Initial Conditions

Questions:

1. How important are initial conditions ?
   
2. How important are final conditions ?
   

Purpose

For example: In order to predict the position of the planets using Newton's Law, you need for the Sun and each of the planets their masses and initial conditions i.e. position and velocity.
All you need to do is to solve the differential equations that are subject of Newton's Law and with the initial conditions you can predict the position of each of the planets in the future.

I doubt if it is that simple

An example to solve the question

Consider the following sequence of measurements:

98, 102, 103, 97, 95, 105, 100, 95, 105, 103, 97, 98, 102

Questions:

1. What is the next value?
2. What is the differential equation that describe this phenomena?
3. What are the initial conditions?

Maybe this sequence of values sounds rather abstract. A more practicle application is if you consider those values as the distance R between two objects.

• For the first question, the best estimate value is 100. It is important to remark that in order to calculate this estimate all measurements are used.
• The differential equation that describes this is:
  dx/dt = 0
  Suppose that the next value is 103.
  Then the differential equation is:
  dx/dt = 0.02
  In general it is better to describe this the differential equation as:
  dx/dt = c1 with c1 = 0
  The important point is that in order to calculate the constant c1, you must include all the measurements.

  If you look to the sequence of numbers the first value is 98. If you use that as your initial condition in your differential equation than the best estimate next value becomes 98
  If you use 100 as initial condition, which is the average of all values, than your best estimate is 100.

What you can learn from above is that in order to predict the future each of every measurement that you have done is of equal importance.

A different example

Consider the following two sequences of numbers:

98, 102, 103, 97, 95, 105, 100, 95, 105, 103, 97, 98, 102 102, 98, 97, 103, 105, 95, 100, 105, 95, 97, 103, 102, 98

For example: two sequences of numbers which represent the distance to the centre of mass (gravity) of two objects m1 and m2

What do those sequences tell us?

In case when those sequences represent the distance to the centre of mass they tell us that the masses of the two objects m1 and m2 are equal
Because, and that is important, the average value of both sequences of measurements is equal.

If any one value would be different than m1 is not equal to m2.

Reflection

Those measurements are translated in the parameters and intial values of the differential equations used in order to calculate the future.
The (lack of) accuracy of those measurements is directly refelected in the equations and in the corresponding predictions.

In case of Newton's Law to calculate the future solely based on observations is relativ easy.
In case of General Realtivity to calculate the future solely based on observations is very difficult. IMO starting from scratch it is (almost) impossible.

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