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Introduction

物理学

  • 物质
    • microscopic: elementary particles （大数粒子组成体系中个别粒子的行为）
    • mesoscopic(介观)
    • macroscopic：大数粒子组成的体系整体行为
    • cosmological
  • 运动：机械运动、热运动、微观粒子运动等（基于时间和空间）
  • 相互作用：由场传递，引力、弱相互作用、电磁相互作用、强相互作用
  • orders, symmetry, symmetry-breaking, conservation laws or invariance
  • 物理学发展历史
    • 19 世纪中叶前：实验科学
    • 20 世纪初：狭义和广义相对论，量子力学
    • 20 世纪中叶：实验+理论科学
    • 如今三足鼎立：实验物理、理论物理、计算物理
  • 基本理论
    • 牛顿力学/经典力学
    • 热力学
    • 电磁学
    • 相对论
    • 量子力学
  • 科学研究方法
    • 观测、实验、模拟得到事实和数据
    • 用已知原理分析事实和数据
    • （理论）形成假说和理论用以解释
    • （理论）预言新的事实和结果
    • （理论）用新的事例修改和更新理论
  • 基本量（操作性定义）：长度、质量、时间、电流、热力学温度、物质的量、发光强度
  • 非决定论性质
    • 微观客体量子力学不确定性关系
    • 多粒子系统中个别粒子统计不确定性
    • 非线性动力系统中的不可预言性

Kinematics

  • diplacement: $\Delta\vec r=\vec r(t+\Delta t)-r(\vec t)$
  • velocity: $\overline{\vec v}=\frac{\Delta\vec r}{\Delta t}$
  • speed: $\overline{v}=\frac{\Delta s}{\Delta t}$
  • acceleration: $\vec a=\frac{d\vec v}{dt}$
  • angular velocity vector: $\vec \omega =\frac{d\vec\theta}{dt} = \dot{\varphi}\vec k$
  • velocity in circular motion: $\dot{\vec \rho}=\vec \omega\times\vec\rho$
  • Oscillation: temporal periodicity and spatial repetitiveness
  • SHM: simple harmonic oscillation:
    • $x = A\sin\omega t+B\cos\omega t$
    • $x = A\cos(\omega t+\varphi)$
    • $x = Ae^{i\omega t}+Be^{-i\omega t}$
    • amplitude: A
    • phase: $\omega t+\varphi =\Phi$
    • initial phase: $\varphi$
  • sum up waves of same frequency: $A^2=A_1^2+A_2^2+2A_1A_2cos(\varphi_2-\varphi_1)$
    • constructive : $0,A=A_1+A_2$
    • destructive: $\pi,A=|A_1-A_2|$
    • quadrature: $\frac{\pi}{2},A=\sqrt{A_1^2+A_2^2}$
  • sum up waves of no initial phase
    • $A^2=A_1^2+A_2^2+2A_1A_2\cos(\omega_1-\omega_2)t$
    • waves of same amplitude: $2A\cos(\frac{\omega_1-\omega_2}{2}t)\cos(\frac{\omega_1+\omega_2}{2}t)$
      • modulating factor: $cos(\frac{\omega_1-\omega_2}{2}t)$
      • beat
  • phase diagram: trajectories of phase space(or in state space)
    • center, node, asddle, spiral
  • Galilean transformation: $\dot{\vec r}’=\dot{\vec r} - \vec u$
  • Coriolis acceleration: $2\vec\omega\times\frac{d\vec r}{dt’}$
    • $\frac{d}{dt}=\frac{d}{dt’}+\vec\omega\times$
    • $\frac{d\dot{\vec r}}{dt} = \frac{\dot{d\vec r}}{dt’} + \omega\times\dot{r}=\frac{d^2\vec r}{dt’^2} + 2\vec\omega\times\frac{d\vec r}{dt’}+\vec\omega\times(\vec\omega\times\vec r)$

Particle Dynamics

  • Newtow’s second law: $\vec F=m\vec a=\frac{d\vec p}{dt}$
  • $\vec F_{AB}=\vec F_{BA}$
  • Gravitation: $\vec F_{21}=-G\frac{m_1m_2}{r_{12}^3}\vec r_{12}$
  • Elastic (restoring) force: $\vec F=-k\vec x$
  • Intermolecular force: $\vec F\sim 2(\frac{\sigma}{r})^{13}-(\frac{\sigma}{r})^7$
  • static friction: $F_f\leq\mu_s F_N$
  • sliding friction: $F_f=\mu_k F_N$ (Amonton-Coulomb law)
  • Frictional drag: $F_d=\frac{1}{2}C_dA\rho v^2$
  • viscosity force: $F_\eta=6\pi\eta rv$ (Stoke’s law)
  • Reynolds number: $Re=\frac{\rho vd}{\eta}$, Stoke’s law holds when Re=0~10
  • noninertial frame:
    • inertial force: $\vec F_{in}=-m\ddot{\vec r}$
    • $(\vec F+\vec F_{in})=m\vec{a}'$
  • inertial centrifugal force:
    • $\vec F=-mr\dot{\varphi}\vec e_\rho$
  • Linear momentum: $\vec p=m\vec v$
  • angular momentum: $\vec L=\vec r\times\vec p$
  • torque: $\vec M=\vec r\times\vec F=\frac{d\vec L}{dt}$
  • Work: $dW=d\vec F\cdot d\vec r$
  • power: $P=\frac{dW}{dt}=\vec F\cdot\vec v$
  • kinetic energy: $T=\frac{1}{2}mv^2$
  • work-energy relation: $W=T_f-T_i$
  • Conservation of mechanical energy
  • equilibrium: stable, unstable, neutral

Gravitation

  • Kepler’s Law:
    • The orbit of each planet is an ellipse with the Sun at one focus
    • The line joining any planet and the Sun sweeps out areas in equal ties
    • The square of the period of revolution of a planet is proportional to the cube of the planet’s mean distance of the Sun
  • Gravitation: $\vec F_{21}=-G\frac{m_1m_2}{r_{12}^3}\vec r_{12}$
  • potential energy: $U=-G\frac{mm’}{r}+U_0$
  • accretion: $\Delta E=G\frac{m’m}{R}$

Dynamics of Many-Partical System

  • COM(center of mass): $\vec r_C=\frac{1}{m_C}\sum_{i}m_ir_i$
  • C-frame:
    • $\vec p’=0$
    • $\frac{d}{dt}\vec L=\vec M_{ext}$
  • system of variable mass: $m\frac{d\vec v}{dt}=\vec F_{ext}+(\vec u+\vec v)\frac{dm}{dt}$
  • collisions
    • head-on
  • equation of continuity: $\rho Av=Constant$
  • Bernoulli’s equation: $p+\frac{1}{2}\rho v^2+\rho gz=Constant$

Dynamics of a Rigid Body

  • principal axis of inertia: $\vec L=\vec I\omega$
  • rotational inertial: $I=\int\rho^2dm$
  • parrallel axis theorem: $I=I_C+md^2$
  • perpendicular axis theorem: $I_z=I_x+I_y$
  • kinetics: $T=\frac{1}{2}I\omega^2$
  • power: $P=\vec M\cdot \vec \omega$
  • precession: $\Omega=\lim_{\Delta t\rightarrow0}\frac{\Delta\phi}{\Delta t}=\frac{mgr}{I\omega}$

Oscillation

  • Hooke’s law: $F=-kx$
  • Equation of motion: $\ddot{x}+\omega_0^2x=0$, $\omega_0=\frac{k}{m}$
  • Energy: $E=\frac{1}{2}m\dot{x}_0^2+\frac{1}m\omega_0^2x_0^2$
  • Damped oscillation
    • $\ddot{x}+\frac{\eta}{m}\dot{x}+\omega_0^2x=0$
    • $x=Ae^{-\gamma t}cos(\omega t+\varphi)$
    • $\gamma=\frac{\eta}{2m},\omega^2=\omega_0^2-\gamma^2$
    • critical damping: $\eta=2m\omega_0$
    • $\langle E\rangle=\frac{1}{2}mA^2\omega_0^2e^{-2\gamma t}$
    • quality factor: $Q=\frac{m\omega}{\eta}$
  • Nonlinear oscillation:
    • hard-spring
      • $F=-(1+Bx^2)kx$
      • $\ddot{x}+\alpha x+\beta x^3=0$
    • soft-spring
      • $F=-(1+Bx)kx$
    • attractor
      • equilibrium point
      • periodic otionor a limit cycle $\Gamma$
      • quasi-periodic motion
  • Forced osillation friction
    • $\ddot{x}+2\gamma\dot{x}+\omega_0^2x=\frac{F_0}{m}cos\omega t$

Waves

  • Wave function: $\frac{\partial^2u}{\partial x^2}-\frac{1}{v^2}\frac{\partial^2u}{\partial t^2}=0$
  • solution form:
    • $u(x,t)=Acos(kx\mp\omega t-\varphi)$
    • $u(x,t)=Re(Ae^{i(kx-\omega t)})$
    • $\frac{\omega}{k}=v$
    • Amplitude: $A$
    • frequency: $f=\frac{\omega}{2\pi}$
    • phase: $\Phi=kx-\omega t-\varphi$
    • wave vector: $\vec k$
    • wave vector: $k$
    • phase speed: $v_p=\frac{\omega}{k}$
    • wavelength: $\lambda=\frac{2\pi}{k}=Tv_p$
    • wavefront: a surface with constant phase
  • AM wave(amplitude-modulated wave)
    • $u_1=Acos((k+\Delta k)x-(\omega+\Delta\omega)t)$
    • $u_2=Acos((k-\Delta k)x-(\omega -\Delta\omega)t)$
    • $u_1+u_2=2Acos(\Delta kx-\Delta\omega t)cos(kx-\omega t)$
    • group velocity: $v_g=\frac{d\omega}{dk}$
  • standing wave
    • $u_1=Acos(kx-\omega t-\phi)$
    • $u_2=Acos(kx+\omega t)$
    • $u_1+u_2=2Acos(kx-\frac{\phi}{2})cos(\omega t+\frac{\phi}{2})$
    • half-wavelength loss
    • fundamental: $f_1=frac{v}{2L}$
    • harmonics:$f_n=nf_1$
  • interference
    • $u_1=cos(kx-\omega t-\phi)$
    • $u_2=cos(kx-\omega t)$
    • $u_1+u_2=2Acos\frac{\phi}{2}cos(kx-\omega t-\frac{\phi}{2})$
    • $cos^2\frac{\phi}{2} = 1,\phi=m\pi,\Delta m\lambda$
    • $cos^2\frac{\phi}{2} = 0,\phi=(2m+1)\pi,\Delta (m+\frac{1}{2})\lambda$
  • Huygen’s principle
  • diffraction
  • Doppler effect

Relativistic Mechanics

  • Sepcial theory of relativity
    • The principle of relativity
    • The principle of the constancy of the speed of light
  • Lorentz transformation:

$$ \left[\begin{matrix} x’\ y’\ z’\ ict' \end{matrix}\right] = \left[\begin{matrix} \lambda & 0 & 0 &i\beta\gamma\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ -i\beta\gamma & 0 & 0 & \gamma \end{matrix}\right]\left[\begin{matrix} x\ y\ z\ ict\end {matrix}\right] $$

  • $\gamma=\frac{1}{\sqrt{1-\beta^2}},\beta=\frac{u}{c}$
  • spacetime interval of event pair: $(\Delta s)^2=(c\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2$
    • timelike: $(c\Delta)^2>(\Delta x)^2$, $\Delta\tau=\sqrt{(\Delta t)^2-(\frac{\Delta x}{c})^2}$
    • spacelike: $\Delta\sigma=\sqrt{(\Delta x)^2-(c\Delta t)^2}$
    • lightlike: $(c\Delta)^2=(\Delta x)^2$
  • time dilation: $\Delta t=\gamma\Delta t'$
  • Lorentz contraction
  • velocity transformation
  • $E^2=c^2p^2+m_0^2c^4$
  • $E=mc^2$