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Use results of NDSolve for other computations?

Posted 1 month ago
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I have been working on a model for a pendulum(using a Lagrangian) and I am using NDSolve. The solver gives me outputs for alpha an l yet I canĀ“t us these to recalculate x and y.

In[2]:= ClearAll["Global`*"]

Definition of parameters

In[3]:= R = 0
Subscript[l, 0] = 0.5
m = 1
M = 2.394
g = 9.81
Subscript[\[Alpha], 0] = 1/2*Pi

Expression of position(x,y) as a function of [Alpha]

In[9]:= x[\[Alpha]_, l_, t_] := -Cos[\[Alpha][t]]*R + 
  Sin[\[Alpha][t]]*(l[t] - \[Alpha][t]*R)
y[\[Alpha]_, l_, t_] := 
 Sin[\[Alpha][t]]*R + Cos[\[Alpha][t]]*(l[t] - \[Alpha][t]*R)

Calculation of the components and the norm of the velocity Computation of kinetic energy, potential energy and Lagrangian

In[11]:= dx[\[Alpha]_, l_, t_] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\((x[\[Alpha], l, t])\)\)
dy[\[Alpha]_, l_, t_] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\((y[\[Alpha], l, t])\)\)
v[\[Alpha]_, l_, t_] := 
  Sqrt[dx[\[Alpha], l, t]^2 + dy[\[Alpha], l, t]^2] // FullSimplify;
Ekin[\[Alpha]_, l_, t_] := 1/2*m*v[\[Alpha], l, t]^2 + 1/2*M*(l'[t])^2
Epot[\[Alpha]_, l_, t_] := m*g*y[\[Alpha], l, t] + M*g (l[t] - Subscript[l, 0])
L1[\[Alpha]_, l_, t_] := Ekin[\[Alpha], l, t] - Epot[\[Alpha], l, t]

Euler-Lagrange equation

In[17]:= eq1[\[Alpha]_, l_, t_] = 
  D[D[L1[\[Alpha], l, t], \[Alpha]'[t]], t] - 
    D[L1[\[Alpha], l, t], \[Alpha][t]] // FullSimplify;
eq2[\[Alpha]_, l_, t_] = 
  D[D[L1[\[Alpha], l, t], l'[t]], t] - D[L1[\[Alpha], l, t], l[t]] // 
   FullSimplify;
eq = {eq1[\[Alpha], l, t] == 0, eq2[\[Alpha], l, t] == 0}

Out[19]= {l[t] (-9.81 Sin[\[Alpha][t]] + 
     2 Derivative[1][l][t] Derivative[1][\[Alpha]][t] + 
     l[t] (\[Alpha]^\[Prime]\[Prime])[t]) == 0, 
 23.4851 + 9.81 Cos[\[Alpha][t]] - l[t] Derivative[1][\[Alpha]][t]^2 + 
   3.394 (l^\[Prime]\[Prime])[t] == 0}

In[20]:= 

Solution of equation of motion [AliasDelimiter]with given initial conditions

In[22]:= sol = First@
  NDSolve[{eq, \[Alpha][0] == Subscript[\[Alpha], 0], \[Alpha]'[0] == 0, 
    l[0] == Subscript[l, 0], l'[0] == 0}, {\[Alpha][t], l[t]}, {t, 0, 20}]

Plot

In[23]:= Plot[\[Alpha][t] /. sol, {t, 0, 20}]

In[24]:= Plot[l[t] /. sol, {t, 0, 20}]

In[29]:= 
ParametricPlot[{x[\[Alpha], t]/.sol, y[\[Alpha], t]/.sol}, {t, 0, 20}, PlotRange -> All]
4 Replies

You can try NDSolveValue instead of NDSolve. It is easier to use. Be careful how you copy-and-paste your code. The way you did it is useless.

It's better to post your code using the code sample icon (the first icon)

I found my problem. x and y are functions of alpha, t and l. I forgot to add the l

Welcome to Wolfram Community! Please make sure you know the rules: https://wolfr.am/READ-1ST

The rules explain how to format your code properly. If you do not format code, it may become corrupted and useless to other members. Please EDIT your post and make sure code blocks start on a new paragraph and look framed and colored like this.

int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]

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