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pages/Math401/Math401_T5.md

@@ -12,19 +12,15 @@ The theory of dynamics is the study of properties of orbits.
 
 #### Definition of measure-preserving map
 
-Let $P$ be a probability measure on a $\sigma$-algebra $\mathscr{F}$ of subsets of $\Omega$. A measurable transformation $T:\Omega\to\Omega$ is said to be measure-preserving if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is:
+Let $P$ be a probability measure on a $\sigma$-algebra $\mathscr{F}$ of subsets of $\Omega$. (that is, $P:\mathscr{F}\to$ anything) A measurable transformation $T:\Omega\to\Omega$ is said to be measure-preserving if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is:
 
 $$
 \int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
 $$
 
-#### Definition of ergodic map
+Example:
 
-A measurable transformation $T:\Omega\to\Omega$ is said to be ergodic if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is:
-
-$$
-\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
-$$
+The doubling map $T:\Omega\to\Omega$ is defined as $T(x)=2x\mod 1$, is a Lebesgue measure preserving map on $\Omega=[0,1]$.
 
 #### Definition of isometry
 
@@ -34,4 +30,32 @@ The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure
 
 The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure preserving map defined on $\mathscr{H}=L^2(\Omega,\mathscr{F},P)$ is unitary of $\mathscr{H}$ if $U$ is an isometry and $T$ is invertible with measurable inverse.
 
-## Section 2: Continuous time (classical) dynamical systems
+## Section 2: Continuous time (classical) dynamical systems
+
+### Spring-mass system
+
+![Spring-mass system](https://notenextra.com/Math401/Spring-mass_system.png)
+
+The pure state of the system is given by the position and velocity of the mass. $(x,v)$ is a point in $\mathbb{R}^2$. $\mathbb{R}^2$ is the state space of the system. (or phase space)
+
+The motion of the system in its state space is a closed curve.
+
+$$
+\Phi_t(x,v)=\left(\cos(\omega t)x-\frac{1}{\omega}\sin(\omega t)v, \cos(\omega t)v-\omega\sin(\omega t)x\right)
+$$
+
+Such system with closed curve is called **integrable system**. Where the doubling map produces orbits having distinct dynamical properties (**chaotic system**).
+
+> Note, some section is intentionally ignored here. They are about in the setting of operators on Hilbert spaces, the evolution of (classical, non-dissipative e.g. linear spring-mass system) system, is implemented by a one-parameter group of unitary operators.
+>
+> The detailed construction is omitted here.
+
+#### Definition of Hermitian operator
+
+A linear operator $A$ on a Hilbert space $\mathscr{H}$ is said to be Hermitian if $\forall \psi,\phi\in$ **domain of $A$**, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$.
+
+It is skew-Hermitian if $\langle A\psi,\phi\rangle=-\langle\psi,A\phi\rangle$.
+
+
+
+