Dynamics of a particle in a central field
From Department of Theoretical and Applied Mechanics
Virtual laboratory > Dynamics of a particle in a central field
Developer: A.M. Krivtsov
Developer: A.M. Krivtsov
Interactive application, shown below, allows us to study trajectories of a particle in the central power-law potential field. Interaction force is a power function of distance :
where is some real dimensionless exponent. The attraction center is at point.
v0 = v1
n =
zoom =
tmax = T
Номер эксперимента
The text of the program is written in JavaScript:
File"FC.js"
1 //The movement of a material point in the central field
2 //developer A.M. Krivtsov
3 //18-21.05.2014
4 //Internet: tm.spbstu.ru/FC
5
6 function MainFC(canvas) {
7
8 // main parameters
9
10 const m = 1.; // weight
11 const a = 1.; // radius
12 const P = 1.; // gravitation force
13
14 // Derivative dimensional parameters
15
16 const T = 2 * Math.PI * Math.sqrt(m * a / P); // period of the movement on a circular orbit
17 const v1 = Math.sqrt(P * a / m); // speed of the movement on a circular orbit (the 1st space)
18
19 // Variables
20
21 var n, v0, zoom, t_max, dt;
22
23 // setting of sliders
24
25 Slider_01.min = 1; Slider_01.max = 22; Slider_01.step = 0.1; // v0 / v1 - initial velocity
26 Slider_02.max = 6; Slider_02.min = -Slider_02.max; Slider_02.step = 0.01; // n - degree indicator
27 Slider_03.min = 0; Slider_03.max = 6; Slider_03.step = 0.01; // zoom
28 Slider_04.min = 0; Slider_04.max = 100; Slider_04.step = 0.1; // t_max/T - time for calculation
29 Slider_05.min = 1; Slider_05.max = set_exp(0); Slider_05.step = 1; // N_exp - experiment number
30
31 Slider_05.focus();
32
33 // values of variables
34
35 dt = T / 200; // integration step
36 set_exp(6); // variables are set for experiment 6
37
38 // display
39
40 draw();
41
42 // the functions which are started at change of values of sliders and text fields
43
44 this.set_01 = function(input) { v0 = Number(input) * v1; draw(); }
45 this.set_02 = function(input) { n = Number(input); draw(); }
46 this.set_03 = function(input) { zoom = Number(input); draw(); }
47 this.set_04 = function(input) { t_max = Number(input) * T; draw(); }
48 this.set_05 = function(input) { set_exp(input); draw(); }
49
50 // display
51
52 function draw()
53 {
54 // optimizing
55
56 var n1 = (1 - n) / 2;
57 var dt1 = -P / m * Math.pow(a, -n) * dt;
58
59 // Area of creation of the schedule
60
61 const X_max = canvas.width, Y_max = canvas.height;
62 var x_max, y_max, sx, sy, X0, Y0;
63
64 x_max = y_max = Math.pow(2, zoom) * a; // size of area of display
65 x_min = y_min = -x_max;
66
67 sx = sy = Y_max / (y_max - y_min); // scale on an axis y
68 X0 = Y0 = Y_max + y_min * sy; // the provision 0 of an axis y in screen coordinates
69
70 // Initialization of graphics
71
72 var context = canvas.getContext("2d"); // context - for drawing
73 context.clearRect(0, 0, X_max, Y_max); // clear screen
74
75 // horizontal axis
76 context.strokeStyle = 'lightgrey';
77 context.beginPath();
78 context.moveTo(0, Y0); context.lineTo(X_max, Y0);
79 context.moveTo(X0, 0); context.lineTo(X0, Y_max);
80 context.moveTo(X0 + a * sx, Y0); context.arc(X0, Y0, a * sx, 0, 2 * Math.PI);
81 context.stroke();
82
83 // inscription
84 context.fillStyle = 'black';
85 context.font = "italic 20px Times";
86 context.fillText("0", X0 - 15, Y0 - 7);
87
88 // schedule
89 context.strokeStyle = 'black';
90 context.beginPath();
91 var vx = v0, vy = 0;
92 var x = 0, y = a;
93 context.moveTo(X0, Y0 - a * sy);
94 for (var t = 0; t < t_max; t += dt)
95 {
96 var r2 = x * x + y * y;
97 var rn = Math.pow(r2, n1);
98 vx += x / rn * dt1;
99 vy += y / rn * dt1;
100 x += vx * dt;
101 y += vy * dt;
102 var X = X0 + x * sx;
103 var Y = Y0 - y * sy;
104 context.lineTo(X, Y);
105 }
106 context.stroke();
107 }
108
109 // Experiment choice
110
111 function set_exp(N_exp)
112 {
113 var k = Number(N_exp);
114
115 // degree indicator initial velocity zoom max number of steps
116
117 if (!--k) { n = -3; v0 = 1.004 * v1; zoom = 2.98; t_max = 17.4 * T; } // spiral
118 if (!--k) { n = -2.9; v0 = 1.023 * v1; zoom = 2.98; t_max = 67.6 * T; }
119 if (!--k) { n = -2.87; v0 = 1.029 * v1; zoom = 2.57; t_max = 21.6 * T; }
120 if (!--k) { n = -2.87; v0 = 1.03 * v1; zoom = 2.96; t_max = 94.4 * T; }
121 if (!--k) { n = -2.5; v0 = 1.135 * v1; zoom = 2.98; t_max = 31.1 * T; } // 2
122
123 if (!--k) { n = -2; v0 = 1.25 * v1; zoom = 2; t_max = 20 * T; } // ellipse
124 if (!--k) { n = -1; v0 = 1.25 * v1; zoom = 1; t_max = 26 * T; }
125 if (!--k) { n = -1; v0 = 1.36 * v1; zoom = 1.18; t_max = 10.8 * T; } // 10
126 if (!--k) { n = -1; v0 = 1.63 * v1; zoom = 1.87; t_max = 25.3 * T; }
127 if (!--k) { n = -1; v0 = 1.93 * v1; zoom = 2.8; t_max = 8 * T; } // 3 !
128 if (!--k) { n = -1; v0 = 2.31 * v1; zoom = 3.92; t_max = 64.1 * T; } // 11
129 if (!--k) { n = -1; v0 = 2.43 * v1; zoom = 4.29; t_max = 61.8 * T; } // 8
130 if (!--k) { n = -1; v0 = 2.74 * v1; zoom = 5.5; t_max = 85.5 * T; } // 5
131
132 if (!--k) { n = -0.74; v0 = 2.665 * v1; zoom = 3.81; t_max = 18.8 * T; } // 5
133
134 if (!--k) { n = 0; v0 = 1.66 * v1; zoom = 1.39; t_max = 5.1 * T; } // 7
135 if (!--k) { n = 0; v0 = 2.7 * v1; zoom = 2.34; t_max = 8.9 * T; } // 9
136 if (!--k) { n = 0; v0 = 3.46 * v1; zoom = 3.03; t_max = 13.3 * T; } // 11
137
138 if (!--k) { n = 1; v0 = 2 * v1; zoom = 1.3; t_max = 1 * T; } // 11
139
140 if (!--k) { n = 2; v0 = 2.39 * v1; zoom = 1.18; t_max = 4.0 * T; } // 11
141
142 if (!--k) { n = 3; v0 = 3.58 * v1; zoom = 1.28; t_max = 1.8 * T; } // 7
143 if (!--k) { n = 3; v0 = 6.97 * v1; zoom = 1.87; t_max = 1.7 * T; } // 9
144 if (!--k) { n = 3; v0 = 11.28 * v1; zoom = 2.13; t_max = 1.6 * T; } // 11
145 if (!--k) { n = 3; v0 = 16.13 * v1; zoom = 2.50; t_max = 1.6 * T; } // 13
146
147 if (!--k) { n = 3.29; v0 = 1.3 * v1; zoom = 0.48; t_max = 1.8 * T; } // 13
148
149 if (!--k) { n = 4; v0 = 3.36 * v1; zoom = 1.12; t_max = 1.1 * T; } // 5 !
150 if (!--k) { n = 4; v0 = 6.08 * v1; zoom = 1.44; t_max = 1.9 * T; } // 12
151 if (!--k) { n = 4; v0 = 9.52 * v1; zoom = 1.66; t_max = 0.9 * T; } // 7 !
152 if (!--k) { n = 4; v0 = 18.45 * v1; zoom = 2.18; t_max = 0.728 * T; } // 9
153
154 if (!--k) { n = 5; v0 = 7.15 * v1; zoom = 1.55; t_max = 0.6 * T; } // 5 !
155
156 if (!--k) { n = 4; v0 = 3.304 * v1; zoom = 1.12; t_max = 36.4 * T; } // 5 ~~~
157 if (!--k) { n = 4; v0 = 9.394 * v1; zoom = 1.66; t_max = 19.6 * T; } // 7 ~~~
158
159 if (N_exp)
160 {
161 Text_01.value = v0 / v1; Slider_01.value = Text_01.value; // initial velocity
162 Text_02.value = n; Slider_02.value = Text_02.value; // degree indicator
163 Text_03.value = zoom; Slider_03.value = Text_03.value; // zoom
164 Text_04.value = t_max / T; Slider_04.value = Text_04.value; // time for calculation
165 Text_05.value = N_exp; Slider_05.value = Text_05.value; // number of experiment
166 }
167
168 return -k; // Если N_exp = 0 returns total of experiments, in other cases returns 0
169 }
170 }
File "FC.html"
1 <canvas id="canvasGraph" width="600" height="600" style="border:1px solid #000000;"></canvas>
2
3 <!--Installation of parameters of interaction (text fields and sliders)-->
4 <div>
5 <font face= "Times New Roman"><I>
6 v</I><SUB>0</SUB> = <input id="Text_01" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
7 // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
8 if (!this.checkValidity()) return;
9 app.set_01(this.value);
10 document.getElementById('Slider_01').value = this.value;
11 "><I> v</I><SUB>1</SUB>
12 <input type="range" id="Slider_01" style="width: 100px;" oninput="app.set_01(this.value); document.getElementById('Text_01').value = this.value;">
13 n = <input id="Text_02" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
14 if (!this.checkValidity()) return;
15 app.set_02(this.value);
16 document.getElementById('Slider_02').value = this.value;
17 ">
18 <input type="range" id="Slider_02" style="width: 100px;" oninput="app.set_02(this.value); document.getElementById('Text_02').value = this.value;">
19 </font>
20 </div>
21 <div>
22 <font face= "Times New Roman">
23 zoom = <input id="Text_03" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
24 // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
25 if (!this.checkValidity()) return;
26 app.set_03(this.value);
27 document.getElementById('Slider_03').value = this.value;
28 ">
29 <input type="range" id="Slider_03" style="width: 100px;" oninput="app.set_03(this.value); document.getElementById('Text_03').value = this.value;">
30 <I>t</I><SUB>max</SUB> = <input id="Text_04" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
31 if (!this.checkValidity()) return;
32 app.set_04(this.value);
33 document.getElementById('Slider_04').value = this.value;
34 "><I> T</I>
35 <input type="range" id="Slider_04" style="width: 100px;" oninput="app.set_04(this.value); document.getElementById('Text_04').value = this.value;">
36 </font>
37 </div>
38 <div>
39 <font face= "Times New Roman">
40 Number of experiment <input id="Text_05" style="width: 4.2ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
41 // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
42 if (!this.checkValidity()) return;
43 app.set_05(this.value);
44 document.getElementById('Slider_05').value = this.value;
45 ">
46 <input type="range" id="Slider_05" style="width: 100px;" oninput="app.set_05(this.value); document.getElementById('Text_05').value = this.value;">
47 </div>
48
49 <script type="text/javascript">var app = new MainFC (
50 document.getElementById('canvasGraph')
51 );</script>
The application allows to set the following parameters interactively:
- is the initial velocity of the point. The initial velocity is directed perpendicular to the radial vector. Velocity is measured relatively to the circular speed at the initial distance from the center (the first cosmic speed).
- is the index of the interaction law ( corresponds to the gravitational interaction, is the elastic interaction).
- "zoom" is the logarithmic scale (the logarithm to base 2).
- is the integration time expressed in terms of periods of circular motion at the initial distance from the center.
Besides all, it is possible to set "a number of the experiment". Every number has a separate set of four parameters named above and characterized by a specific type of motion.
Research ideas
- To find all the possible trajectories
- To define the dependence of the distance to the epicenter (the most distant point) of the orbit on and .
- To find the closed trajectories of different topologies (for example, 5th, the 7th pointed stars, etc.) and to determine their position on the plane of parameters and .
- To find a method for integrating the equations of motion with a variable step allowing to model the motion of a point at a large values of effectively.
- Fit the parameters values (and develop a handy method to do so), to obtain "beautiful" curves, which can be considered as Science Art objects (see in particular, the last numbers of experiments).
