First look at Symplectic Topology

Brief Recap of Lagrangian and Hamiltonian Mechanics

Let a $2n+1$-variable function $L(t,x_n,v_n):\mathbb{R}^{2n+1}\rarr\mathbb{R}$ be given, such that:

• it’s twice continously differentiable, so partial derivatives like $\frac{\partial L}{\partial x_i}, \frac{\partial^2 L}{\partial v_i^2}$ are well defined.
• $t$ represent time.
• $x$ is a point on some space $S$, e.g. $x \in \mathbb{R}^n$
• $v_n \in T_x S$ is a possible derivative of $x$, along any possible curves $\tau$ that passes through $x$, i.e. $v_i=\frac{\partial x}{\partial \tau}$ for any curve $\tau$

We call this function the Lagrangian.

Most things we encounter in physics, their time-evolution is governed by a second-order differential equation.

Note that most systems we encounter in physics are first order Most systems in physics can be describe by some value, plus the derivate of that

So Lagrangian takes a configuration of some system and outputs a number.

Usually, $x_i$ are the value of positions of a system, and It might be non-intuitive to think about a function that takes derivate as an input.

, and following the rules of Lagrangian mechanics, it completely describes a system .

Lemma 1.1.1 A minimal path

LaTex test: inline Latex:

inline Latex 2: $E=mc^2$

Block latex

Block latex 2:

$$E=mc^2$$

Sources

McDuff, Dusa, and Dietmar Salamon, Introduction to Symplectic Topology, 3rd edn