Mechanics

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:

• $\gamma=\frac{1}{\sqrt{1-\beta^2}},\beta=\frac{u}{c}$
• 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
• $E^2=c^2p^2+m_0^2c^4$
• $E=mc^2$