Re: Varying a constant in an ODE to Manipulate solution

*To*: mathgroup at smc.vnet.net*Subject*: [mg126970] Re: Varying a constant in an ODE to Manipulate solution*From*: Narasimham <mathma18 at hotmail.com>*Date*: Thu, 21 Jun 2012 05:18:08 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201206130857.EAA03730@smc.vnet.net> <jreout$nqu$1@smc.vnet.net> <jrlfnb$c2q$1@smc.vnet.net>

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 space -- > > 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, selecting some > > pair of the 3 components of the solution function? > > Yes, that was my query. Bob Hanlon pointed out my error. (Evaluating > together was needed). > > > Or a 3D-parametric plot of the trajectory over a fixed-duration time in= = > > terval, but dynamic > > > with c as control variable? > > Yes, in fact that is going to be my next question now, and thanks that > 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 Hanlon > > > > All variables or a single variable are easily pocked out from sol[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, Thick}, > > > AspectRatio -> Automatic, > > > PlotRange -> {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}] > > > Manipulate[ > > > Plot[Evaluate[sol[c][[1]]], {t, -3, 3}, PlotStyle -> {Red, Thi= ck= > }, > > > 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 -> Automatic, > > > PlotRange -> {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}] > > > -- > > Murray Eisenberg murrayeisenb..= .@= > gmail.com > > 80 Fearing Street phone 413 549-= 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 to gas law P V / T = const. when T is varied. The general form is more 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},Mesh- >{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,PlotRange->{{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- >Automatic,PlotRange->{{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 chosen domain) into a variable, from state X1 ( with one independent variable 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 continuous 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 well 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 to 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. The 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. Or 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,PlotRange->{{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- >Automatic,PlotRange->{{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!! *)

**References**:**Varying a constant in an ODE to Manipulate solution***From:*Narasimham <mathma18@hotmail.com>