## Ordinary Differential Equations

The program is able to display solutions ("orbits") of initial value problems for ODE of first or second order in one, two, or three dimensions. There is an ODE submenu of the Category menu to choose the order and number of dependent variables. Keep in mind that when working with the three dimensional categories, you will see what is going on much better if you switch to stereo (by selecting Stereo Vision from the View menu).

After you have chosen a particular category of ODE and selected a particular pre-programmed example from the Main menu (or set up your own using the User Defined... feature) you can select "ODE Settings..." from the Settings menu, and this will bring up a dialog that will allow you to set various data needed to compute and display an orbit, namely:

• the initial time,
• the time-span,
• the step-size (used for Runge-Kutta discretization of the problem),
• the initial values off x, y, and z.
• dot spacing (the time between showing successive dots along the orbit).

Normally, for two-dimensional systems, the orbit is drawn dotted, with a constant time interval between dots. This gives a valuable visual clue concerning the velocity at which the orbit is traced out, but if you wish to turn this feature off, just set Dot Spacing to zero. Choosing Create from the Main menu will then display the solution of the currently selected differential equation.

When you select a pre-programmed ODE from the Main menu, then as usual, the program will choose reasonable default values of the initial time, the time-span, the step-size, the parameters, and the initial values of x,y,z and immediately draw an orbit.

In any of the ODE categories the main display shows the evolution of an orbit in the phase space. By default, (except for a first order equation in one variable) the program also shows projections of the orbit on the various coordinate axes (using different colors to distinguish the projections). This display occurs in a second pane of the graphics window that opens automatically below the main pane. You can switch between showing and hiding this pane by selecing "Project ODE Orbits" from the Action menu.. There is a rectangular button at the right edge of the screen where the two panes meet. If you press on this button, the button itself will disappear and be replaced by a horizontal line. Drag the horizontal line to where you would like the new boundary between panes and release the mouse. (At least twenty percent of the total screen height must be devoted to each pane.)

For a first order ODE category in one or two variables and a second order ODE in one variable the program by default displays the direction field of the vector field defining the ODE. This display of the direction field can be turned on or off with in the Action menu. Fields submenu. (For the case of a first order ODE in three variables, showing the direction field is also possible when the orbit is being displayed in stereo vision. For second order ODE in three variables, for the special case of a charged particle in a magnetic field, it is possible to display the magnetic field by choosing Show Direction Field in the Action menu, but note, it is the direction of the magnetic field, not that of the Lorentz force that is displayed.

Note that, while most of the pre-programmed ODE are autonomous (i.e., their "right-hand sides" are not functions of the time t) the Runge-Kutta integration is set up to handle non-autonomous ODE too, and you may use t in the formulas of the User Defined... dialog box. (You can even show the direction field for the time-varying case.)

It is fairly easy to do a rough "phase space analysis" by keeping the other data fixed and varying the initial values. This is particularly easy in the 2-dimensional case. There, if you select "IC by mouse" from the main menu, you will be able to click the mouse at a point on the screen, and the program will figure out the x and y coordinates of the mouse location and use these as the initial conditions for an orbit that it will immediately draw. (To speed things up even more, you can type Option-Command-C instead of selecting "IC by mouse" from the menu).

An ODE control panel opens by default just below the Graphics Window. This has buttons to do more easily things you can also do with menu selections (Create, Erase, Continue, double or half the scale, and bring up the dialog to set initial conditions, step-size, time-span, and dottedness. In addition there are buttons for single-stepping the ODE forward or backward, and there is a read-out of the current time, position, and velocity. This control panel can be made visible or invisible by selecting Show (or Hide) ODE Controls in the Action menu