The Nature of Classical Physics Determinism, reversibility, states, cycles, conservation laws, and precision 1 Nature of Classical Physics TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A

The Job of Classical Mechanics… is to predict the future. Nature of Classical Physics 2

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. Nature of Classical Physics 3

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. Nature of Classical Physics 4

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. Nature of Classical Physics 5

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: S 1,…S 6. An eight-bit color pixel:S 1,…,S 256. The distance over time to Voyager 1 in meters: S 1,… Nature of Classical Physics 6

State Machines (cont) Nature of Classical Physics 7 a function n an integer

Reassurance Don’t Panic™. Nature of Classical Physics 8

A Simple Example of States Nature of Classical Physics 9  just means identity, a sort of extra-strong equal sign

State Law of Motion Nature of Classical Physics 10

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. Nature of Classical Physics 11

A Simple Function Nature of Classical Physics 12

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. Nature of Classical Physics 13

State Functions (cont) So in our example it might be the case that  (S 112 ) returns the value S 67 Q: What determines the return value? A: The definition of  (). Nature of Classical Physics 14

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 S 1,…,S 136 ? If so return the next sequential state. 3)x must be S 137. Return S 1. Nature of Classical Physics 15

Back to Physics and States Nature of Classical Physics 16

Graphical Forms of the Rules Nature of Classical Physics actual value of the state States: σ(1) = 1 σ(2) = 3 σ(3) = 7 σ(4) = 1 σ(5) = 3 …

Determinism Says There is exactly one arrow leaving each state box. There is a unique future. Nature of Classical Physics

Reversibility Nature of Classical Physics arrows reversed States: σ(5) = 3 σ(4) = 1 σ(3) = 7 σ(2) = 3 σ(1) = 1 …

Reversibility Says There is exactly one arrow entering each state box. There is a unique past. Nature of Classical Physics

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 its future - Laplace Nature of Classical Physics

One Arrow In and One Arrow Out Could be the same arrow: a system in stasis Nature of Classical Physics 22 87