The Action is defined as $S(q(t)):=\int_{[t_{1},t_{2}]}L(q,q',t)dt$. For example, in absence of potential, the Lagrangian equals $L=\frac{1}{2}\cdot|\gamma|^{2}= \frac{1}{2} m(\dot{r}^{2} + r^{2} \dot{\phi}^{2} ).$

So the $\partial S=0\Rightarrow$EL equations are $\partial_{t\dot{\phi}}L=\partial_{\phi}L$ and  $\partial_{t\dot{r}}L=\partial_{r}L$. For our example,$m\cdot r^{2}\ddot{\phi} =0$ and $2m \ddot{r}=m\cdot r\dot{\phi}\Rightarrow$ with solutions $re^{i\phi}=(at+b)+i(ct+d).$

Intuitively: Action is the distance in configuration space like the absolute metric is the distance in Euclidean space; it gives the shortest path between states.

## Gelfand–Naimark theorem and Probability

summarize connections with probability

In random matrix theory

## Buffon’s noodle problem

I will summarize steps in

Proof of Buffon’s noodle problem

## Large deviations and entropy

Details on

1)relation of microstate and rate functions

2)Kullback–Leibler divergence