Re: Varying a constant in an ODE to Manipulate solution

*To*: mathgroup at smc.vnet.net*Subject*: [mg127012] Re: Varying a constant in an ODE to Manipulate solution*From*: Narasimham <mathma18 at hotmail.com>*Date*: Sat, 23 Jun 2012 04:18:11 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201206130857.EAA03730@smc.vnet.net> <jreout$nqu$1@smc.vnet.net>

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 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, 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 -> Automatic, > > > > PlotRange -> {{-3, 3}, {-10, 10}}], {c, -0.5, 2, 0.2}] > > > > -- > > > Murray Eisenberg murrayeisenb= ..= > .@= > > gmail.com > > > 80 Fearing Street phone 413 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 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,Plo= 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->Automatic,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 chose= n > 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,Plo= 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->Automatic,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}},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,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

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