SOLVING SOME PHYSICAL PROBLEMS USING THE METHODS OF NUMERICAL MATHEMATICS WITH THE HELP OF SYSTEM MATHEMATICA

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1 SOLVING SOME PHYSICAL PROBLEMS USING THE METHODS OF NUMERICAL MATHEMATICS WITH THE HELP OF SYSTEM MATHEMATICA Pavel Trojovský Jaroslav Seibert Antonín Slabý ABSTRACT Ever future teacher of mathematics must know smbolic approaches for example for finding derivations and integrals solving algebraic and differential equations ver well But most of the problems from practice are not solvable b a smbolic wa and therefore numerical ones must be used That is wa the basic knowledge of some numerical methods for solving problems of the real world is a component of the complex training of teachers of mathematics Different computer sstems for smbolic and numerical mathematics as MatLab Maple Mathematica or Derive enable their effective use The posibilit to join numerical and smbolic computation in these sstems makes them ver well suited for expressing numerical algorithms testing them out and postprocessing the results for example in a graphical form Their abilit for high-precision or exact calculations is also important This contribution deals with a concrete illustration of using the sstem Mathematica for solving several tpical phsical problems b differential equations or their sstems KEYWORDS Sstem Mathematica Runge-Kutta method the simple pendulum pendulum phslet movement of projectile orbits of satelite INTRODUCTION Mathematica is a powerful mathematical software sstem for students researchers and someone finding an effective tool for mathematical analsis As the portable electronic calculator made doing simple arithmetic unnecessar so powerful and sophisticated smbolic processing programs such as Mathematica do calculus unnecessar b hand Algebraic manipulation finding roots of equations derivatives of functions solutions of differential equations and ver large number of other mathematical operations have been done in a similar wa and the solutions appear in smbolic form Mathematica also has extensive and powerful graphic capabilities Such a smbolic processing programs have profound implications for mathematicall rich fields (for example phsics) where a great deal of time and much accumulated skill goes into master handling about complicated mathematics With the advent of Mathematica educators have endeavored to make use of the mathematical manipulation power of this software sstem to improve the traditional wa of teaching mathematics phsics and engineering Mathematica has been incorporated in a number of universities as a productivit tool to help students in analtical performance numerical and graphical work Mathematica provides facilities for doing both of smbolic and numerical mathematics or mixing the two as required 59

2 Mathematica contains an extensive collection of algorithms for the users covering almost ever field of mathematics including calculus differential equations matrix algebra etc In our contribution we take two tpes of problems the problems solvable smbolicall using build-in commands in the sstem Mathematica (commands Integrate[] DSolve[] and so on) and the problems whose solution can be found onl b numerical wa In this second case we use the fourth order Runge- Kutta method for numerical integration of autonomous sstems of ordinar first-order differential equations For example we take solving the problem of movement projectile with resistance of air the motion of satellite of the Earth in its gravitational field and oscillating of a simple pendulum in vacuum and with resistance of environment too In all these examples we are going to show a mathematical description of the problem a numerical solution the graphs of functional dependencies of various quantities and computer animations of the movements THE RUNGE-KUTTA METHOD An autonomous first-order sstem of differential equations of n variables n with the initial conditions at ( x0 ) = 0 ( x0 ) = 0 n ( x0 ) = n0 in the sstem ' = f( x n) = f ( x ) ' n n ' = f n ( x n ) where i ' i = n denote derivative of i with respect to a variable x Puting = ) ( n = ( and f f ( x ) f ( x ) f n ( x )) we can rewrite the equations and the initial conditions simpl as a sstem starts from the initial conditions () ' = f ( x ) (0) ( x = 0 ) (0) A numerical method for solving such ( 0 (0) The given value = x ) leads to find the value at the point x = x 0 + h then this value leads to find the value at x = x0 + h and so on The constant h > 0 is called the step size There are man different formulaes for this purpose We take the fourth order Runge-Kutta method in solutions of the following problems (i+ ) This method computes (i) from in the following wa: ( ) = i k k k f ( x i h h = f ( xi + k ) ) h h = f xi + k ( ) () 60

3 k i = f ( x + h h k ) ( i) h = + ( k + k + k + k 6 ( i+ ) ) This Runge-Kutta method can be easil programmed in Mathematica because it contains a special tpe of object for vectors and man functions for operations with them If we use built-in functions SetAttributes ClearAttributes Flatten Append While and Module we get this Mathematica s function for implementation of Runge-Kutta method: RungeKuttaMethod[f_{x_x0_}{_0_}b_h_]:= Module[{kkkk xk=x0 k=0 res={ Flatten[{x00}] } } SetAttributes[Rule Listable]; While[xk<b k=f/x->xk / ->k; k=f/x->xk+h/ / ->k+h/ k; k=f/x->xk+h/ / ->k+h/ k; k=f/x->xk+h / ->k+h k; k=k+h/6 (k+(k+k)+k); xk=xk+h; res=append[res Flatten[{xkk}] ] ]; ClearAttributes[Rule Listable]; res ] THE SIMPLE PENDULUM The pendulum should be thought of as a weight hung on a rigid rod of negligible mass from a pivot without friction in a medium which offers no resistance to things moving through it The state of the motion at an time is defined completel b the position and velocit at that time If we know those values we can calculate its position and velocit for an time in the future or past Figure 6

4 Using the second Newton law the differencial equation for a description of the movement of the pendulum is d Θ ml = mg sin Θ where Θ is the angle of the pendulum from the vertical m is the mass l is the length and g is the dθ acceleration due to gravit Introducing the angular velocitω = we can rewrite this equation as a sstem of two first order ordinar differential equations dθ = ω dω g = sin Θ l Analticall the motion of a pendulum is given in an implicit form b elliptic integrals We will solve the sstem of equations numericall using sstem Mathematica The computed displacement of the value Θ of the pendulum versus time t is shown in the following Figure : Figure Now we relax some of our unrealistic requirements for our sstem and we allow an air consistence We get the similar secondar-order equation: d Θ dθ lm = mg sin Θ c We can transform it into two first order equations with the variables The equations become: = Θ dθ = ' = g ' = sin l c ml The displacement of the pendulum is in the Figure Figure 6

5 MOVEMENT OF PROJECTILE Now we will stud elements of a projectile motion with the resistence of air if it is shot with a initial velocit v 0 and the mass is m Using the second Newton law we again obtain the differential equations for x and coordinates positions of the projectile: d x m = c vx cvx vx + v d m = mg c v cv vx + v The sstem can be rewritten to the four first-order equations with variables d = We obtain ' = c c ' = + m m ' = c c ' = g + m m = x dx = = with the initial conditions (0) = 0 ( 0) = v0x (0) = 0 ( 0) = v0 Fig shows the displacement of the projectile and Fig 5 the graph of the velocit versus time ( v 0 x = 00 v 0 = 80 ) Figure Figure 5 6

6 MOVEMENT OF A SATELITE OF PLANETS In this part we find satelite orbits for the case of one planet and then for the case of a couple of planets Using the second Newton law and Newton s classical law of gravit we get an equation for the description of orbits of satelite mm m a = κ r r where ( d x d a = ) is the acceleration of the satelite m the mass of it M the mass of the planet κ the universal constant of gravit and r = ( x ) is a position vector of the satelite We get d x = κ M d = κ M x + / ( x ) + / ( x ) which can be rewritten to the four first-order equations with variables d = ' = : ' = κ M ( ' = ' = κ M ( + + / ) / ) with (0) = 0 ( 0) = v0x (0) = H (0) = 0 = x dx = = Figure 6 contains the orbits of the satelite of one planet for two different initial velocities of the satelite (circular resp parabolical orbits): Figure 6 6

7 Similarl we get the equation for the orbit of the satelite of a couple of planets which are in the distance D: m mm mm a = κ r κ r r r where r = ( x D ) and r = ( x + D ) and we can again rewrite it to four first-order differential equations The computed orbits of the satelite of the couple of planets for different velocities of the satelite are in Figure 7 Figure 7 CONCLUSION We showed an approach of finding solutions of several phsical problems of using function RungeKuttaMethod[] which we created The sstem Mathematica contains from version 0 built-in command NDSolve[] which can be used to solve out majorit of the solved problems Practical use of the theor of numerical methods is on this wa more important for our students 65

8 REFERENCES Maeder R Programming in Mathematica Adison-Wesle Publishing Compan 99 Wolfram S Mathematica A sstem for doing mathematics b computer Adison-Wesle Publishing 99 Pavel Trojovský Universit of Hradec Králové Department of Mathematics Víta Nejedlého Hradec Králové Czech Republic paveltrojovsk@uhkcz Jaroslav Seibert Universit of Hradec Králové Department of Mathematics Víta Nejedlého Hradec Králové Czech Republic jaroslavseibert@uhkcz Antonín Slabý Universit of Hradec Králové Department of Informatics Víta Nejedlého Hradec Králové Czech Republic antoninslab@uhkcz 66