Review of Classical Mechanics

施工中，不知道什么时候能写完。

这一部分主要为理论力学部分。然而半理半工的鼠鼠并没有学过，最后选择三小时速通最小作用量原理。这一部分原书就写的比较简略，可能会写一点个人浅薄的理解，当然和我一样没有学过这一部分想要速通的可以考虑考虑知乎。

2.1 The Principle of Least Action and Lagrangian Mechanics

mass ​m, potential ​V(x), along the ​x axis:

m\frac{d^2x}{dt^2}=-\frac{dV}{dx}
x_{\mathrm{cl}}\left(t_i+\Delta t\right)=x(t_i)+\dot{x}(t_i)\Delta t

we need to specify a unique ​x_{cl}(t)——Lagrangian approach.

confuguration space——Cartesian coordinates: ​m_j\frac{d^2x_j}{dt^2}=-\frac{\partial V}{\partial x_j}

Lagrangian function ​\mathscr{L}=T-V, notice that ​T is kinetic, ​V is potential energies, and it is velocity independent. Thus ​\mathscr{L}=\mathscr{L}(x,\dot{x},t).

For each path ​x(t) cormecting ​(x_i,t_i) and ​(x_f,t_f), the action ​S[x(t)]=\int^{t_f}_{t_i}\mathscr{L}(x,\dot{x})dt. The ​S here is functional.

principle of least action: The classical path is one on which ​S is a minimum.

Euler-Lagrange equation:

\{\frac{\partial\mathscr{L}}{\partial x(t)}-\frac{d}{dt}[\frac{\partial\mathscr{L}}{\partial\dot{x}(t)}]\}_{x_{cl(0)}}=0\quad\quad for\quad t_i\le t\le t_f

if we need into it ​\mathscr{L}=T=V, we will get Newton's Second Law.

cyclic coordinate: Suppose the Larangian depends on a certain velocity ​\dot{q_i} but not the corresponding coordinate ​q_i.

2.2 The Electomagnetic Lagrangian