View source for Dynamics of a particle in a central field

[[Virtual laboratory]] > [[Dynamics of a particle in a central field]] <HR>
Developer:  [[А.М. Кривцов]]<HR>

Interactive application, shown below, allows us to study the trajectories of a particle in the central power-law potential field. Interaction force <math>F</math> is a polynomial function of distance <math>r</math>:

:<math>F \sim r^n,</math>

where <math>n</math> is some real dimensionless degree exponent. The attraction center is at the <math>0</math> point. 

<htmlet nocache="yes">Krivtsov/FC_TM</htmlet>

<div class="mw-collapsible mw-collapsed" style="width:100%" >
'''The text of the program is written in JavaScript:''' <div class="mw-collapsible-content"> 
Файл '''"FC.js"'''
<syntaxhighlight lang="javascript" line start="1" enclose="div">
//The movement of a material point in the central field
//A.M developer. Krivtsov 
//18-21.05.2014 
//Internet: tm.spbstu.ru/FC

function MainFC(canvas) {

    // main parameters
    
    const m = 1.;   // weight
    const a = 1.;   // radius
    const P = 1.; 	// gravitation force

    // Derivative dimensional parameters

	const T = 2 * Math.PI * Math.sqrt(m * a / P);	// period of the movement on a circular orbit
	const v1 = Math.sqrt(P * a / m);				// speed of the movement on a circular orbit (the 1st space)
	
	// Variables
	
	var n, v0, zoom, t_max, dt;
	
	// setting of sliders 
	
	Slider_01.min = 1;			Slider_01.max = 22;					Slider_01.step = 0.1;		// v0 / v1 - initial velocity
	Slider_02.max = 6;			Slider_02.min = -Slider_02.max; 	Slider_02.step = 0.01;		// n - degree indicator	
	Slider_03.min = 0;       	Slider_03.max = 6;					Slider_03.step = 0.01;		// zoom	
	Slider_04.min = 0;       	Slider_04.max = 100;				Slider_04.step = 0.1;		// t_max/T	- time for calculation	
	Slider_05.min = 1;       	Slider_05.max = set_exp(0);			Slider_05.step = 1;			// N_exp	- experiment number
	
	Slider_05.focus();

	// values of variables
	
	dt = T / 200;       // integration step
	set_exp(6);			// variables are set for experiment 6

	// display
	
	draw();

    // the functions which are started at change of values of sliders and text fields

    this.set_01 = function(input) { v0 		= Number(input) * v1; 	draw(); }  
    this.set_02 = function(input) { n 		= Number(input); 		draw(); }  
    this.set_03 = function(input) { zoom 	= Number(input); 		draw(); } 
    this.set_04 = function(input) { t_max 	= Number(input) * T; 	draw(); } 	
    this.set_05 = function(input) { set_exp(input);					draw(); } 	
	
	// display
	
	function draw() 
	{ 
		// optimizing
		
		var n1 = (1 - n) / 2;
		var dt1 = -P / m * Math.pow(a, -n) * dt;
		
		// Area of creation of the schedule
		
		const X_max = canvas.width, Y_max = canvas.height;
		var x_max, y_max, sx, sy, X0, Y0;
	
		x_max = y_max = Math.pow(2, zoom) * a;		// size of area of display
		x_min = y_min = -x_max;  

		sx = sy = Y_max / (y_max - y_min); 			// scale on an axis y
		X0 = Y0 = Y_max + y_min * sy;  				// the provision 0 of an axis y in screen coordinates
	   
	   // Initialization of graphics
		
		var context = canvas.getContext("2d");  	// context - for drawing
		context.clearRect(0, 0, X_max, Y_max); 	// clear screen
        
        // horizontal axis
        context.strokeStyle = 'lightgrey';
        context.beginPath();
        context.moveTo(0, Y0);	context.lineTo(X_max, Y0);
        context.moveTo(X0, 0);	context.lineTo(X0, Y_max);
        context.moveTo(X0 +  a * sx, Y0);	context.arc(X0, Y0, a * sx, 0, 2 * Math.PI);
        context.stroke();

        //  inscription
        context.fillStyle = 'black';
        context.font = "italic 20px Times";
        context.fillText("0", X0 - 15, Y0 - 7);
		
		// schedule 
		context.strokeStyle = 'black';
		context.beginPath();
		var vx = v0, vy = 0;
		var x = 0, y = a;
		context.moveTo(X0, Y0 - a * sy);
		for (var t = 0; t < t_max; t += dt)
		{
			var r2 = x * x + y * y;
			var rn = Math.pow(r2, n1);
			vx += x / rn * dt1;
			vy += y / rn * dt1;
			x += vx * dt;			
			y += vy * dt; 
			var X = X0 + x * sx; 
			var Y = Y0 - y * sy; 
			context.lineTo(X, Y);
		}
		context.stroke();			
	}
	
	// Experiment choice
	
	function set_exp(N_exp)
	{	
		var k = Number(N_exp);
		
//			degree indicator	initial velocity	 zoom		max number of steps
	
		if (!--k) { n = -3;		v0 = 1.004 * v1; 	zoom = 2.98;	t_max = 17.4 * T; }  	// spiral			
		if (!--k) { n = -2.9;	v0 = 1.023 * v1; 	zoom = 2.98;	t_max = 67.6 * T; }  	 		
		if (!--k) { n = -2.87;	v0 = 1.029 * v1; 	zoom = 2.57;	t_max = 21.6 * T; }  	 		
		if (!--k) { n = -2.87;	v0 = 1.03 * v1; 	zoom = 2.96;	t_max = 94.4 * T; }  	 		
		if (!--k) { n = -2.5;	v0 = 1.135 * v1; 	zoom = 2.98;	t_max = 31.1 * T; }  	// 2

		if (!--k) { n = -2;		v0 = 1.25 * v1; 	zoom = 2;	t_max = 20 * T; }  			// ellipse
		if (!--k) { n = -1;		v0 = 1.25 * v1; 	zoom = 1;		t_max = 26 * T; }
		if (!--k) { n = -1;		v0 = 1.36 * v1; 	zoom = 1.18;	t_max = 10.8 * T; }		// 10
		if (!--k) { n = -1;		v0 = 1.63 * v1; 	zoom = 1.87;	t_max = 25.3 * T; }
		if (!--k) { n = -1;		v0 = 1.93 * v1; 	zoom = 2.8;		t_max = 8 * T; }		//  3 !
		if (!--k) { n = -1;		v0 = 2.31 * v1; 	zoom = 3.92;	t_max = 64.1 * T; }		// 11
		if (!--k) { n = -1;		v0 = 2.43 * v1; 	zoom = 4.29;	t_max = 61.8 * T; }		//  8
		if (!--k) { n = -1;		v0 = 2.74 * v1; 	zoom = 5.5;		t_max = 85.5 * T; }		//  5

		if (!--k) { n = -0.74;	v0 = 2.665 * v1; 	zoom = 3.81;	t_max = 18.8 * T; }		//  5 
		
		if (!--k) { n = 0;		v0 = 1.66 * v1; 	zoom = 1.39;	t_max = 5.1 * T; }		//  7 
		if (!--k) { n = 0;		v0 = 2.7  * v1; 	zoom = 2.34;	t_max = 8.9 * T; }		//  9 
		if (!--k) { n = 0;		v0 = 3.46 * v1; 	zoom = 3.03;	t_max = 13.3 * T; }		// 11 
	
		if (!--k) { n = 1;		v0 = 2 * v1; 		zoom = 1.3;		t_max = 1 * T; }		// 11 	

		if (!--k) { n = 2;		v0 = 2.39 * v1; 	zoom = 1.18;	t_max = 4.0 * T; }		// 11 	
	
		if (!--k) { n = 3;		v0 = 3.58 * v1; 	zoom = 1.28;	t_max = 1.8 * T; }		//  7 
		if (!--k) { n = 3;		v0 = 6.97 * v1; 	zoom = 1.87;	t_max = 1.7 * T; }		//  9
		if (!--k) { n = 3;		v0 = 11.28 * v1; 	zoom = 2.13;	t_max = 1.6 * T; }		// 11 	
		if (!--k) { n = 3;		v0 = 16.13 * v1; 	zoom = 2.50;	t_max = 1.6 * T; }		// 13 	

		if (!--k) { n = 3.29;	v0 = 1.3 * v1; 		zoom = 0.48;	t_max = 1.8 * T; }		// 13 	
		
		if (!--k) { n = 4;		v0 = 3.36 * v1; 	zoom = 1.12;	t_max = 1.1 * T; }		//  5 !	
		if (!--k) { n = 4;		v0 = 6.08 * v1; 	zoom = 1.44;	t_max = 1.9 * T; }		// 12 	
		if (!--k) { n = 4;		v0 = 9.52 * v1; 	zoom = 1.66;	t_max = 0.9 * T; }		//  7 !	
		if (!--k) { n = 4;		v0 = 18.45 * v1; 	zoom = 2.18;	t_max = 0.728 * T; }	//  9	

		if (!--k) { n = 5;		v0 = 7.15 * v1; 	zoom = 1.55;	t_max = 0.6 * T; }		//  5 !	

		if (!--k) { n = 4;		v0 = 3.304 * v1; 	zoom = 1.12;	t_max = 36.4 * T; }		//  5 ~~~
		if (!--k) { n = 4;		v0 = 9.394 * v1; 	zoom = 1.66;	t_max = 19.6 * T; }		//  7 ~~~	

		if (N_exp)
		{
			Text_01.value = v0 / v1; 		Slider_01.value = Text_01.value;		// initial velocity	    			
			Text_02.value = n;     			Slider_02.value = Text_02.value;        // degree indicator
			Text_03.value = zoom;     		Slider_03.value = Text_03.value;	    // zoom
			Text_04.value = t_max / T;     	Slider_04.value = Text_04.value;		// time for calculation 			
			Text_05.value = N_exp;          Slider_05.value = Text_05.value;        // number of experiment
		}
		
		return -k;	// Если N_exp = 0 returns total of experiments, in other cases returns 0
	}
}
</syntaxhighlight>
Файл '''"FC.html"'''
<syntaxhighlight lang="html5" line start="1" enclose="div">
    <canvas id="canvasGraph" width="600" height="600" style="border:1px solid #000000;"></canvas>

    <!--Installation of parameters of interaction (text fields and sliders)-->
    <div>
        <font face= "Times New Roman"><I>
        v</I><SUB>0</SUB> = <input id="Text_01" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
            // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
            if (!this.checkValidity()) return;
            app.set_01(this.value);
            document.getElementById('Slider_01').value = this.value;
        "><I> v</I><SUB>1</SUB> 
		<input type="range" id="Slider_01" style="width: 100px;" oninput="app.set_01(this.value); document.getElementById('Text_01').value = this.value;">
		n = <input id="Text_02" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
            if (!this.checkValidity()) return;
            app.set_02(this.value);
            document.getElementById('Slider_02').value = this.value;
        ">
		<input type="range" id="Slider_02" style="width: 100px;" oninput="app.set_02(this.value); document.getElementById('Text_02').value = this.value;">
		</font>
	</div>
    <div>
        <font face= "Times New Roman">
        zoom = <input id="Text_03" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
            // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
            if (!this.checkValidity()) return;
            app.set_03(this.value);
            document.getElementById('Slider_03').value = this.value;
        "> 
		<input type="range" id="Slider_03" style="width: 100px;" oninput="app.set_03(this.value); document.getElementById('Text_03').value = this.value;">
		<I>t</I><SUB>max</SUB> = <input id="Text_04" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
            if (!this.checkValidity()) return;
            app.set_04(this.value);
            document.getElementById('Slider_04').value = this.value;
        "><I> T</I>
		<input type="range" id="Slider_04" style="width: 100px;" oninput="app.set_04(this.value); document.getElementById('Text_04').value = this.value;">
		</font>
	</div>
     <div>
        <font face= "Times New Roman">
        Number of experiment <input id="Text_05" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
            // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
            if (!this.checkValidity()) return;
            app.set_05(this.value);
            document.getElementById('Slider_05').value = this.value;
        "> 
		<input type="range" id="Slider_05" style="width: 100px;" oninput="app.set_05(this.value); document.getElementById('Text_05').value = this.value;">
	</div>
	
	<script type="text/javascript">var app = new MainFC	(
		document.getElementById('canvasGraph')
	);</script>
</syntaxhighlight>
</div>
</div>

The application allows to set the following parameters interactively: 
* <math>v_0</math> — initial velocity of the point. The initial velocity is directed tangentially (perpendicular to the radial direction).Velocity is measured in relation to the <math>v_1</math> - speed circular motion on the initial distance from the center (the first cosmic speed).  
* <math>n</math> — is an index in the law of interaction(<math>n=-2</math> corresponds to the gravitational interaction, <math>n=1</math> - the linear-elastic).  
* zoom is the logarithmic scale of display of the schedule (the logarithm on the basis 2 is used). 
* <math>t_{\rm max}</math> — integration time expressed in <math>T</math> periods of circular motion at the initial distance from the center. 

Besides, it is possible to set "number of the experiment". Every number has a separate set of four parameters named above and characterized by some specific kind of movement. Experiments are ordered by <math>n</math>(first), then by <math>v_0</math>.

== The proposed research areas ==

* To define possible variants of motion 
* To define the dependence of the distance to the epicenter (the most distant point) of the orbit from <math>n</math> and <math>u=v_0/v_1</math>.
* To find closed trajectories of different topologies (for example, 5th, the 7th terminating stars, etc.) and to determine their position on the plane of the parameters <math>n</math>, <math>u</math>.
* To find a method for integrating of the equations of motion with a variable step allowing to model the movement of a point at large <math>|n|</math> effectively.
* •	To pick up values of parameters (and to develop a method of such selection), allowing to receive "beautiful" schedules which can be considered as objects of Science Art (see in particular, last numbers of experiments). 

[[Category: Virtual laboratory]]
[[Category: Programming]]
[[Category: JavaScript]]