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From: Gareth Russell on 3 Dec 2009 06:16
Hi,

Does anyone know how to use NDSolve to create the trajectory of motion
of a point following positive gradients on an arbitrary landscape (a
ListStreamPlot-like path), but with 'momentum'? The idea is that the
trajectory would not terminate on smaller local maxima, depending on
the momentum term.

I don't know exactly how ListStreamPlot does its thing -- I would guess
it interpolates the array of gradient pairs, and then uses NDSolve to
find the paths. If so, is there way to modify that basic idea to
include a momentum-like term? Or would I need to do a full physics-type
model as if I were modelling a ball with mass, gravity, friction, etc.?

Thanks,

--
Gareth Russell
NJIT


From: dh on 4 Dec 2009 04:31


Gareth Russell wrote:

> Hi,

>

> Does anyone know how to use NDSolve to create the trajectory of motion

> of a point following positive gradients on an arbitrary landscape (a

> ListStreamPlot-like path), but with 'momentum'? The idea is that the

> trajectory would not terminate on smaller local maxima, depending on

> the momentum term.

>

> I don't know exactly how ListStreamPlot does its thing -- I would guess

> it interpolates the array of gradient pairs, and then uses NDSolve to

> find the paths. If so, is there way to modify that basic idea to

> include a momentum-like term? Or would I need to do a full physics-type

> model as if I were modelling a ball with mass, gravity, friction, etc.?

>

> Thanks,

>

Hi Garteh,

if I understand correctly, you want something like simulated annealing.

Thsi can be donme by: NMinimize[f, vars, Method -> "SimulatedAnnealing"].

On the other hand, if you want the momentum-friction approach, here is

an example:



Clear[mass, friction, x, y, pos, position, pot, grad];

mass = 1;

friction = 0.3;

tmax = 10;

pot[pos_] := pos.pos;

grad[{x_, y_}] =

Evaluate[{D[pot[{#1, #2}], #1], D[pot[{#1, #2}], #2]}] & @@ {x, y};

position =

pos /. NDSolve[{-grad[pos[t]] - friction (pos'[t].pos'[t]) ==

pos''[t] mass, pos[0] == {1, 0}, pos'[0] == {0, 1}},

pos, {t, 0, tmax}][[1]]

ParametricPlot[position[t], {t, 0, tmax}, AspectRatio -> Automatic,

PlotRange -> All]



Daniel



From: Gareth Russell on 4 Dec 2009 04:38
Hi,

Thanks to the response from "Daniel", I was able to get the answer to
my own question, so I'm posting it here. Daniel provided the basic
model of motion (F=ma) implemented in NDSolve, with an example on a
parametric surface. I figured out how to apply it to to an arbitrary
surface that starts life as an array of height values.

I may have some of the physics oversimplified, esp. friction, but it's
ok, I'm not tring to actually model a ball rolling!

Gareth


Create an interesting surface, based on something in the help function,
plus a bowl shape to help the "ball" stay within the bounds.

ifun = ListInterpolation[RandomReal[1, {5,5,5,5}], {{0,1},{0,2},{0,3},{0,4}}];
integ = Integrate[ifun[x1, x2, x3, x4], {x1,0,1},{x4,0,4}];
array=Table[integ,{x2,0,2,0.01},{x3,0,3,0.01}];
bowlArray=Table[{x,y}.{x,y},{x,-1,1,0.01},{y,-1,1,0.006666666}];
smoothSurface=ListInterpolation[array+bowlArray,{{0,2},{0,2}}];
Plot3D[smoothSurface[x,y],{x,0,2},{y,0,2}]

Create derivative surfaces in the x and y directions.

dX=Derivative[1,0][smoothSurface];
dY=Derivative[0,1][smoothSurface];

Clear some variables and set up constants

Clear[mass,friction,x,y,pos,position,pot,grad];
mass=1;
friction=0.2;
gravity=9.81;

A function that returns the gradient at a point by sampling from the
two surfaces. This was used in Daniel's original version.

grad[{x_,y_}]={dX[x,y],dY[x,y]};

For a moving ball, though, there is a problem in that there should be a
maximum acceleration due to gravity even at an infinitely steep
gradient. A function that (I think) converts the gradient into a force
under the principle that F=m g where g=0.981.

grav[{x_,y_}]=gravity*Sin[2*Pi+Map[ArcTan,{dX[x,y],dY[x,y]}]];

A function used to plot the trajectory on the surface.

trajectory3D[t_]:=Append[position[t],Apply[smoothSurface,position[t]]]

Simulate the dynamics!

tmax;
position=pos/.NDSolve[{-grav[pos[t]]-pos'[t]*0.5==pos''[t]
mass,pos[0]=={1.2,1.7},pos'[0]=={1,0}},pos,{t,0,tmax},MaxSteps->100000][[1]];

Plot

trajectory on surface

Show[Plot3D[smoothSurface[x,y],{x,0,2},{y,0,2}],ParametricPlot3D[trajectory3D[t],{t,0,tmax},PlotStyle->Thickness[0.003]],ViewPoint->{1.3,-2.4,2}]
trajectory3D[tmax]

A

Manipulate[] function to animate the rolling ball.

Manipulate[Show[Plot3D[smoothSurface[x,y],{x,0,2},{y,0,2}],ListPointPlot3D[{trajectory3D[t]+{0,0,0.2}},PlotStyle->PointSize[0.05]],Axes->False,Boxed->False,ViewPoint->{1.3,-2.4,2}],{t,0,tmax}]


On

2009-12-03 06:16:16 -0500, Gareth Russell <russell(a)njit.edu> said:

> Hi,
>
> Does anyone know how to use NDSolve to create the trajectory of motion
> of a point following positive gradients on an arbitrary landscape (a
> ListStreamPlot-like path), but with 'momentum'? The idea is that the
> trajectory would not terminate on smaller local maxima, depending on
> the momentum term.
>
> I don't know exactly how ListStreamPlot does its thing -- I would guess
> it interpolates the array of gradient pairs, and then uses NDSolve to
> find the paths. If so, is there way to modify that basic idea to
> include a momentum-like term? Or would I need to do a full physics-type
> model as if I were modelling a ball with mass, gravity, friction, etc.?
>
> Thanks,


--
Gareth Russell
NJIT


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