Solving equations for a specified variable in terms of other variables

I am trying to solve euqation (13) and (16) as listed on this website https://www.myphysicslab.com/pendulum/double-pendulum-en.html

This is what I have attempted so far however I am running into errors

syms g x1 y1 x2 y2 l1 l2 m1 m2 theta1 theta2 omega1 omega2 alpha1 alpha2 dy1 dy2 dx1 dx2 d2y1d d2y2 d2x1 d2x2

x1= l1*sin(theta1);
y1= -l1*cos(theta1);
x2 = x1+l2*sin(theta2);
y2 = y1-l2*cos(theta2);

dx1 = omega1*l1*cos(theta1);
dy1 = omega1*l1*sin(theta1);
dx2 = dx1 + omega2*l2*cos(theta2);
dy2 = dy1 + omega2*l2*sin(theta2);

d2x1 = -omega1^2*l1*sin(theta1)+alpha1*l1*cos(theta1);
d2y1 = omega1^2*l1*cos(theta1)+alpha1*l1*sin(theta1);
d2x2 = d2x1-omega2^2*l2*sin(theta2)+alpha2*l2*cos(theta2);
d2y2 = d2y1+omega2^2*l2*cos(theta2)+alpha2*l2*sin(theta2);

solve('sin(theta1)*(m1*d2y1+m2*d2y2+m2*g+m1*g)==-cos(theta1)*(m1*d2x1+m2*d2x2)', 'alpha1')
solve('sin(theta2)*(m2*d2y2+m2*g)==-cos(theta2)*m2*d2x2', 'alpha2')

Any help is much appreciated

Answer 1

[Eugenio Grabovic]

 syms g   l1 l2 m1 m2 theta1 theta2 omega1 omega2 alpha1 alpha2 
x1= l1*sin(theta1);
y1= -l1*cos(theta1);
x2 = x1+l2*sin(theta2);
y2 = y1-l2*cos(theta2);
dx1 = omega1*l1*cos(theta1);
dy1 = omega1*l1*sin(theta1);
dx2 = dx1 + omega2*l2*cos(theta2);
dy2 = dy1 + omega2*l2*sin(theta2);
d2x1 = -omega1^2*l1*sin(theta1)+alpha1*l1*cos(theta1);
d2y1 = omega1^2*l1*cos(theta1)+alpha1*l1*sin(theta1);
d2x2 = d2x1-omega2^2*l2*sin(theta2)+alpha2*l2*cos(theta2);
d2y2 = d2y1+omega2^2*l2*cos(theta2)+alpha2*l2*sin(theta2);
eq1 = sin(theta1)*(m1*d2y1+m2*d2y2+m2*g+m1*g)==-cos(theta1)*(m1*d2x1+m2*d2x2);
eq2 =  sin(theta2)*(m2*d2y2+m2*g)==-cos(theta2)*m2*d2x2;
solutions = solve([eq1 eq2],[alpha1 alpha2])

If u check solver helper its written that it doesn't accept char vectors anymore and requires just symbolycs. Should be working now. In addition a system of equations requires to be solved within the same solver.

[Alastair Poore]

When using your edit it returns ans=alpha1 and ans=alpha2, whereas I'm looking for it to return alpha1 and alpha2 in terms of all the other variables (as given on the website I linked).

[madhan ravi]

which release are you using?

[Alastair Poore]

R2018a

[Eugenio Grabovic]

"When using your edit it returns ans=alpha1 and ans=alpha2, whereas I'm looking for it to return alpha1 and alpha2 in terms of all the other variables (as given on the website I linked)."

My apologies, i din't actually verify the results. I edited my main answer with the solution.

[Alastair Poore]

After trying your code I am now recieving these error messages

[Alastair Poore]

The first error message finishes with "is not valid equation or expression"

Missed it off my screenshot sorry

[Eugenio Grabovic]

solutions = 
  struct with fields:

    alpha1: [1×1 sym]
    alpha2: [1×1 sym]
    
>> solutions.alpha1
ans =
-(g*m1*cos(theta2)^2*sin(theta1) + g*m2*cos(theta2)^2*sin(theta1) + g*m1*sin(theta1)*sin(theta2)^2 - l2*m2*omega2^2*cos(theta1)*sin(theta2)^3 + l2*m2*omega2^2*cos(theta2)^3*sin(theta1) - g*m2*cos(theta1)*cos(theta2)*sin(theta2) - l1*m2*omega1^2*cos(theta1)*sin(theta1)*sin(theta2)^2 + l1*m2*omega1^2*cos(theta2)*sin(theta1)^2*sin(theta2) + l2*m2*omega2^2*cos(theta2)*sin(theta1)*sin(theta2)^2 + l1*m2*omega1^2*cos(theta1)*cos(theta2)^2*sin(theta1) - l1*m2*omega1^2*cos(theta1)^2*cos(theta2)*sin(theta2) - l2*m2*omega2^2*cos(theta1)*cos(theta2)^2*sin(theta2))/(l1*(m1*cos(theta1)^2*cos(theta2)^2 + m1*cos(theta1)^2*sin(theta2)^2 + m1*cos(theta2)^2*sin(theta1)^2 + m2*cos(theta1)^2*sin(theta2)^2 + m2*cos(theta2)^2*sin(theta1)^2 + m1*sin(theta1)^2*sin(theta2)^2 - 2*m2*cos(theta1)*cos(theta2)*sin(theta1)*sin(theta2)))
>> solutions.alpha2
ans =
-(g*m1*cos(theta1)^2*sin(theta2) + g*m2*cos(theta1)^2*sin(theta2) - l1*m1*omega1^2*cos(theta2)*sin(theta1)^3 + l1*m1*omega1^2*cos(theta1)^3*sin(theta2) - l1*m2*omega1^2*cos(theta2)*sin(theta1)^3 + l1*m2*omega1^2*cos(theta1)^3*sin(theta2) - g*m1*cos(theta1)*cos(theta2)*sin(theta1) - g*m2*cos(theta1)*cos(theta2)*sin(theta1) + l1*m1*omega1^2*cos(theta1)*sin(theta1)^2*sin(theta2) + l1*m2*omega1^2*cos(theta1)*sin(theta1)^2*sin(theta2) + l2*m2*omega2^2*cos(theta1)*sin(theta1)*sin(theta2)^2 - l2*m2*omega2^2*cos(theta2)*sin(theta1)^2*sin(theta2) - l1*m1*omega1^2*cos(theta1)^2*cos(theta2)*sin(theta1) - l1*m2*omega1^2*cos(theta1)^2*cos(theta2)*sin(theta1) - l2*m2*omega2^2*cos(theta1)*cos(theta2)^2*sin(theta1) + l2*m2*omega2^2*cos(theta1)^2*cos(theta2)*sin(theta2))/(l2*m1*cos(theta1)^2*cos(theta2)^2 + l2*m1*cos(theta1)^2*sin(theta2)^2 + l2*m1*cos(theta2)^2*sin(theta1)^2 + l2*m2*cos(theta1)^2*sin(theta2)^2 + l2*m2*cos(theta2)^2*sin(theta1)^2 + l2*m1*sin(theta1)^2*sin(theta2)^2 - 2*l2*m2*cos(theta1)*cos(theta2)*sin(theta1)*sin(theta2))

This is what i am getting after running it. The only difference is that I am using R2018b... im sorry i actually don't know where is the problem then.

[madhan ravi]

+1 Eugenio , appreciate your effort,

you can also change your last line to:

[alpha1 alpha2] = solve([eq1 eq2],[alpha1 alpha2])

[Alastair Poore]

It's ok. Thanks for trying, ill try figure another way out