基础物理学

Mechanics and Introduction to Spacial Relativity

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Outline of the Course - Newtom’s Formalism

1. Basic Math: Calculus - differential method, vector - Linaer Algebra
2. Newton’s Law (Equation of Motion)
3. Conservation: Momentum, Energy, Angular Momentum (Rotation)
4. Non-inertial
5. Two important Models: Central field and Oscllation
6. Wave - Electro-megnetic (Light): Description of Wave, Interference of Wave (Young’s)
7. Special Relativity: Kinematics (Event $(x,t)$ Transform), Dynamics(Momentum and Energy)

Mechanics

Q&A:

1. What is the simplest question trying to solve? Motion of an object.
2. The simplest object? Point (Mass - Intuitive).
3. Motion? Change of the position over time. Trajectiory $x(t),y(t),z(t)$
   1. Coordinate System: Origin + Base vectors (change of coordinate sys)
   2. Reference Frame
   3. Other Physical Properties:

      $v_x = \dot{x}$

      $a_x = \ddot{x}$

      $\vec{p_x} = M\dot{x}$

      $K=\frac{1}{2}m|\vec{v}|^2$

      $\vec{L}=\vec{r}\times\vec{p}$

• Newton’s Differential Thinking (1-Dimensional)
  Want to predict x(t) at same later, what are the initial information needed?
  • Initial position $x_0$
  • Initial velocity $\dot{x}_0$
    $t_{1}=\Delta t$ can be predicted.
    $t_2=2\Delta t$?
  • Initial acceleration $\ddot{x}_0$
    $3\Delta t$? Any $\Delta t$?$\frac{d^{n}x}{dt^{n}}$?
  • Force provided $F$ - Power of Newton’s Law of Motion: $F=m\ddot{x}$
  • State of Motion: $(x,\dot{x})$
• Fundamental Symmetries in Classical Mechanics
  • Homogeneous and Isotropic of Space and Time (We have to believe in them)
    • Homogeneous of space: Translational Invarience (Symmetry) $\rightarrow$ Freedom choice of origin
    • Homogeneous of time: Translational Symmetry of time
    • Isotropic of space: Rotational Symmetry about space $\rightarrow$ Freedom choice of Base
    • Isotropic in time: Time Reversal Symmetry, Physical laws in particle microscopic scale are time-reversal(But of course, a broken glass seemed to be not time-reversal. It’s about Probability but not about capability)
  • Relativity Principle: Physical laws are same to all inertial frame.(No absolute Motion of ref. frame can e detected by doing experiments in your frame)

  Understand Basic Force (Gravity, Col.) In Physics
  $(x_1, \dot{x}_1),(x_2, \dot{x}_2) \implies F = f(x_1, x_2, \dot{x}_1, \dot{x}_2, t)$
  $F=f(x_1+a,x_2+a)=f(x_1,x_2)=f(x_1-x_2)$
  $F=f(x_1,x_2,t)=f(x_1,x_2)$
  Force is an interaction that time-delayed.
  Newton’s Day: No delay, Non-local
  Our view: time delay,local, $\tau = \frac{L}{c}$
  $F_1=f(x_2'-x_1)$
  $x_2==x_2'+v_2\tau$
  $F = f(x_1-x_2-v_2\tau)$
  Classically Newton thinks $\tau\to0$, force like $F=q\vec{v}\times\vec{B}$ depends on $v$.

  • Realistic System: Superposition Principle.
• Newton’s Mechanics has limitations (incomplete)
  • not too fast, $\beta=\frac{v}{c}<<1$
  • not too small, $\hbar=\frac{h}{2\pi}\to0$
  • not too heavy, $m<<M$
• Reason Stdying Newton
  • simple
  • Correspondence Principle - a test to new theories

1-D Kinematics

• Math: Single Variable Calculus.
  1. Derivative:

     1. $f(x)=\frac{dF}{dx}$

     2. Taylor Expansion
  2. Integration

2-D Kinematics - Vector(Linear Algebra)

1. vector(Geometry): an arrow with a magnitude(Norm, Module,length…) and a direction

• $\hat{a}=\frac{\vec{A}}{|\vec{A}|}$
• in order to define a vector, we need to define the summation rule: Parallelograph

2. analytical expression of a vector in base

$\vec{A}=A_{x}\hat{i}+A_{y}\hat{j}=\begin{bmatrix}A_{x}\\A_{y}\end{bmatrix}$

3. Dot Product & Cross Product

• Wave Function:

  $\phi=e^{ikx-\omega t}$

  General plane wave:

  $\varphi=e^{i\vec{k}\cdot\vec{r}-\omega t}$

• Finding projection

4. Transformation of vector’s expression