerate in the sense that for any ρNρ∈N , the map Iρ:TρNTρN,Iρ(s)=σ(s,)Iρ:TρN→Tρ∗N,Iρ(s)=σ(s,⋅) is bijective.

Example 1. The space R2nR2n has a standard symplectic structure given by the 22 -form σ=dx1dxn+1++dxndx2nσ=dx1∧dxn+1+…+dxn∧dx2n .

Example 2. More generally, if MM is an nn -dimensional CC∞ manifold, then N=TMN=T∗M becomes a symplectic manifold as follows: if π:TMMπ:T∗M→M is the natural projection, there is a 11 -form λλ on NN (called the Liouville form) defined by λρ=πρ,ρTMλρ=π∗ρ,ρ∈T∗M . Then σ=dλσ=dλ is a closed 22 -form. If xx are local coordinates on MM , and if (x,ξ)(x,ξ) are associated local coordinates (called canonical coordinates) on TMT∗M , then in these local coordinates

λσ=ξjdxj,=dξjdxj.λ=ξjdxj,σ=dξj∧dxj.

It follows that σσ is nondegenerate and hence a symplectic form.

It is a theorem of Darboux that if (N,σ)(N,σ) is a symplectic manifold, then near any point of NN there are local coordinates (x,ξ)(x,ξ) so that σ=dξjdxjσ=dξj∧dxj in these coordinates. Hence every symplectic manifold is locally symplectically equivalent to R2nR2n , and all obstructions to having a symplectic structure are global in nature: for instance if NN is a closed manifold having a symplectic form σσ , then the de Rham cohomology groups H2(N)H2(N) and H2n(N)H2n(N) are nontrivial (this follows since the nn -fold wedge product of σσ is non vanishing, more about this later).

Hamilton flows

Definition. Let (N,σ)(N,σ) be a symplectic manifold. Given any function fC(N)f∈C∞(N) , the Hamilton vector field of ff is the vector field HfHf on NN defined by

Hf=I1(df)Hf=I−1(df)

where dfdf is the exterior derivative of ff (a 11 -form on NN ), and II is the isomorphism TNTNTN→T∗N given by the nondegenerate 22 -form σσ .

Example 1. Let MM be an nn -manifold, and let σσ be the standard symplectic form on TMT∗M . If (x,ξ)(x,ξ) are canonical local coordinates, then (using that αβ(v,w)=α(v)β(w)α(w)β(v)α∧β(v,w)=α(v)β(w)−α(w)β(v) for 11 -forms α,βα,β )

Ix,ξ(sjxj+tjξj)(s~jxj+t~jξj)=(dξidxi)(sjxj+tjξj,s~kxk+t~kξk)=i=1n(tis~isit~i)Ix,ξ(sj∂xj+tj∂ξj)(s~j∂xj+t~j∂ξj)=(dξi∧dxi)(sj∂xj+tj∂ξj,s~k∂xk+t~k∂ξk)=∑i=1n(tis~i−sit~i)

which gives that

I(sjxj+tjξj)Hf=tjdxjsjdξj=ξjfxjxjfξj.I(sj∂xj+tj∂ξj)=tjdxj−sjdξjHf=∂ξjf∂xj−∂xjf∂ξj.

Example 2. In particular, in R2nR2n one has I(s,t)=(t,s)I(s,t)=(t,−s)s,tRns,t∈Rn , and

Hf=ξfxxfξ.Hf=∇ξf⋅∇x−∇xf⋅∇ξ.

Definition. Let (N,σ)(N,σ) be a symplectic manifold, and let fC(N)f∈C∞(N) . Denote by φtφt the flow on NN induced by HfHf , that is,

φt:ρ(0)ρ(t) where ρ˙(t)=Hf(ρ(t)).φt:ρ(0)↦ρ(t) where ρ˙(t)=Hf(ρ(t)).

We will later see that any Hamilton flow map is symplectic ((φt)σ=σ(φt)∗σ=σ ) and consequently volume-preserving.

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