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Classical Mechanics Lecture 1 Notes

The Susskind Coin

(Many of Professor Susskin's lecture series begin with, or shortly get down to, a discussion of discrete states which start with a system involving an abstract coin. Included here is a composite from several lectures with a little supplemental material and transitions.)

Consider a (mathematical) space in which in which every possible state of a system is represented by a point. (Configuration is a common synonym for state.)

Susskind's coin is an example of the simplest possible example of a system in such a space (leaving aside a one-sided coin). It is an imaginary (abstract) coin on the table. It is either heads (head side up) or tails (tail side up). This is the only fact about the coin that is relevant to anything.

This depicts the two sides of the actual imaginary coin:

{$$ \Huge \style{color:red} \bigodot_\text{heads} \; \style{color:blue} \bigoplus_\text{tails}$$}

There is just one coin, but it has two sides. The totally imaginary coin is not the state space. The state space is represented as {$ P $} below. ({$ P $} is for ''phase space.")

"Coin States"
Coin States

Now we consider a system with laws of motion over stroboscopic time. Stroboscopic time is just an expression for discrete time, or in other words, For two times {$ t_1, t_2,$}{$$ \left\vert t_1 - t_2 \right\vert = n, \text{ for } n \in \left\{0,1,2,\dots\right\}. $$}


p
Rule 1

One possible law of motion is if the coin is heads it goes to heads and if it is tails it goes to tails.

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:red} H \\ \style{color:blue} T \to \style{color:blue}T \end{matrix} \right.$$}

This is not especially interesting, but it does say something, namely that if we know the state of the coin at any time, we know its state for any time in the future. So if the initial state is {$ \style{color:red} H,$} the history of the world will be {$ \style{color:red} H, \style{color:red} H, \style{color:red} H, \dots,$} and if the initial state is {$ \style{color:blue} T,$} the history of the world will be {$ \style{color:blue} T, \style{color:blue} T, \style{color:blue} T, \dots.$} depending on the initial conditions.


p
Rule 2

Another possible law of motion is if the coin is heads it goes to tails and if it is tails it goes to heads.

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:blue} T \\ \style{color:blue} T \to \style{color:red}H \end{matrix} \right.$$}

Now the history of the world will be {$ \style{color:blue} T, \style{color:red} H, \style{color:blue} T, \style{color:red} H, \dots$} or {$ \style{color:red} H, \style{color:blue} T, \style{color:red} H, \style{color:blue} T, \dots,$}

Rule 1 and Rule 2 are two possible laws of motion for the coin system. They are alternatives, not simultaneously true.

They can be expressed mathematically. {$ \sigma $} is a variable name traditionally used for a two-valued system with typical values 1 and -1, so the rules may be summarized:

{$$ \large \bf \begin{align} \text{Rule 1: } \sigma_{t+1} &= \sigma_t \\ \bf \text{Rule 2: } \sigma_{t+1} &= -\sigma_t \end{align} $$}

In other words, Rule 1 says whichever state the coin is in at time t, it is in the same state at time t+1, and Rule 2 says whichever state the coin is in at time t, it is in the opposite state at time t+1.

These rules, or laws of motion in this world, have been deterministic. They make it possible to predict the state at any time in the future. If the state is known at one time, the state at the next time can be predicted. They are also retrodictive in that if you know what the state is a one time, you know what the state was at the previous time. If you know where you are, you know with certainty where you are going, and you know where you came from with certainty.


p
Bad Rule

Other possible rules can be imagined. One (illustrated as Bad Rule) is

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:blue} T \\ \style{color:blue} T \to \style{color:blue}T \end{matrix} \right.$$}

And another (not illustrated) is:

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:red} H \\ \style{color:blue} T \to \style{color:red} H \end{matrix} \right.$$}

Although these rules are deterministic, in that if you know the state at one time, you know the state in a future time, they are not retrospective. You cannot be sure what the state was at a previous time. These systems have lost information, and this violates the principle of conservation of information.

In sum, we had one object with two possible states. There were four possible rules (laws of motion) for the configuration, but two of them lost information which is not allowable.


Dice
Dice

Die Facts

  • Die is the singular of dice.
  • Opposite faces sum to 7.
  • Western dice are "right handed." If you curl your fingers from 1 toward 2, your thumb points in the direction of 3. Asian dice are left-handed.

Now, we consider the states of an abstract die. We were not flipping the coin, and we are not rolling the die. For present purposes, how the die changes states is a deep and perhaps unknowable mystery. We are just observing the die in discrete, stroboscopic flashes of time. So first we describe the states of the die in a state space. The numbers 1 through 6 are used to label the observable states. So far as the state space is concerned, these numbers are just labels and have no computational implications. They could be labeled Fire, Air, Earth, Water, Holy, and Dark. They do not have any particular order in state space.

"Die States"
Die States


"Die Rule 1"
Die Rule 1

Now we can consider the laws of motion for the states of the die. The simplest is that everything stays the same. Nothing happens. If it was state 5 to begin with, it is state 5 the next time and it was state 5 the previous time.
{$$ \left\{ \begin{matrix} 1 \to 1 \\ 2 \to 2 \\ 3 \to 3 \\ 4 \to 4 \\ 5 \to 5 \\ 6 \to 6 \end{matrix} \right. \tag{Die Rule 1}$$}


Usually my naming of the rules is completely arbitrary, but I have called this one Die Rule 1 because it has much in common with multiplying by 1. These states do not have values yet. Recall that the numbers 1 through 6 are just labels and do not necessarily have anything to do with whatever value we might assign or discover for the states. But if we had the value of the states at one time, we could discover the value of the states at the next time or the previous time by multiplying by 1. The state does not change, so its value does not change, and multiplying by 1 does not change the value. Moreover, it does not matter if we reverse the order of the arrows in the diagram or in the bracket definition of Dice Rule 1. However we look at it, it is still the same rule.


"Die States Hexagon"
Die Rule a

There are numerous possible rules (or laws of motion). One of them is to rotate around the states in the numeric order of their labels almost. The almost is because in the intergers, 1 does not follow 6, but in the rule state 1 does come after state 6.

{$$\left\{ \begin{matrix} 1 \to 2 \\ 2 \to 3 \\ 3 \to 4 \\ 4 \to 5 \\ 5 \to 6 \\ 6 \to 1 \end{matrix} \right. \tag{Die Rule a}$$}

It is important here to distinguish retrospection from reversal. Retrospection is looking back and reversal is going back. Retrospection is looking at the arrows backwards: if you are 1, you can see you came from 6, and see that to get to 6 you had to have been at 5 before that. Time reversal means turning the arrows around. The reversal of Dice Rule a is not the same rule as Dice Rule a. All that we are requiring of the laws of motion for now is that if you know where you are, you know where you are going and where you came from.

"Die Rule b"
Die Rule b

Die Rule b looks as if it ought to be very different from Die Rule a, but logically they are the same. This brings home the point that the numbers on the state diagram are just labels. The real imaginary observation we are making is the number of pips showing on the die. The pips on the die are the phenomena, and we have to answer to them. But there is nothing sacrosanct about the labels. We can rearrange the relation between the labels and the observed the values so that Die Rule b looks like Die Rule a.

We can do this by switching the labels 3 and 5 and then the labels 3 and 4.

If in observations the die really does show the values 1, then 2, then 5, then 3, then 4, then 6, and then continues to cycle through those values in the same order so that in Die Rule b the names of the states (1 through 6) really do correspond to the number of pips on the die, we can just use the label 5 for the observation that 4 pips are showing, the label 4 for the observation that 3 pips are showing, and the label 3 for the condition that 5 pips are showing, leaving labels 1, 2, and 6 to represent the observations of 1, 2, and 6 pips respectively, then we can display a phase space that looks exactly like Die Rule a. This has not changed the phenomena at all, but has only changed the way we label them.

dieLabel bLabel a
111
222
553
334
445
666

There are 6! (720) ways labels can be rearranged so that the state space diagrams look like Die Rule a. These do not change the phenomena or the laws of motion, they only change the labels. All these rules are perfectly good, although they are redundant. If you know the state at one time you know the next state uniquely and you know the previous state uniquely. Information is preserved. So there is no reason they would not be perfectly good laws of motion.

But there are other possible rules which are not equivalent to Dice Rule a.


"Die Rule c"
Die Rule c
"Die Rule d"
Die Rule d

Die Rule c and Die Rule d are not equivalent to Die Rule a, or to each other. No switching of labels can make them look like Die Rule a, but the point of them is that they are example of conservation laws.

A quantity that is preserved regardless of time is conserved. I have mentioned that a principle of physics is that information is conserved. Die Rule c and Die Rule d illustrate conservation laws. In both rules the quantity of being in the cycle 4 goes to 6, 6 goes to 5, 5 goes to 4 is preserved. So state 5 always goes to goes to 4 and always is preceded by state 6, but there is no way state 5 evolves to state 2 no matter how many steps ahead we look and never evolved from state 1 no matter how many steps we look back. Die rule a has two cycles and Die rule b has three cycles, and in both cases the quantity of belonging to a particular cycle is conserved.

"Conserve Evenness"
Conserving Evenness

There is no need to confine the consideration of state to cases in which there are a finite number of states.

There could be a infinite number of states. That does mean an infinite number of things, but could mean one thing capable of having an infinite number of states.

Consider an infinite line and a particle which can occupy any integer position on the line. There could be a law that whatever position the particle is in, it moves to the next position in the positive direction which could be expressed mathematically as {$ n \to n + 1 .$} In other words, if the particle is at n, it next goes to n+1. An obvious conserved quantity is being an integer, but this is not saying much because there are only integer positions in the state space. Position 1.5 just does not exist in the state space.

But instead the law of motion could be that a particle must move to the second next position, {$ n \to n + 2 .$} In this case, the conserved quantity is oddness or evenness. A particle that is in an even position must always go to an even position, and has always been in an even position. If a particle starts odd, it will always be odd, and always been odd.

Local Source

Classical Three Space

Classical Three-Space (also known as Euclidean space) is a way of describing positions in space quantitatively. For classical mechanics, space is considered as being three-dimensional, so a position in space can be identified by three real numbers, usually called {$ x, y, z$} or sometimes {$ x_1, x_2, x_3 .$}

Classical 3 Space
Classical 3 Space

The three numbers refer to distances along the axes of a Cartesian co-ordinate system, constructed with an arbitrary origin, or zero point, and three perpendicular axes.

Right-Hand Rule

Professor Susskind gives the pointy version of the right-hand rule. I prefer the curly finger version: if you curl the fingers of your right hand from the x-axis to the y-axis (that is, counter-clockwise, the same as the positive direction of angles), your thumb will be pointing in the direction of the z-axis. This is the same orientation as most standard screws: if you turn them counter-clockwise, they come out, which toward you.

Some 3D graphing calculators by default show the x-axis as coming toward the viewer, which means the z-axis is pointing up (and the y-axis points to the right). Professor Susskind draws 3D graphs with the z-axis coming out from the board, the y-axis up, and the x-axis to the right. These are completely compatible systems.

Beside an origin and the three axes, the co-ordinate system needs a unit of measure. For space the unit of measure is a measure of length such as meters, inches, miles, and so forth. Then every position in space can be represented by a distance on the x-axis, a distance on the y-axis, and a distance on the z-axis or {$ (x, y, z) .$}

Vectors

A vector is a magnitude with an associated direction. Vectors in general do not have to be tied to the origin, but a position can be represented by a vector with a tail on the origin.

Vectors are represented on the blackboard with a bar or arrow over them {$ \bar v, \overrightarrow v , $} and in print by boldface {$ \mathbf v.$} [More about vectors]

The length or magnitude of a vector is represented by enclosing it in bars or double bars: {$\rVert v \lVert, \rvert v \lvert. $} As might be expected of a length, this value is always a non-negative number (with length unit). The length is found by a three-dimensional extension of the Pythagorean theorem.

{$$ \Large \lVert {\bf v} \rVert = \sqrt{v_x^2 + v_y^2 + v_z^2} \tag{length of a vector}$$}

The components {$ v_x, v_y, v_x $} are not shown as vectors, because they are just numbers, lengths on their respective axes. Because they are just numbers, they can be squared in the ordinary way numbers are squared.

Operations on Vectors

A Vector can be multiplied by an ordinary number, called a scalar.

{$$ \Large k {\bf v} = (kv_x, kv_y, kv_z) \tag{scalar multiplication} $$}

If a vector is multiplied by two, the result is a vector that is twice as long, and if multiplied by three, the result is three times as long. When a vector is multiplied by a negative number, the direction of the vector is changed to the opposite direction, but scalar multiplication cannot change the direction of a vector in any other way.

Two vectors can be added to produce another vector.

Vector addition
Vector Addition: Parallelogram method and tail-to-head method

Geometrically, vectors can be added by placing their tails together and constructing a parallelogram, with the result being the diagonal of the parallelogram between them, or by placing the tail of one vector at the head of the other, with the result being the third side of the triangle thus formed from the tail of the first vector to the head of the second.

Algebraically,

{$$ \Large \mathbf a + \mathbf b = ( a_x+b_x, a_y+b_y, a_z+b_z) \tag{vector sum} $$}

It should be clear from the algebra that it does not matter which vector is taken first: {$ \mathbf a + \mathbf b = \mathbf b + \mathbf a ,$} and also that scalar multiplication is distributive over vector addition {$ k (\mathbf a + \mathbf b) = k\mathbf a + k\mathbf b.$}

The dot product of two vectors is the sum of the arithmetic product of the corresponding coefficients on the basis vectors.

{$$ \Large {\bf A} \cdot {\bf B} = A_x B_x + A_y B_y + A_z B_z \tag{Dot product}$$}

Through some algebra and the Law of Cosines, if {$ \theta $} is the angle between the vectors, the dot product can be shown to be:

{$$ \Large {\bf A} \cdot {\bf B} = \lVert {\bf A} \rVert \lVert {\bf B} \rVert \cos \theta \tag{Dot product} $$}

"Scalar Projection of A onto B"
Scalar projection of A on B

This amounts to the projection of {$ \mathbf A $} onto {$ \mathbf B $} multiplied by the length of {$ \mathbf B ,$} and the form should make clear that it also can be the projection of {$ \mathbf B $} onto {$ \mathbf A $} multiplied by the length of {$ \mathbf A .$}

An immediate and very useful result of this is that if the dot product is zero, the vectors are perpendicular (because the cosign of a right angle is zero).

The Radius Vector

The radius vector, traditionally represented as {$ r $} is the vector from the origin to a particle in space. Almost always, physics is interested in motion, which is how particles move over time, so most of the time {$ r $} is considered as a function of time, and its components are functions of time.

A common convention is that a dot over a function represents the first-order derivative of the function with respect to time and two dots represents the second-order derivative with respect to time.

{$$ \Large \begin{align} \mathbf r (t) &= \left( r_x (t), r_y (t). r_z (t) \right) \tag{position}\\ \mathbf v (t) = \frac{d}{dt} \mathbf r (t) = \mathbf{\dot r} &= \left( \dot r_x, \dot r_y. \dot r_z \right) \tag{velocity} \\ \mathbf a (t) = \mathbf{\ddot r} &= \left( \ddot r_x, \ddot r_y. \ddot r_z \right) \tag{acceleration} \end{align} $$}

Local Source

Circular Motion

Position

Consider a point on the unit circle whose position is given by the radius vector {$ \mathbf r .$} The angle {$ \mathbf r $} forms with the x-axis may be called {$ \theta .$}

{$ \mathbf r $} may be expressed as its x and y components {$ \mathbf r = (r_x,r_y) = (\cos \theta, \sin \theta) .$}

That this is on the unit circle can be verified because the length of {$ \mathbf r $} is 1, and its tail is on the origin. {$$ \lVert \mathbf r \rVert = \sqrt{r_x^2 + r_y^2} = \sqrt{\cos^2\theta + \sin^2\theta} = \sqrt{1} = 1$$}


Acceleration not to scale
Motion

Now, the position of a particle in circular motion is a function of time, and clearly the angle of the vector with the x-axis is a function of time. (By convention, counter-clockwise rotation is taken as positive.)

{$ \large \mathbf r(t) = (\cos \theta(t), \sin \theta(t) ) $}

{$ \omega $} is called the angular frequency. Its units are radians per second. This is the unit physics more commonly uses to express frequency than Hertz (Hz). Hz is cycles/second, so {$ \omega = 2\pi f. $} For {$ \theta $} expressed in radians (as it always is in physics) {$ \theta = \omega t .$}

{$ \large \mathbf r(t) = (\cos \omega t , \sin \omega t ) $}
{$ \large r_x(t) = \cos \omega t $}
{$ \large r_y(t) = \sin \omega t $}

Velocity

Velocity is the instantaneous change in position, so it is the derivative with respect to time of the position.

{$$ \large \dot r_x = \frac{d(\cos \omega t)}{dt} = -\omega\sin\omega t \\ \large \dot r_y = \frac{d(\sin \omega t)}{dt} = \omega\cos\omega t \\ \large \mathbf{v} = \mathbf{\dot r} = (-\omega\sin\omega t,\omega\cos\omega t) $$}

It is intuitively obvious that the velocity is perpendicular to the radius vector, but this can be verified by taking the dot product.

{$$ \large \begin{align} \mathbf{v} \cdot \mathbf{r} &= (-\omega\sin\omega t,\omega\cos\omega t) \cdot (\cos \omega t , \sin \omega t ) \cr &= -\omega\sin\omega t\cos \omega t + \omega\cos\omega t\sin \omega t \cr &= 0 \ \tag*{$ \blacksquare$} \end{align}$$}

The magnitude of the velocity:

{$$ \begin{align} \lVert \mathbf v \rVert &= \sqrt{v_x^2 + v_y^2} \\ &= \sqrt{(-\omega\sin\omega t)^2 + (\omega\cos\omega t)^2} \\ &= \sqrt{\omega^2\sin^2\omega t + \omega^2\cos^2\omega t}\\ &= \sqrt{\omega^2(\sin^2\omega t + \cos^2\omega t)} \\ &= \sqrt{\omega^2(1)} \\ &= \omega \end{align} $$}

Acceleration

Acceleration is the instantaneous change in velocity, so it is the derivative with respect to time of the velocity and the second derivative of position.

{$$ \large a_x = \dot v_x = \ddot r_x = \frac{d^2(\cos \omega t)}{dt^2} = \frac{d(-\omega\sin\omega t)}{dt} = -\omega^2\cos\omega t \\ a_y = \dot v_y = \ddot r_y = \frac{d^2(\sin \omega t)}{dt^2} = \frac{d(\omega\cos\omega t)}{dt} = -\omega^2\sin\omega t \\ \mathbf a = -\omega^2(\cos\omega t, \sin\omega t) = -\omega^2\mathbf r $$}

This last substitution saves the work of calculating the dot product and the magnitude. Since it has a negative sign, it is anti-parallel to the position vector and it has a magnitude of {$ \omega^2 $} because {$ -\omega^2 $} is just a scalar and we can observe its absolute value, which is just {$ \omega^2 .$}

Local Source


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