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--- ebsd2021:tema2 [2021/06/28 16:10] – external edit 127.0.0.1
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-Tema 2
+**Tema 2: Geodesic flows in Hamiltonian dynamics**
+**Symplectic form and symplectic manifold**
+We start recalling some known facts on linear algebra.
+Let $V$ be a vector space.
+//Symplectic bilinear map//: We say that a map
+$\omega:V\times V\to\mathbb R$ is //symplectic// if it is:
+  * bilinear: $\omega$ is linear on each coordinate.
+  * antisymmetric: $\omega(x,y)=-\omega(y,x)$ for all $x,y\in V$.
+  * non-degenerate: the map $x\in V\mapsto \omega_x(\cdot)=\omega(x,\cdot)\in V^*$ is an isomorphism, i.e. $\omega(x,y)=0$ for all $y\in V$ iff $x=0$.
+In this case, we call $(V,\omega)$ a //symplectic vector space//.
+Symplectic bilinear forms have canonical representations:
+if $\omega$ is symplectic, then there is a basis
+$\{x_1,\ldots,x_n,y_1,\ldots,y_n\}$ s.t.
+$$
+\left\{
+\begin{array}{l}
+\omega(x_i,x_j)=\omega(y_i,y_j)=0\\
+\omega(x_i,y_j)=\delta_{i,j}
+\end{array}\right.
+,\ \ \forall i,j=1,\ldots,n.
+$$
+In particular, $V$ has even dimension, equal to $2n$. In this basis, $\omega$ takes the form
+$$
+\omega(x,y)=x^T
+\left[
+\begin{array}{ccc|ccc}
+&&&&&\hspace{.2cm}\\
+&0&&&{\rm Id}&\\
+&&&\hspace{.2cm}&&\\ \hline
+&&&&&\\
+&{\rm -Id}&&&0&\\
+&&&&&\\
+\end{array}\right]
+y.
+$$
+Here is the prototype of a symplectic vector space: given $(E,\langle\cdot,\cdot\rangle)$ a vector space
+with inner product, let $V=E\times E$ and define $\omega:V\times V\to \mathbb R$ by
+$$
+\omega((x_1,y_1),(x_2,y_2))= \langle y_1,x_2\rangle - \langle x_1,y_2\rangle.
+$$
+Then $(V,\omega)$ is a symplectic vector space. The literature usually considers $E=\mathbb R^n$
+for this example, in which case the symplectic bilinear map is defined in $\mathbb R^{2n}$.
+With respect to the symplectic structure, there are some special subspaces.
+//Isotropic and Lagrangian subspaces:// Let $(V,\omega)$ be a symplectic vector space. Given
+a subspace $Y\subset V$, define
+$$
+Y^\omega=\{x\in V:\omega(x,y)=0\text{ for all }y\in Y\}.
+$$
+It is easy to see that ${\rm dim}(Y)+{\rm dim}(Y^\omega)=2n$.
+We call $Y$:
+  * //isotropic// if $Y\subset Y^\omega$, i.e. if $\omega(x,y)=0$ for all $x,y\in Y$. Clearly ${\rm dim}(Y)\leq n$.
+  * //Lagrangian// if it is isotropic and ${\rm dim}(Y)=n$. Alternatively, $Y$ is Lagrangian iff $Y=Y^\omega$.
+The next definition introduces the symplectic structure on manifolds.
+Let $M$ be a $m$-dimensional differentiable manifold, and recall the definition of a symplectic
+form (which is the choice of symplectic bilinear maps on each tangent space of $M$).
+//Symplectic manifold:// We call $(M,\omega)$ a// symplectic manifold// if $M$ is a differentiable manifold and $\omega$ is a symplectic form on $M$.
+In particular, $m=2n$ is even. The simplest example of a symplectic manifold is $(\mathbb R^{2n},\omega)$ as defined above: if $(x_1,\ldots,x_n,$ $p_1,\ldots,p_n)$ denotes the coordinates of $\mathbb R^{2n}$, then
+$$
+\omega=dx_1\wedge dp_1+\cdots+dx_n\wedge dp_n.
+$$
+Compare this with Riemannian manifolds. A Riemannian manifold is a differentiable manifold
+equipped with a Riemannian metric. The difference between metrics and two-forms is that the
+first is symmetric, while the other is antisymmetric. This simple difference makes the two
+contexts extremely different.
+**$TM$ is a symplectic manifold**
+We already know that $(TM,\omega)$ is a symplectic manifold. Let us complete this discussion showing another definition of $\omega$.
+//1--form $\Theta$ on $TM$:// Define the 1--form $\Theta$ on $TM$ taking
+$\Theta_v:TTM_v\to\mathbb R$ by
+$$
+\Theta_v(x,y)=\langle x,v\rangle.
+$$
+Above, $TTM_v\cong TM_p\times TM_p$ via the horizontal and vertical coordinates.
+**Lemma.** We have $d\Theta=\omega$. In particular, $\omega$ is closed.
+**Hamiltonian vector fields**
+Let $(M,\omega)$ be a symplectic manifold, and let $H:M\to\mathbb R$ be a $C^1$ function.
+The derivative $dH$ is a 1--form.
+//Hamiltonian vector field:// The //Hamiltonian vector field// of $H$ is the unique
+vector field $X_H:M\to TM$ defined by
+$$
+\omega_p(\cdot,X_H(p))=dH_p(\cdot).
+$$
+The flow generated by $X_H$ is called the //Hamiltonian flow// of $H$.
+In other words, the contraction 1--form $\omega(\cdot,X_H)$ equals the derivative $dH$.
+Existence and uniqueness follow from the non-degeneracy of $\omega$.
+Let us express the Hamiltonian vector field in the prototypical example of $(\mathbb R^{2n},\omega)$.
+Let $H:\mathbb R^{2n}\to\mathbb R$ be a $C^1$ function.
+We will show that $X_H$ satisfies the //Hamilton equations//.
+For $(v,w)\in\mathbb R^{2n}$, we have
+$$
+dH(v,w)=\sum_{i=1}^n\tfrac{\partial H}{\partial x_i}v_i+\sum_{i=1}^n\tfrac{\partial H}{\partial p_i}w_i.
+$$
+On the other expression, if $X_H=(X,Y)=(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$ then
+$$
+\omega((v,w),X_H)=\sum_{i=1}^n w_iX_i-\sum_{i=1}^n v_iY_i.
+$$
+Comparing the last two equations, we conclude that $X_i=\tfrac{\partial H}{\partial p_i}$
+and $Y_i=-\tfrac{\partial H}{\partial x_i}$ for $i=1,\ldots,n$, and so
+the Hamilton equations
+$$
+\left\{
+\begin{array}{l}
+X=\tfrac{\partial H}{\partial p}\\
+\\
+Y =-\tfrac{\partial H}{\partial x}
+\end{array}
+\right.
+$$
+characterize $X_H$.
+The Halmiltonian vector field has some important invariance properties.
+Let $\phi=\{\phi^t\}$ be flow generated by $X_H$.
+**Theorem.** The following holds:
+(1) $H\circ\phi^t=H$ for all $t\in\mathbb R$, hence $H$ is constant along flow lines.
+(2) $\omega$ is preserved by $\phi$, i.e.
+$\omega(d\phi^t v,d\phi^t w)=\omega(v,w)$ for all $v,w\in TM$ with same basepoint
+and all $t\in\mathbb R$.
+(3) $\phi$ preserves the volume form $\omega^n$.
+**The geodesic flow is a Hamiltonian flow**
+We already know that $(TM,\omega)$ is a symplectic manifold.
+We claim that the geodesic flow $g:TM\to TM$ is the Hamiltonian flow
+of the function $G:TM\to\mathbb R$ defined by
+$$
+G(v)=\tfrac{1}{2}\langle v,v\rangle.
+$$
+To prove this, write $X_G=(X,Y)$. For $\xi=(x,y)\in TTM$, we have
+$$
+\omega(\xi,X_G)=\langle y,X\rangle-\langle x,Y\rangle.
+$$
+If $x,y\in TM_v$, let $Z:(-\varepsilon,\varepsilon)\to TM$ s.t. $Z(0)=v$ and
+$Z'(0)=(\dot{c}(0),\nabla_{c}Z(0))=\xi$ where $c=\pi\circ Z$. Then
+$$
+dG_v(\xi)=\tfrac{d}{dt}|_{t=0}(G\circ Z)=\tfrac{d}{dt}|_{t=0}\tfrac{1}{2}\langle Z(t),Z(t)\rangle=\langle Z,\nabla_{\dot{c}}Z\rangle(0)=
+\langle v,y\rangle=\langle y,v\rangle
+$$
+and so $(X,Y)=(v,0)$. This later vector is the generator of the geodesic flow, and so the assertion is proved.
 ~~DISCUSSIONS~~