User will merry - MathOverflow

Answer by Will Merry for Splitting of the double tangent bundle into vertical and horizontal parts, and defining partial derivatives

2011-04-06T08:36:36Z

Let us write the isomorphism

$T_{(x,v)}TM = H_{(x,v)}TM \oplus V_{(x,v)})TM \cong T_xM \oplus T_xM$

$\xi \simeq (\xi^h,\xi^v)$,

so that $\xi^h \in T_xM$ and $\xi^v \in T_xM$.

Here the identification $H_{(x,v)}TM \cong T_xM$ is given by the restriction of $d_{(x,v)}\pi$ to $H_{(x,v)}TM$ (where $\pi:TM \rightarrow M$ is the projection), and the isomorphism $V_{(x,v)}TM \cong T_xM$ is canonical.

The key point is that under these identifications, if $z$ is a curve on $TM$, say $z(t)=(\gamma(t),u(t))$ then

$\dot{z}(0) \simeq (\dot{\gamma}(0),(\nabla_tu)(0))$.

So suppose $x \in M$ and $v,w,y \in T_xM$. Let $\gamma$ be a curve in $M$ such that $\gamma(0)=x$ and $\dot{\gamma}(0)=w$, and let $u$ be a vector field along $\gamma$ such that $u(0)=v$ and $(\nabla_tu)(0)=y$. Let $z(t)=(\gamma(t),u(t))$.

Think of $A$ as a map $TM \rightarrow TM$, so that the differential $dA$ is a map

$d_{(x,v)}A:T_{(x,v)}TM \rightarrow T_{(x,v)}TM$.

Then given $w\in T_xM$, if $\xi_w$ is the unique vector whose horizontal component is $w$ and whose vertical component is zero (i.e. $\xi^h = w$ and $\xi^v = 0$), then we define

$(\nabla_xA)(x,v)(w):=(d_{(x,v)}A(\xi_w))^v$,

and similarly if $\zeta_w$ is the unique vector whose horizontal component is zero and whose vertical component is $w$ (i.e. $\zeta^h = 0$ and $\zeta^v = w$), then we define

$(\nabla_vA)(x,v)(y):=(d_{(x,v)}A(\zeta_w))^v$.

Then it follows that

$d_{(x,v)}A(\xi) \simeq ((\nabla_xA)(x,v)(w),(\nabla_vA)(x,v)(y))$,

and these two maps have the properties you're looking for.

Answer by Will Merry for A $C^2$ small autonomous Hamiltonian has only constant 1-periodic orbits

2011-02-15T19:28:14Z

See Hofer and Zehnder's book "Symplectic invariants and Hamiltonian dynamics" p184-185 for a proof.

Answer by Will Merry for Books you would like to read (if somebody would just write them...)

2011-01-24T11:00:38Z

I would have killed for this a couple of years ago: a big book on Floer homology, written to be understandable for graduate students. Includes all the analytical details.

Answer by Will Merry for Why does the group act on the right on the principal bundle?

2010-12-27T18:28:40Z

Recall one way of defining a fibre bundle is a quintuple $(p,E,B,F,G)$, where $E$ is the total space, $B$ is the base, $p:E \rightarrow B$ is the projection, $F$ is the typical fibre, and $G$ is the structure group - that is, $G$ is a Lie group which acts on $F$ on the left.

Then a principal bundle is simply the special case where $G=F$ and the action is given by left translation. It is then an (easy) lemma that such a bundle $(p,E,B,G,G)$ carries a right action of $G$ on the total space $E$.

Answer by Will Merry for Book on Symplectic Geometry

2010-09-17T07:57:24Z

My favourite book on symplectic geometry is "Symplectic Invariants and Hamiltonian Dynamics" by Hofer and Zehnder. It's wonderfully written. Another lovely book (which has just been reissued as an AMS Chelsea text) is Abraham and Marsden's book "Foundations of Mechanics" which covers a lot of symplectic geometry as well as so much more...

Answer by Will Merry for When is a symplectic manifold equivalent to a cotangent bundle?

2010-08-19T09:05:00Z

In response to your last paragraph, the so-called "twisted" cotangent bundles provide examples where different symplectic forms exhibit very different dynamics with the same Hamiltonian.

Suppose $\omega=d\alpha$ is the standard symplectic form on a cotangent bundle $\pi:T^{*}X\to X$, where $X$ is a closed manifold. Let $\sigma$ denote a closed non-exact two-form on $X$, and consider a new family of two-forms $\omega_{s}$ for $s\in [0,\infty)$ defined by $\omega_{s}:=\omega-s\pi^{*}\sigma$. It's easily checked that $\omega_{s}$ is again a symplectic form on $T^{*}X$ for each $s\in [0,\infty)$ (it's closed as $\sigma$ is closed and non-degenerate as $d\pi$ vanishes on "vertical" tangent vectors).

Fix a Riemannian metric $g$ on $X$, and let $H:T^{*}X\to X$ denote the standard "kinetic energy" Hamiltonian defined by $H(x,p):=\frac{1}{2}|p|^{2}$, and let $\xi_{s}$ denote the symplectic gradient of $H$ with respect to $\omega_{s}$ (i.e. $i_{\xi_{s}}\omega_{s}=-dH$). Let $\phi_{s}$ denote the flow of $\xi_{s}$.

Let $S^{*}X$ denote the unit cosphere bundle of $X$. Since $H$ is autonomous, the flow $\phi_{s}$ preserves $S^*{X}$ for each $s\in[0,\infty)$. The point is that the dynamics of $\phi_{s}$ on $S^{*}X$ can vary dramatically depending on $s$.

As a concrete example of this, consider a closed hyperbolic surface $X=\mathbb{H}^{2}/\Gamma$, where $\Gamma$ is a cocompact lattice of $\mathrm{PSL}(2,\mathbb{R})$. Let $\sigma$ denote the area form on $X$. Note that for $s=0$, $\phi_{0}$ is just the cogeodesic flow. For $0\le s<1$, the dynamics of $\phi_{s}$ is Anosov and conjugate (after rescaling) to the cogeodesic flow. All closed orbits are non-contractible. In this case the unit cosphere bundle is a contact type hypersurface in the symplectic manifold $(T^{*}X,\omega_{s})$. For $s=1$ we get the horocycle flow. There are no closed orbits at all, and the unit cosphere bundle is not of contact type (in fact, it's not even stable). For $s>1$ all the orbits are closed and contractible. The unit cosphere bundle is again of contact type, but with the opposite orientation.

Perhaps the best place to read about this is Ginzburg's survey article "On closed trajectories of a charge in a magnetic field: An application of symplectic geometry", which is in the book "Contact and symplectic geometry" (CUP,1994). The recent paper "Symplectic topology of Mane's critical values" by Cieliebak, Frauenfelder and Paternain contains lots of examples of this sort of behaviour.

Comment by Will Merry

2011-10-22T09:52:06Z

Let me add two more references - Appendix C of McDuff and Salamon's big book explains why the index of the operator on the disc obtained by capping off the Hamiltonian orbit end of the half cylinder is given by the Maslov index of the Lagrangian loop $u^*T\Lambda$ (wrt the chosen trivialization) - and Matthias Schwarz' thesis has a very detailed proof (see Section 3.3) as to why the index of the operator on the cap is given by the CZ index. Since the index is additive, as Sam says the index on the half cylinder is [index of capped cylinder] - [index of cap], which gives the required answer.

Comment by Will Merry

2011-10-15T10:30:14Z

This question is answered here: math.stackexchange.com/questions/45923/…

Comment by Will Merry

2011-05-24T08:23:36Z

This question was actually asked on my behalf, so thanks Tim for your answer (and apologies for the delay). Let me also add an answer of my own: in the non-compact case (which is what we were primarily interested in), the existence of such length bounds seems to be essentially equivalent to the existence of $L^{\infty}$ bounds on gradient flows lines. One direction uses Tim's answer - as once $L^{\infty}$ bounds are established one can invoke Gromov compactness as in the closed case. The converse can be made explicit at least in the case of quadratic Hamiltonians on cotangent bundles, say.

Comment by Will Merry

2011-04-18T15:18:07Z

All the prime numbers less than or equal to 2 are even, and all the prime numbers greater than or equal to 3 are odd :)

Comment by Will Merry

2011-01-24T14:32:00Z

I should say that very recently such a book has been written by Audin and Damian - "Théorie de Morse et homologie de Floer", which is a beautiful and comprehensive introduction to the easiest parts of Floer homology. My only complaint with this book is that it doesn't go quite far enough - I guess I'm thinking more of a book the size of McDuff and Salamon's wonderful "J-holomorphic curves and symplectic topology" - but written specifically for Floer theory.