Samples

 

This software comes with the folder Samples containing numerous ready to use samples of Initial Value Problems (IVPs) in files of two kinds:

 

Certain scripts can be loaded directly from Demo by choosing the desired sample via a hierarchical menu – this is the recommended way for first exploration of the software. 

 

The selection of the samples covers various branches of applied mathematics: from purely parametric curves (where the ODE section contains merely a trivial ODE  t'=1) to many samples in celestial mechanics, general mechanics (single and double pendulums, rolling disk), examples of special functions such as the Bessel function (which are regular, but their ODEs are singular), and some others examples. 

 

The full description for all of them (called Exploratorium) is in preparation.

 

The samples run via the main menu item Demo


 

Spirals    ®      Cornu

Curls

Double spiral (as an x, y-extraction of the complex tangent. Each lap takes the same time to run when Played)

 

3 Bodies  ®      Lagrange case  ®        1 IVP

                                                            3 IVPs

 

Disturbed          ®        2D

                                    3D

 

Periodic (closed            I2A "Butterfly"

 chain-like orbits)       VIII15B "Spaghetti"    
                                                 . . . . .

                                     A list of 203

 

Periodic (open               #1

free fall orbits              #18

with rest points)           A list of 30

 

Relatively periodic                     

 

Choreographies             #1   (eight)      

                                    #2
                                                #3

                                    . . . . . .

                                    A list of 345

 

Non-Newtonian ®        Precession        ®        Power -1    > -1.5 (Newtonian)

                                                                        Power -1.7 < -1.5 (Newtonian)

                                    Choreography   ®        Power -1    > -1.5 (Newtonian)

                                                                        Power -1.7 < -1.5 (Newtonian)  

4 Bodies (3D) ®          Torus-like trajectories  

                                    Trajectories inscribed in a cube

 

7 Bodies 5 Lagrange points ®   m6 = 0.1,     m7 = 0.9

                                               m6 = 0.001, m7 = 0.999 

Möbius outline (3D)

Lorenz attractor (3D)

Linked tori (3D)

Rolling disk (3D)  (visualized with several fixed points in it)

 

These and many other samples may be opened either as ODEs or scripts from the File menu navigating to the subfolder Samples.

 

The folder Samples contains subfolders considered separately:

 

SpecPoints – containing samples of integration beginning with a singular point of ODEs yet having a regular solution, for which the Taylor expansion in at the special point was brought from other sources.


Periodic – containing samples of Choreographies and other special periodic cases of the 3 body problem.

 

LagrangePointsWorkshop – containing the algebraic equations whose solutions are the Lagrange points, and ODEs for the inverse functions whose integration delivers those points.  

 

Here is the list and minimal explanations for the 72 samples in the folders. For every script file with the extension .scr there always exists the respective file .ode (not listed here if it accompanies the scr file). Go to File menu to…

 


1) Go to Demo/3 bodies/... and one of the lists.
 
2) Enter a number of the desired sample. The program automatically loads the ODEs for the 3 body problem with the initial values corresponding to the selected orbit, compiles, and opens the Graph window for visual integration. At that, the length of the period of this sample is entered into the termination condition and the program is set for integration until this termination point. (The period of the orbit is visible also in the Front panel in the Constant section as a comment line for constant  a).

3) By default, in the Graph window you will see already a result of integration of 10 steps. Typically 10 steps are not enough for reaching the terminal point. You may need to click the button More several times changing the number of steps to something big, say 1000. For some samples with long periods you may need more than 1000 steps. (By default the storage is reserved for 1000 points, therefore when the program asks you to append the storage, do it adding say 3000 or so). You may notice that graph has not enough room (in default setting). Any moment you can adjust the room clicking at the button Adjust.

4) Finally when you have reached the termination point (the period) obtaining the full graph of this orbit, you may wish to Play it dynamically. The default time 5 sec is too short. Enter a longer interval. Depending on the complexity of the curve, it may be something like 60-80 seconds. Enjoy the show, and then repeat everything from step 1 for another sample.

5) If you like the curves that you have obtained and played, you may wish to save this session as the script so that next time they integrate ready for playing automatically.  In order to do this, in the label below
note how many steps n the integration took till reaching the termination point. Enter this n under the button More, proceed to the Main Panel and under File Save as Script into your location of choice (available only in the licensed version). Next time you will be able to open this script and play it.


Many of the samples below and few others are considered in a frame of teaching here and here.

 

Pure parametric curves samples      

 

CornuSpiral.scr

Cornu spiral

CurlySpiral.scr

Curves looking like curls

Mobius.scr   

Möbius outline (3D)

MobiusLarge.scr

Möbius outline of a larger stripe

MobiusArray.ode

Möbius outlineas as an array of IVPs (must be unfolded)

Tori.scr 

Linked tori outlined with screw lines (3D)

ToriArray.ode

Linked tori outlined with latitudes and longitudes as an array of IVPs (must be unfolded)

TrefoilKnot3D.scr

Trefoil knot (3D)

KnotChain3D.scr

Knot chain (3D)

CoshVsPar.scr

The hyperbolic cosine vs. a parabola

 

 

Mechanics

 

Pendulums

 

Pendulum2D.scr

The mathematical pendulum 2D

DoubePendulum.scr

The double pendulum 2D

PendulumApple.scr

The mathematical pendulum 3D

PendulumFlower.scr

The mathematical pendulum 3D

  

Rolling disk

Credit to Prof. Dmitry Garanin (page 25-26)

 

RollingDisk.scr

Rolls along a straight line

RollingDisk01.scr

Rolls along a circular line

RollingDisk01-03.scr

Rolls along multiple circular lines

RollingDisk1-03.scr

Rolls chaotically

RollingDisk0103.scr

Rolls along a semi-arch

 

 

The Newtonian n-body problems

 

2 bodies

 

2BodiesProbe.scr

the simplest 2 body problem one of which is a probe (of near zero mass)

2BodiesProbe3D.scr

the same as above in 3D

2plusProbe.ode

2 bodies and a probe (of near zero mass) in the linear formation

2plusSmall.ode 

2 bodies and a small body in the linear formation

 

 

3 bodies

 

3Bodies2D.scr

The Lagrange case

3Bodies2D3IVPs.scr

The Lagrange case, 3 aggregated IVPs with different initial velocities

3Bod9995.scr

A slightly disturbed Lagrange case – a chaotic dance of three

3EqBodEuler.scr

The Euler case of 3 bodies of equal masses in linear formation

3NonEqBodEuler.scr

The Euler case of 3 bodies with the small central mass

(folder Samples/Periodic)

Simo1.scr
Simo2.scr
Simo3.scr
...............

Scripts of 7 choreographies from the 345 Carles Simo's list.
All 345 of them are available as .ode files to be compiled and played as explained here.

(folder Samples/Periodic)

I2A.scr
HudomalVIII15B.scr

Scripts of 2 periodic orbtis from the list of 204 by Ana Hudomal.
All 204 of them are available as .ode files to be compiled and played as explained here.

 

4 bodies

 

4BodiesCubic.scr

Four bodies in 3D space each having near a plane orbit inscribed in a cube (credit to Cris Moore & Michael Nauenberg)

4BodiesTorus.scr

Four bodies in 3D space whose orbits outline a flattened torus

4BodiesTetra.scr

Four bodies in 3D space placed at vertices of the regular tetrahedron. It was proved that only with radial initial velocities the formation (similar to the Lagrange case in plane) may hold (until the collision or escape)

4BodiesTetraCollision.scr

Four bodies in 3D space placed at vertices of the regular tetrahedron with zero velocities on collision course

4BodiesPlane.scr

Four bodies in plane (the Lagrange case)

 

   

5 bodies

5Bodies3D.scr

Five bodies in 3D in the Lagrange formation plane

5BodiesEuler.scr

Five bodies (including 2 probes) in 3D in the linear Euler formation in plane

 

 

7  bodies at the 5 Lagrange points

5 probes at the 5 Lagrange points with different proportions of the masses m6 and m7

 

LagrangePoints001-999.scr

m6 = 0.001; m7 =0.999

LagrangePoints05-95.scr

m6 = 0.05;   m7 =0.95

LagrangePoints10-90.scr

m6 = 0.1;     m7 =0.9

LagrangePoints50-50.scr

m6 = 0.5;     m7 =0.5

 

 

20 bodies

 

20BodiesCirc.scr

20 bodies Lagrange case circular

20BodiesEll.scr 

five bodies Lagrange case elliptic

 

 

Non-Newtonian 3 body problems for the central force

other than the inverse square law whose exponent in the ODEs is  1.5 = 0.5 – 2

 

NonNewton-0.5.scr

Exponent = -0.5, precession of the orbits

NonNewton-1.0.scr

Exponent = -1, precession of the orbits

NonNewton-1.7.scr

Exponent = -1.7, precession of the orbits

NonNewton-1.0Choreo.scr

Exponent = -1, orbits get closed as a choreography

NonNewton-1.7Choreo.scr

Exponent = -1.7, orbits get closed as a choreography

 

 

A chemical reaction 

 

WineGlass.scr

Oscillating chemical reactions with a graph looking like a wineglass (courtesy of the late Prof. Robert L. Borrelli, Courtney S. Coleman)

 

 

Special functions

 

Bessel2.scr

The Bessel function J2 from the initial value x0=0.00001>0 because the Bessel ODE is singular at x=0. The initial value of the function at point x0 is computed (in the section Constants) employing a segment of the special Taylor expansion for J2 at zero. There are more examples of the Bessel functions in the folder SpecPoints from the initial point x=0 utilizing the special expansions of Jp at zero.

LorenzAttractor.scr

Lorenz attractor (3D)

DoubleSpiral.scr

Double spiral is an x, y -extraction of the complex tangent. Each lap takes the same time to run when Played

Harley_t.ode

HugeRad_y7x5_in_t.ode

xm/m + yn/n = const  for even m, n (studied by the late Prof. Harley Flanders) – "flattened circle-like" curves. They are helpful for integration with the associated ODEs in x and y. They demonstrated near infinite heuristic convergence radius.

EqSumTo3Points_x.ode

A curve equidistant to three points: requires switching to y in order to close the curve  

Xyyx.scr

The commuting powers solution y(x) (satisfying  xy=yx ) vs. special hyperbola. In this example the holomorphic solution y(x) integrated from the initial point x0=2  jumps over the point of singularity x=y=e of the ODE

 

 

Holomorphic functions – solutions of singular ODEs

(folder Samples/SpecPoints)

 

This folder contains a selection of samples all illustrating the case when singular rational ODEs have holomorphic solutions. For some ODEs it was proven that no rational regular ODE at this special point can have those functions as solutions because the point is a special point of violation of elementariness by those functions. For some other ODEs there is no such proof, and it is not known yet whether regular rational ODEs may have these solutions at the special points – see here and here.   

 

Bernoulli.scr

Integration of the function t / (et -1) begins with t=0, which is possible because the special expansion at this point is provided and automatically entered (as in the samples below).

Bessel-0-2.scr

In all the Bessel functions here of the index 0-4 integration begins with t=0. The digit "2" in the names means that the special expansion was provided for the derivative of order 2 rather than for the function proper.  

 

Bessel-1-2.scr

Bessel-2-2.scr

Bessel-3-2.scr

Bessel-4-2.scr

Cauchyn!.ode

Cauchyn!array10.scr

This is the only example where also the solution is singular at t=0 in the Cauchy ODE  t2x' + (t-1)x + 1 = 0 whose formal Taylor coefficient at t=0 is an = n!

cos(sqrt(t))-2.scr

 Function cos(sqrt(t))

InvBernoulli.scr

 Function (et -1) / t

Ln(tplus1)divt.scr

 (Ln(t + 1)) / t

sin(t)Divt.scr

 sin(t) / t

x^y=y^x1.scr

Various forms of the function of the commuting power y(x) satisfying the equation xy=yx 

 

x^y=y^x2.scr

x^y=y^xParam.scr

x2=xDivt.scr

 Solution of the equation  x'' = x/t

z^(y^x)=(z^y)^x.scr

Various forms of the associating powers satisfying the equation  z^(y^x)=(z^y)^x 

z^(y^x)=(z^y)^x1.ode

 

 See more explanations here.