Easiest way to Mathematically Model

[ALine]

Is there an easy straightforward way to produce a mathematical model for anything that I can imagine? The more that I think about it the more and more that I see that maybe everything could be written as a function with distinct variables. I just want to know how I can make mathematics seem extremely intuitive so that if I see a leaf fall on the ground I could fully model that falling in all case scenarios with either a simple algebra equation or with a more complex differential and multi-variable calculus equation.

I have had the urge for years so if anyone could help that would be great.

 

[Sensei]

The easiest to model is linear evolution of physical variable.

Read some physical variable x0 at time t0,

Read physical variable x1 at time t1..

Equation can look like:

f(t)=x0+(x1-x0)/(t1-t0)*t

or

f(t)=x0+dx/dt*t

where dx=x1-x0 and dt=t1-t0

 

Similar easy to model is constant acceleration, and constant deceleration.

 

14 minutes ago, ALine said:

I just want to know how I can make mathematics seem extremely intuitive so that if I see a leaf fall on the ground I could fully model that falling in all case scenarios with either a simple algebra equation or with a more complex differential and multi-variable calculus equation.

Calculation of leaf falling from tree would be extremely hard, because each object has different terminal velocity, which depends on e.g. mass and shape of object, and angle of object while traveling down..

https://en.wikipedia.org/wiki/Terminal_velocity

It would be easy to model only in scenario with vacuum.

 

[ALine]

What if I were to model it in layers, like making the model smaller and smaller, zooming in on each component of the thing that is being modeled and then after this looking at each subsection and finding which variables are similar? Maybe something like this.

I guess something like this.

So if I were in system 2 and the leaf was in system 1 on the I could determine its relative orientation whereas because both I and the leaf were in system 0 then we would not know our orientation in respect to system 0. Maybe this does not just apply to orientation but instead to any variable or function. So if system 0 is applying gravity to both me and the leaf then we are both going to experience it along with all systems that are inside of our separate systems. 

Edited July 30, 2018 by ALine
removed the last part. Did not want to seem like an asshole

[studiot]

There is currently a discussion going on where a mathematical modelling process is emerging.

Perhaps you could use it as a real example for your question?

https://www.scienceforums.net/topic/115474-table-tennis-lawn-tennis-badminton-smashes/?tab=comments#comment-1062510

 

In you box partitioned into system1 and system 2, how would you introduce dynamic similarity?

There is a whole detailed (mathematical) theory of this.

How also would you introduce statistics and probability, for example in a Latin squares experiment?

Edited July 30, 2018 by studiot

[ALine]

3 minutes ago, studiot said:

In you box partitioned into system1 and system 2, how would you introduce dynamic similarity

I am not exactly sure, If the definition I derived from a quick search in dynamic similarities, should have gone more in depth with this, but if the definition that Google gives. That being the comparison between a prototype or model vs. the fully scaled up version then maybe you could say that the size of the system would be a factor as well. So very small systems inside very large systems would need to consider where the position of the system is in respect to its size. So if you have a model car vs an actual car then the model car would experience different scaled down forces vs. the larger car. I can also presume that because the smaller car is smaller it's individual systems within the model car system would be different than the larger car system. I probably also say that I could just take the larger car system and just make it smaller and then just use the same calculations for the larger car system as for the smaller car system.

Not exactly sure about the latin squares experiment I would need more time to think about that one.

Maybe this box inside box idea could be used to represent energy instead of lightning mequeen.

Ok so I have been thinking about it and here is what I came up with.

So maybe the systems all have a smaller and smaller grid which represent their probability of actually being in that position. I don't know, not sure. 

[studiot]

Mathematical modelling is a huge and vastly incredibly important topic.

It is also incredibly varied.

Dynamic similarity is about using or not using the same equations to compare the activity of two systems.

You mentioned the falling leaf.

Would you use the same equations for a leaf and a handful of buckshot?

How about the leaf or buckshot falling through treacle?

The point here is that the are no equations in Physics that produce the values of all the necessary coefficients for each situation.

Some are available and dynamic similarity is what that is all about.

But some have to be measured.

These coefficients determine what actually happens.

 

Have you used a simulator as a form of modelling?

I did a lot of this in my applied Maths degree - it was great fun.

 

But another area of modelling is finite element analysis.

Here we establish a grid or network of points in space or on a graph (like my avatar)
And calculate values of the mathematical property of interest at the grid points.

Now we don't usually use the actual mathematical formula from Physics for this, but a much simplified one or the calculation may be to onerous, even for todays supercomputers.

We than have to decide how well our calculations will match the 'real' values by 'calibration'.
We have to decide for instance whether just the values match or the values and first derivatives match or the values, first and second derivatives and so on.
At every bounday (eg the beginning and the end) we have to invent fictitious points , values and derivatives because there are no real ones outside the boundary.

We can also sometimes call on the fundamental theorems of calculus, Gauss's , Stoke's and Green's theroems to link boundary values to interior values and reduce the calculation burden.
This is then called boundary element analysis.

 

>>>>>>>>>>>>>>EDIT>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Here is a very simple example of mathematical modelling and (non dynamic) similarity.

I have a lawn of size Lmetres by Wmetres.

I pay a contractor £p per linear metre to trim the edges and £q per square metre to cut the middle.

My neighbour has a lawn twice as wide and twice as long.

What factor should be applied to my costs to calculate the cost for my neighbour's cost if he employed the same contractor?

Edited July 30, 2018 by studiot
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