construction principles

IITU
Department of Information Systems
and Mathematical Modeling

be the rational order and harmony which God gave to

the world and revealed it to us in the language of mathematics. Johannes Kepler
Mathematics is radically different from physics, biology, history, psychology and other sciences. Its object are the abstract concepts like numbers and functions, equations and sets, which are strange products of the human brain, however mathematics is surprisingly adapted to comprehend all the phenomena of nature and society.
Striking applicability of mathematics for analyzing various world events is the result of the link between objective reality and the abstraction of the mathematical structures.

which is able to translate the laws of nature studied

by individual sciences into mathematical language, to see real-life events behind the strict mathematical formulas.
Our goal is to learn how to connect mathematics and the world, make sure that different natural phenomena can be described mathematically with the same methods. There is strict mechanics, thoroughly imbued with mathematical ideas, and psychology, which is not very amenable to formalization in amazingly equal position.

and interferes in them by asking certain questions and getting

appropriate answers. After examining some information about the object of the study, the subject forms his idea about ​​the objects of interest and his view of the phenomenon. The view which is based on the available objective information about the object under study but representing a subjective point of view of the researcher is called a model of the object.
Thus, the process of cognition, which is essentially reduced to the collection, storage and processing of all kinds of information, is at the same time the process of modeling.

cognition
Model of the object

interpretation and
specification
representation
question
answer

some common laws in the construction of mathematical models. First

of all, we should understand what exactly is the object of study.
After that it is necessary to identify functions of the system condition (or system functions) which are the values ​​which according to the researcher fully reflect the progress of the process. Mathematical models usually represent equations for the selected system functions.
Exploring the resulting mathematical model and solving the relevant equations we try to understand type of dependence of the system functions on various parameters. First of all, these are independent variables, which are usually time and space variables.

have to specify the coordinate system within which a mathematical

model will be constructed.
Now we can try to go directly to the construction of mathematical relations that characterize the law of variation of the system functions. We must find out what exactly influenced the course of events and specify the reason of the system evolution.
After indication the reasons that led to the events which are of our interest, we must identify the cause-and-effect relationship and determine how exactly each of these reasons influenced the simulated process. Basis of mathematical model is cause-and-effect relationship which expresses the laws of variation of the system functions. It is formed on the basis of underlying regularities established by particular science.

phenomenon, we will find out that besides the system functions

and independent variables mathematical model includes characteristics of completely different nature. We are talking about the parameters of the system, allowing to choose the particular case which is of direct interest to us at this stage of the study.
Formally, the system parameters can be changed arbitrarily. However, not all the variants of setting these values ​​have physical meaning. Conditions of applicability of the mathematical model, range of the parameters (independent variables and parameters of the system), in which a model makes sense should be specified.

Parameters of the process in each case are fixed quantities

which compose the input data. Independent variables are not fixed, they change within a certain range and form the domain of the system functions. Thus, the mathematical model is most often a problem of restoring the system functions dependence on the corresponding variables in the allowed set of process parameters.
After determination necessary mathematical relations, we have to find what information about the process is practically interesting for us besides the system functions. We have to specify the output information.

study
Functions of the system condition / System functions
Independent variables
Coordinate system
Reasons

for the system evolution
Cause-and-effect relations
Input parameters of the system
Conditions of the mathematical model applicability
Output parameters of the system

on the basis of various quantitative and qualitative methods. Conducting

appropriate analysis using only mathematical tools (usually computer), we return to the studied phenomena in order to make physical interpretation of the results, refine our tasks and summarize relevant outcomes.

mathematical models, we can specify some of the principles of

classification.
Lumped parameter system, in which the system function depends only on time.
Distributed parameter systems, which allow dependence of the system function on space variables.
Ordinary differential equations most often represent lumped parameter systems. Distributed parameter systems are usually characterized by partial differential equations.

factors.
Deterministic models, in which such impact is ignored, and system

state is unambiguously determined.
Continuous systems in which independent variables are changing continuously.
Discrete systems with independent variables that vary in pace (fixed or not).
Dynamic systems with the system functions that change with the time.
Stationary systems which characteristics do not change over time.

all academic requirements is associated with the name of Galileo

and describes the process of falling bodies under their own weight.
When the body is falling, its distance to the ground will be change over time. Thus, we can describe the process using function y = y (t) which characterizes the height of the body at any given time t.
Thus, the motion of the body is necessarily accompanied by changing its height. In order to find the required function y, we can estimate velocity of its change.

obtain the relation
= v (l.l)
Equation (1.1), we suppose, should

describe the process under study, as its mathematical model.
The second stage of the study involves the extracting using mathematical tools information which is contained in the resulting model in a latent form. To find the unknown functional relation we have to solve the resulting equation. If the velocity does not change over time, equation (1.1) will be satisfied by any function of the following form:
y(t) = vt + c, (1.2)
where c is an arbitrary constant. Relation (1.2) gives the general solution of the equation (1.1).

to interpret the results. First of all, we have to

find out whether constructed model describes the phenomenon in question. To do this, we compare the results of analysis of the mathematical model and experiment.
The ambiguity of the found solution means that the information about the process is still not enough for a full description. Therefore, proposed mathematical model needs considerable refinement. This means that we have to return to the first step and try to fill a gap in the understanding of the process under study.

the exact position of the body at some initial time.

Suppose that body has initial height:
y(t0)=y0 (1.3)
It is called the initial condition.
Refined mathematical model is characterized by equations (1.1), (​​1.3), which constitute the Cauchy problem for the equation. We define a solution of the equation (1.1), which satisfies the initial condition (1.3). We put in the formula (1.2) t = t0.
y(t0)=vt0+c=y0
From this we find the constant: с = у0 – vt0. Then the solution of equation (1.1) with the initial condition (1.3) has final form:
y(t) = y0 + v(t-t0) (1.4)

calculated height of falling body to its experimental value. Indeed,

a simple experiment shows that in the period of falling the body velocity is changing. Specifying the model, we get the same Cauchy problem, but with a variable v.
Integrating equation (1.1) from t0 to some value t and using the initial condition (1.3), we find the following law of body height change:
y(t)=y0+ (1.5)
In the case of constant velocity, this formula takes the familiar form (1.4).

Velocity of change of the function v, that is its

derivative (or second derivative of the coordinate ÿ) in mechanics is called acceleration and is denoted by а. As a result, to find the velocity we obtain another differential equation.
= а (1.6)
We can assume, for example, that the body at the time t = t0 just starts to move, and therefore it has zero velocity:
v(t0)=0 (1.7)
So, we have a system of differential equations (1.1), (1.6) with initial conditions (1.3) and (1.7).

derivative (velocity). For calculation of velocity is required to find

its derivative (acceleration). Reliable experiment shows that the acceleration of the falling body does not change over time, and it does depend on the specific characteristics of the body under study, as a universal physical constant. This striking fact allows us, finally, to complete the research and get the result.
Note that the falling body height is reduced in the motion, in accordance with equation (1.1) it is only possible in case of the negative velocity v.

and it is an absolute constant which is independent from

process conditions. Thus, the relation (1.6) can be written as:
= -g (1.8)
Differential equations (1.1), (1.8) with initial conditions (1.3), (1.7) are the mathematical model of the free falling process of the body.

that the velocity of a falling body will be changed

by the following law:
v(t)=-gt
Substituting this value in equation (1.5) with t0=0, we establish the following dependence of the falling body height on time for various values ​​of initial height
y(t)=y0-gt2/2
Comparing the variation law of the falling body height with the experimental results, we conclude that constructed model is quite satisfactory.

model we can get some additional information on the behavior

of the system under study. In particular, it is important for the process to set the time T, in which the body hits the ground, and therefore will be y(T) equal to zero.
We find the value у0=gТ2/2. From this we can determine the value of the falling body, depending on its initial height
(1.9)

the body velocity at the falling time. For a given

initial height of the body, it is equal to:

mathematical model. The model describes the studied process only in

the time interval from zero to T, as long as the body does not reach the ground. Thus, equations (1.1), (1.8) make sense only for a limited time interval 0 у(Т) = 0 (1.10)
Finally, the parameter у0, included in the initial condition (1.3), must be necessarily positive:
у0>0 (1.11)

state equations (1.1), (1.8) on the time interval 0

process

particular level of study of the process, and if necessary

can be refined. Thus, we have completely ignored the effect of air resistance, the effect of other forces. We have absolutely no interest in the shape, size and weight, which is also under certain conditions themselves, may change. Actually, the immutability of the free fall acceleration will be recognized as true only for relatively small discontinuity of height.