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Re: Varying a constant in an ODE to Manipulate solution
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127133] Re: Varying a constant in an ODE to Manipulate solution
*From*: Narasimham <mathma18 at hotmail.com>
*Date*: Mon, 2 Jul 2012 05:23:55 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <201206130857.EAA03730@smc.vnet.net> <jreout$nqu$1@smc.vnet.net>
On Jul 1, 11:14 am, Narasimham <mathm... at hotmail.com> wrote:
> On Jun 23, 1:26 pm, Narasimham <mathm... at hotmail.com> wrote:> On Jun 21, =
2:20 pm, Narasimham <mathm... at hotmail.com> wrote:
>
> > > On Jun 18, 1:46 am, Narasimham <mathm... at hotmail.com> wrote:
>
> > > > On Jun 15, 12:41 pm, Murray Eisenberg <mur... at math.umass.edu> wrote=
:
>
> > > > > It's not clear to me what you want to do.
>
> > > > I hope this reply clarifies, and my broader question too.
>
> > > > > Do you want to create 3D-parametric plot of the trajectory in spa=
ce--
> > > > > possibly as a dynamic with the time t as dynamic parameter?
>
> > > > Yes. I can get this plot OK simply as:
>
> > > > Manipulate[ParametricPlot3D[Evaluate[sol[c][[1]]], {t, -3, 3},
> > > > PlotStyle -> {Red, Thick},
> > > > AspectRatio -> Automatic, PlotRange -> {{-3, 3}, {-10, 10}}],=
{c,
> > > > -0.5, 2, 0.2}]
>
> > > > Also I have asked in a separate but related question now under
> > > > clearance by moderators
> > > > regarding Clairaut's Equation_
>
> > > > How to include traces of these lines in the plot and not have them
> > > > flying in the 3space
> > > > on the Manipulate command. Call them t parameter lines as t is
> > > > continuous and c is discrete.
>
> > > > > And/or do one of those same things but do it just in 2D, selectin=
g one
> > > > > pair of the 3 components of the solution function?
>
> > > > Yes, that was my query. Bob Hanlon pointed out my error. (Evaluatin=
g
> > > > together was needed).
>
> > > > > Or a 3D-parametric plot of the trajectory over a fixed-duration t=
ime in
>
> > > > terval, but dynamic
>
> > > > > with c as control variable?
>
> > > > Yes, in fact that is going to be my next question now, and thanks t=
hat
> > > > you have foreseen it !
>
> > > > Call these lines c parameter lines as c would be the continuous
> > > > control variable and t's would be discrete.
>
> > > > In this case, each time dot is extruded or curvilinearly dragged
> > > > across the surface, cutting the
>
> > > > t lines. Bringing together action of control variables t and c, we
> > > > have a surface sol[c_,t_] which
>
> > > > can be computed and plotted. t and c are two parameters for this
> > > > patch.
>
> > > > How can this be done? I am almost sure this is doable. I shall try =
for
> > > > Clairaut's 2D problem and send result when done.
>
> > > > Regards
>
> > > > Narasimham
> > > > > On 6/13/12 4:57 AM, Narasimham wrote:
> > > > > > Same topic is continued. Thanks to Murray Eisenberg and Bob Han=
lon
>
> > > > > > All variables or a single variable are easily pocked out from s=
ol[c_]
> > > > > > list for plotting.
> > > > > > But how to pick out two out of them for ParametricPlot ( 2D)=
?
>
> > > > > > sol[c_] := {x[t], y[t], z[t]} /.
> > > > > > First[NDSolve[{y''[t] + Sin[y[t]/c] == 0, y'[0] ==
=
> 0,
> > > > > > y[0] == 1/(1 + c), x'[t] == t, x[0] == =
c^2,
> > > > > > z'[t] == 2 c x[t] - y[t], z[0] == 2}, {x, y=
, z}, {t, -3, 3}]]
> > > > > > Manipulate[
> > > > > > Plot[Evaluate[sol[c]], {t, -3, 3}, PlotStyle -> {Red, Th=
ick},
> > > > > > AspectRatio -> Automatic,
> > > > > > PlotRange -> {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}=
]
> > > > > > Manipulate[
> > > > > > Plot[Evaluate[sol[c][[1]]], {t, -3, 3}, PlotStyle -> {Re=
d, Thick},
> > > > > > AspectRatio -> Automatic,
> > > > > > PlotRange -> {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}=
]
> > > > > > " 2 parameter Dynamic manipulation not OK"
> > > > > > Manipulate[
> > > > > > Plot[{Evaluate[sol[c][[1]]], Evaluate[sol[c][[1]]]}, {t, -3=
, 3}
> > ,
> > > > > > PlotStyle -> {Green, Thick}, AspectRatio -> Automa=
ti=
> c,
> > > > > > PlotRange -> {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}=
]
>
> > > > > --
> > > > > Murray Eisenberg murrayei=
se=
> nb=
> > ..=
> > > .@=
> > > > gmail.com
> > > > > 80 Fearing Street phone 41=
3 =
> 54=
> > 9-=
> > > 10=
> > > > 20 (H)
> > > > > Amherst, MA 01002-1912
>
> > > With some conviction I make the following suggestion.
>
> > > { a Cos[t], a Sin[t] } with perturbation of a gives an annular ring.
> > > Boyles Law P*V = const for given const temp T becomes generalised t=
o
> > > gas law P V / T = const. when T is varied. The general form is m=
or=
> e
> > > value in scientific work.
>
> > > The same case illustrates my point.
>
> > > c=1 ;
> > > CL1=Manipulate[ParametricPlot3D[{c Cos[t],0.6 t,c Sin[t]},{t,
> > > 0,Pi},PlotStyle->{Red,Thick},PlotRange->{{-1.5,1.5},{0,2},{0,1.5}}],{=
c,
> > > 1,1.5,.1}]
> > > CL2=ParametricPlot3D[{c Cos[t],0.6 t,c Sin[t]},{t,0,Pi},{c,1,1.5},M=
es=
> h-=
> > >{15,4}]
>
> > > sol[c_]:={x[t],y[t],z[t]}/.First[NDSolve[{y''[t]+Sin[y[t]/
> > > c]==0,y'[0]==0,y[0]==1/(1+c),x'[t]==t,x[0]==c^2,z=
'[=
> t]=
> > =
> > > ==2 c x[ t]-
> > > y[t],z[0]==2},{x,y,z},{t,-3,3}]];
>
> > > DIFF1=
> > > Manipulate[ ParametricPlot3D[Evaluate[{sol[c][[1]] ,sol[c][[2]] ,
> > > sol[c][[3]] }],{t,-2,2},PlotStyle->{Red,Thick},AspectRatio->Automatic=
,P=
> lo=
>
> > tRange->{{0,5},{-2.5,2.5}, {-2.5,2.5}},PlotRange-> >{{0,5},{-2.5,2.5=
},=
>
> {-2.5,2.5}},PlotLabel->Style[Framed[ "c PARAMETRIC
>
> > > LINES EXIST BUT INVISIBLE"] ]],{c,-0.5,2,0.2}]
>
> > > DIFF2=
> > > ParametricPlot3D[Evaluate[{sol[c][[1]] ,sol[c][[2]] , sol[c][[3]] =
}]=
> ,
> > > {t,-2,2},{c,-0.5,2,0.2}, PlotStyle->{Red,Thick},AspectRatio->Automati=
c,=
> Pl=
>
> > otRange->{{0,5},{-2.5,2.5}, {-2.5,2.5}},PlotRange-
>
> > > >{{0,5},{-2.5,2.5}, {-2.5,2.5}}]
>
> > > (* the [% ] above would certainly produce error diagnostics as of
> > > now!! *)
>
> > > Is it possible to push up or upgrade a constant (along with a c=
ho=
> se=
> > n
> > > domain) into a variable, from state X1 ( with one independent variabl=
e
> > > t ) to CL2 ( with two independent variables t and c) thereby
> > > spreading/dragging/extruding a line into a surface?
>
> > > It is possible to generalize both in closed form (analytically
> > > definable functions )and in differential form (for smooth continuo=
us
> > > manifolds ) with automatically generated boundary conditions added ,
> > > from surfaces to solids/volumes and further on to 4D objects .. by
> > > proper assembly or stacking into 3- space objects.
>
> > > It is possible to implement a form in CL2 for every form CL1 to
> > > accommodate c seeding among determining differential equations as wel=
l
> > > as in Boundary Conditions as an add-on/ generalization function by
> > > variation of parameter c, sharing same boundary conditions between
> > > them.
>
> > > The manner of range specification in code available to the user is=
t=
> o
> > > put c and t ranges simply together inside square bracket rather than
> > > on either side of the closing square bracket.
>
> > > So I tend to suggest/ request for making a provision, incorporating
> > > whatever it takes to realize it, to enable plot of DIFF2 without
> > > error diagnostics and output the same DIFF1 as DIFF2 but now as a
> > > surface .
>
> > > The logic is the same, to me at least, in such a result depiction. Th=
e
> > > DIFF2 should show _a surface_ just as CL2 does.
>
> > > This may seem too long drawn out, A request for its acceptance or
> > > implementation in further Mathematica development may look a tall
> > > order, I took liberty to make the comment. Please comment about your
> > > view and problems involved in what looks as a crass generalization. O=
r
> > > a simple code to make it routinely available be please indicated.
>
> > > Regards
> > > Narasimham
>
> > > sol[c_]:={x[t],y[t],z[t]}/.First[NDSolve[{y''[t]+Sin[y[t]/
> > > c]==0,y'[0]==0,y[0]==1/(1+c),x'[t]==t,x[0]==c^2,z=
'[=
> t]=
> > =
> > > ==2 c x[ t]-
> > > y[t],z[0]==2},{x,y,z},{t,-3,3}]];
> > > DIFF1=
> > > Manipulate[ ParametricPlot3D[Evaluate[{sol[c][[1]] ,sol[c][[2]] ,
> > > sol[c][[3]] }],{t,-2,2},PlotStyle->{Red,Thick},AspectRatio->Automatic=
,P=
> lo=
>
> > tRange->{{0,5},{-2.5,2.5}, {-2.5,2.5}},PlotRange-> >{{0,5},{-2.5,2.5=
},=
>
> {-2.5,2.5}},PlotLabel->Style[Framed[ "c PARAMETRIC
>
> > > LINES EXIST BUT INVISIBLE"] ]],{c,-0.5,2,0.2}]
>
> > > DIFF2=
> > > ParametricPlot3D[Evaluate[{sol[c][[1]] ,sol[c][[2]] , sol[c][[3]] =
}]=
> ,
> > > {t,-2,2},{c,-0.5,2,0.2}, PlotStyle->{Red,Thick},AspectRatio->Automati=
c,=
> Pl=
>
> > otRange->{{0,5},{-2.5,2.5}, {-2.5,2.5}},PlotRange-> >{{0,5},{-2.5,2.=
5}=
> , {-2.5,2.5}}]
>
> > > (* the % would certainly produce error diagnostics!! *)tRange->{{0,5}=
,{=
> -2=
>
> > .5,2.5}, {-2.5,2.5}},PlotRange-> >{{0,5},{-2.5,2.5}, {-2.5,2.5}},Plo=
tL=
>
> abel->Style[Framed[ "c PARAMETRIC
>
> > > LINES EXIST BUT INVISIBLE"] ]],{c,-0.5,2,0.2}]
> > > DIFF2=
> > > ParametricPlot3D[Evaluate[{sol[c][[1]] ,sol[c][[2]] , sol[c][[3]] =
}]=
> ,
> > > {t,-2,2},{c,-0.5,2,0.2}, PlotStyle->{Red,Thick},AspectRatio->Automati=
c,=
> Pl=
>
> > =
>
> > otRange->{{0,5},{-2.5,2.5}, {-2.5,2.5}},PlotRange-
>
> > > >{{0,5},{-2.5,2.5}, {-2.5,2.5}}]
> > > (* the % would certainly produce error diagnostics!! *)
>
> > If DIFF2 cannot work without error diagnostics (regarding pushing
> > domain of c inclusively into square brackets together with domain of
> > t), please indicate a dynamic adapter code module lines that may be
> > appended so that one need not write own code lines with Tooltip or
> > whatever it takes to compute/display the generalized function [t,c].
>
> > Took liberty to suggest this as the prospect of computing and
> > viewing generalized functions (numerically at least to begin with)
> > whereby multiple constants can get embedded into 3-space and
> > even into a muti-dimensional spaces by the power of functional
> > definition flexibility and graphical depiction in Mathematica, is
> > is indeed exciting.
>
> > Regards
> > Narasimham
>
> The suggestion actually is for solid modeling,could not express it
> earlier.
>
> To appreciate the invariantly fixing up any one among three variables
> and making the
> three component surfaces which can be morphed/ Manipulated/ clubbed
> back/integrated
> in 3D solid modelling,I have written this code ( fwiw, instructional)
> in mathematical solid
> parametric 3D modeling thus:
>
> c = 1.2; " 3 PARAM DECOMPN THICK CYL Solid modelling "
> " Take two out of 3 parameters {a,t,v } at a time for surface \
> generation and
> one out three parameters at a time for single generating parameter =
\
> line. "
> Clear[a, t, v] ;
> " vary v & t. It makes a cylinder surface, Dilatation of each cyl \
> surface at a =1 generates a solid. "
> a = 1.;
> aa = ParametricPlot3D[ {a Cos[t + v], c t , a Sin[t + v] }, {t, 0,
> 2 Pi}, { v, 0, 2 Pi}, Mesh -> {18, 4}] ;
> at = ParametricPlot3D[ {a Cos[t + 2.5], c t , a Sin[t + 2.5] }, {t,
> 0, 2 Pi}, PlotStyle -> {Orange, Thick}] ;
> Clear[a, t, v] ;
> " vary a & t. It makes a helical fin surface, entire fin rigid body
> \
> Rotation at v = Pi /2 generates a solid. "
> v = Pi/2. ;
> vv = ParametricPlot3D[ {a Cos[t + v], c t , a Sin[t + v] }, {t, 0,
> 2 Pi}, { a, 1, 1.6}, Mesh -> {18, 4}] ;
> va = ParametricPlot3D[ {a Cos[4 + v], c 4. , a Sin[4 + v] }, {=
a,
> 1, 1.6}, PlotStyle -> {Orange, Thick}] ;
> Clear[a, t, v] ;
> " vary v & a. It makes an annulus surface , entire annulus \
> Translation at t = Pi generates a solid. "
> t = Pi ;
> tt = ParametricPlot3D[ {a Cos[t + v], c t , a Sin[t + v] }, { v, 0,
> 2 Pi}, {a, 1, 1.6}, Mesh -> {18, 4}] ;
> tv = ParametricPlot3D[ {1.25 Cos[t + v], c t ,
> 1.25 Sin[t + v] }, { v, 0, 2 Pi}, PlotStyle -> {Orange, Thick}=
] ;
> Surfs = Show[{aa, vv, tt}, PlotRange -> All, Axes -> None,
> Boxed -> False]
> Lines = Show[{at, va, tv}, PlotRange -> All, Axes -> None,
> Boxed -> False]
> Show[Surfs, Lines]
>
> I hope the need for development of general command in solid
> modeling is clearer from above closed parametric form example.
>
> Regards
> Narasimham
I imagined a 3 parameter inclusive command like :) :
Manipulate[ ParametricSolid[ { f[x[u, v, w], y[u, v, w], z[u, v,
w]] }, {u, 0.1, 3}, {v, .8, 1.2}, {| w, -3, 3, .2 |},Mesh -> {20, 20,
5}, PlotRange -> {{-0, 20}, {-10, 10}, {-10, 10}}], Appearance ->
"Labeled", ControlPlacement -> Left]
where domain of w may be indicated with a special symbol to indicate
its discrete character, u and v being continuous.
Narasimham
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