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csThe Nature of Classical PhysicsDeterminism, reversibility, states, cycles, conservation laws, and precision

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The Job of Classical Mechanics…

is to predict the future.

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Determinism

We may regard the present state of the universe as the effect of its past and the cause of its future. —Pierre-Simon Laplace, A Philosophical Essay on Probabilities

Given the present there can be only one future.

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Reversibility

• The equations of motion are unchanged by reversing the direction of time and the directions of all velocities.• Sufficient knowledge of the present

(particle positions and velocities) can be used to reconstruct the past with any desired accuracy.• Given the present, there can be only one past. 4

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State Machines

We will use the notion of state machines to provide a simple analogy to examine the structure of physical laws.

The analogy is imperfect and we should be careful not to push it too far.

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State Machines (cont)

Imagine some system that at any moment is in exactly one of a (possibly infinite) set of states. (A state is whatever we say it is.)

Examples:• A six-sided die at rest on a table: S1,…S6.

• An eight-bit color pixel:S1,…,S256.

• The distance over time to Voyager 1 in meters: S1,…

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Now suppose we have a law that, given the current state of the system, predicts the next state.Preliminaries:

Evolutionary Law:

State Machines (cont)

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a function

n an integer

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Reassurance

Don’t Panic™.

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Imagine that the states are just integers:S1 1,

S2 2, …

S137 137

σ(92) could be any of 1,…,137, such as σ(92) = S17 17

A Simple Example of States

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just means identity, a sortof extra-strong equal sign

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State Law of Motion

The law of motion:(something) is a function, a mathematical machine. It has• a domain – all of the things we can

legally feed into its parentheses, and• a range – all of the things it can

possibly return to us. 10

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A Function Is A Rule, A Machine.

We stuff an object from the function’s domain, such as a number, into its parentheses,

and it returns an object from its range, such as a different number.

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A Simple Function

Given x: 1. take away the sign of x,2. double it, and3. return it.

Domain: all real numbers.Range: all non-negative real numbers. 12

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State Functions

In our case, the input to () is one of the system’s states, so

()’s domain is the set of system states.

When we give () a state as input, it gives us back a state as output so

()’s range is (also) the set of system states. 13

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State Functions (cont)

So in our example it might be the case that

(S112)

returns the valueS67

Q: What determines the return value?A: The definition of (). 14

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State Functions (cont)

A state machine example of ():

Given: x from (x)1) Is x in the domain of ()? If not, the

world ends.2) Is x one of S1,…,S136? If so return the

next sequential state.3) x must be S137. Return S1.

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Back to Physics and States

Consider once again

Note that only knows about Then can only depend upon .

That is, if we know the current state of the system, we know everything there is to know about its future. That’s determinism.

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Graphical Forms of the Rules

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actual value ofthe state

States:σ(1) = 1σ(2) = 3σ(3) = 7σ(4) = 1σ(5) = 3…

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Determinism Says

There is exactly one arrow leaving each state box.

There is a unique future.

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Reversibility

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arrows reversed

States:σ(5) = 3σ(4) = 1σ(3) = 7σ(2) = 3σ(1) = 1…

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Reversibility Says

There is exactly one arrow entering each state box.

There is a unique past.

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Determinism and Reversibility Say

There is exactly one arrow entering each state box and exactly one arrow leaving.

the present state of the universe is the effect of its past and the cause of itsfuture

- Laplace21

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One Arrow In and One Arrow Out

Could be the same arrow:

a system in stasis

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87Λ (87 )=87