Susskind Classical Mechanics Lecture 1

Source

• YouTube

Overview

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Consider:

• how surprising was this?
• possible links to other concepts
• what gaps in my knowledge does this expose?

Summary

Classical Mechanics | Lecture 1 - YouTube

0:02

Introduction to classical mechanics, including a reminder of basic vector arithmetic

0:24

Classical mechanics as a set of rules about allowable laws of motion.

• what are the specific laws of motion for a specific system?
• what are the rules for allowable laws in general?

7:24

Examples in a simplified world of discrete time steps (a stroboscopic world) using simple system consisting of a coin with two states of:

Initial Conditions - what is the starting state of the system? (e.g. H or T)

Laws of Motion - how do state change from one time interval to the next. e.g. for a coin that flips each iteration

given assigned state values of $\sigma =1$ (H) or $\sigma =-1$ (T), $\sigma (t+1)=-\sigma (t)$

State Space is the set of all possible states of the system

9:56

In a closed system, in classical mechanics, laws of motion are Deterministic

16:14

Examples in systems with more states (e.g. a die) of Conserved Quantity

e.g. if your die-based system has two closed cycles e.g. 1->2->3 and 4->6->5 which cannot interact, then if you were to label one of these $Q=0$ and the other $Q=1$, then $Q$ would be a Conserved Quantity of this system

22:19

Don't need to have finite number of states - e.g. infinite line of integer numbers, you can still have a law of motion creating one or more cycles

e.g. $\sigma (t+1)=\sigma (t)+2$ which gives you two cycles (odds and evens)

25:59

Example of unallowable law

H->T, T->E, E->T

This is unallowable because it is not Reversible - if you are at T you cannot tell how you got there.

32:29

Limits on Predictability

In the real world, which is continuous, you can never perfectly know the initial conditions.

41:49

Sidebar on co-ordinate systems (specifically 3D Cartesian)

43:31

Introduction to Vectors

Vector has length and direction

Notation - bar with arrow

Adding vectors

55:09

Discussion on vector dot product

$\overrightarrow{A}.\overrightarrow{B}$ as the component of $\overrightarrow{A}$ along the axis of $\overrightarrow{B}$ x the component of $\overrightarrow{B}$ along axis of $\overrightarrow{B}$:

$\overrightarrow{A}.\overrightarrow{B}=|A||B|cos\theta$

Therefore test for perpendicularity is $\overrightarrow{A}.\overrightarrow{B}=0$

component version

$\overrightarrow{A}.\overrightarrow{B}=A_x{B}_x+A_y{B}_y+A_z{B}_z$

1:02:43

Law of cosines

1:08:04

Using vectors to analyse position and velocity of a particle

Velocity is time derivative of position

$V_{xyz}=\frac{d{r}_{xyz}}{dt}$

1:12:06

Dot notation for time derivative

1:16:15

Example of uniformly-accelerated particle moving on a line with law of motion

$x(t)=a+bt+c{t}^2$

$v=\dot{x}=b+2ct$

$a=\ddot{x}=2c$

1:19:03

Example of particle moving around the unit circle

$\theta =\omega t$

$\frac{2\pi }{\omega }=$ period

$x(t)=\cos \omega t$

$y(t)=\sin \omega t$

$V_x=-\omega \sin \omega t$ $V_y=\omega \cos \omega t$

The velocity vector can be proved to be perpendicular to the position vector:

$\overrightarrow{V}\cdot \overrightarrow{r}=0$ :

$\overrightarrow{V}\cdot \overrightarrow{r}=r_x{V}_x+r_y{V}_y$

$=-(\omega \cos \omega t\sin \omega t)+(\omega \sin \omega t\cos \omega t)=0$

Acceleration $\overrightarrow{a}$ can be shown to be parallel but opposite to the position vector:

$a_x={\dot{V}}_x=-{\omega }^2\cos \omega t$

$a_y={\dot{V}}_y=-{\omega }^2\sin \omega t$

$\overrightarrow{a}=-{\omega }^2\overrightarrow{r}$

See also

• Basic Trigonometry