ATRACTOR

The applet consists of what we call the viewing area (with a gray background) and menus with buttons, text boxes and value control bars (with a blue background). However, it is worth noting that, as we will see later, in some cases it is possible to also enter data directly into the display area.

Click on the image where you have questions.

Note: The following figures are non-manipulable images. In the figures marked with , if you hover the mouse over them you will see an animation.

The display zone is divided into seven areas:

• in the two leftmost areas are representations of the potential, kinetic and total energies of the pendulum:
  • at each instant, in the upper area.

  Dragging the mouse in this area vertically allows you to zoom;

  • drawing the graphs as a function of time, in the lower area.

• in the central area:
  • on the top is a pendulum formed by a ball of mass m, suspended by a rigid rod of length L (supposed to have mass 0); the pendulum is assumed to be subject to friction and the action of gravity, with values q and g, respectively. Drag the ball with the mouse to the desired starting position;
  • below is the graph of the function x(t) where x(t) represents the measurement (in R) of the angle between the pendulum and the vertical direction at time t.

• in the remaining three areas on the right are represented (from top to bottom, respectively):
  • in the plane, the phase curves and vector fields for the system of differential equations that describe the motion of this pendulum. The green dot determines the position and speed of the pendulum at each moment. Click on this area (since the cursor has the form : more details) to see the phase curve starting at that point;

   

  • the phase curves and the vector field, but now represented on the cylinder (which corresponds to parameterizing the pendulum angle based on a circle and not a straight line);

• the graph of the function x(t) where x(t) is the point on the circle corresponding to the angle of the pendulum at time t (note that in this graph, the coordinate axis corresponding to the independent variable t is precisely the axis of symmetry of the cylinder).

The initial parameters of the pendulum can be modified:

• in the case of the value of the initial angle between the pendulum and the vertical direction,
  • directly in the text box ;
  • or dragging the pendulum ball;
  • or by clicking on one of the two areas in the upper right part of the applet (as long as the cursor has the shape : more details). The x-coordinate of the green point determines the new value of x(0).

• in the case of the initial velocity value,
  • directly in the text box ;
  • or by clicking on one of the two areas at the top right (as long as the cursor has the shape : more details). The y-coordinate of the green point determines the new value of x'(0).

• in the case of length (L), mass (m), friction(q) and gravity (g) values,
  • entering the values in the text boxes;

  

  • dragging the control bars.

  

Once these values have been chosen, click on the button , to see the simulation of the pendulum movement.

The menu has the following options:

• : to reset the time value (t = 0);
• : to return to the initial configuration.

The following options, when selected, allow you to see:

• :
  • the phase curves for the system of differential equations that describe the movement of this pendulum;
• :
  • the vector fields for the system of differential equations that describe the movement of this pendulum;
• 
  • the velocity vector (in yellow) of the pendulum (without friction) at each moment;
• 
  • forces acting on the pendulum (without friction) at each moment
    • centrifugal force () ;
    • weight () :
      • radial component () ;
      • tangential component () ;
    • sum of external radial and centrifugal forces;
    • symmetric of the previous vector (=bar reaction force).

Important note: to choose a point in one of the two areas in the upper right part of the applet, always check that the mouse cursor has the following shape:. If it has another shape, like , click first. If it is in the shape and you want to change to the shape simply right-click and in the window that appears choose the Rotate, Orbit option.

In the five areas at the bottom and right of the applet, try clicking on them with the right mouse button: you get a menu with other options for manipulating objects.

Among the options, it allows you to have a resizable window of the area where you clicked - for more information, click here.

The flag, at the top of the applet, indicates the language used. If you wish, click on it and select another language:

By clicking , you can choose between:

• : access this instruction page;
• : accesses information about the applet version*.