How to use the variational method to solve this problem

Question

I see this mechanical problem here.

I want to solve this problem with the variational method. The Lagrangian of this system is obtained by subtracting potential energy from kinetic energy.

m = 1;
g = 9.8;
R = 1;
EulerEquations[
 m*g*R (1 - Cos[θ[t]]) - m*g*R*Cos[θ[t]], θ[
  t], t](*L=T-V or L=kinetic energy - potential energy*)

But the result is $19.6 (\sin (\theta [t]))=0$, which has no significance to solve this problem.

I know that by listing the Lagrangian of this system, we can get the correct equation of motion:

<< VariationalMethods`
L = 1/2 m (R θ'[t])^2 - m g R Cos[θ[t]];
EulerEquations[L, θ[t], t] // 
 FullSimplify[#, Assumptions -> R > 0 && m > 0] &

But before the object moves along the sphere and does not separate, the kinetic energy of the system $ m g R (1-\cos (\theta (t)))=\frac{1}{2} m \left(R \theta '(t)\right)^2$.

I want to know what mistakes I made in listing the following Lagrangian of this system(L= m*g*R (1 - Cos[θ[t]]) - m*g*R*Cos[θ[t]]), and how to use the variational method to solve this problem correctly.

Answer 1

This is the problem I solved sometime ago. Same problem as you show. The angle $\theta$ used is measured from x-axis, positive anti-clockwise, as it was simpler to do so, but it does not affect the solution ofcourse. Since there is no Mathematica stuff in this, I can add a Manipulate later on if needed to show the ball falling of?

There are two coordinates $r,\theta$ (polar) which is the position vector of the ball, and one constraint \begin{equation} f\left( r,\theta\right) =r-R=0 \tag{1} \end{equation}

$R$ above is the radius of the hemisphere.

Now we set up the equations of motion for $m$ \begin{align*} T & =\frac{1}{2}m\left( \dot{r}^{2}+r^{2}\dot{\theta}^{2}\right) \\ U & =mgr\sin\theta\\ L & =T-U\\ & =\frac{1}{2}m\left( \dot{r}^{2}+r^{2}\dot{\theta}^{2}\right) -mgr\sin\theta \end{align*} Hence the Euler-Lagrangian equations are (we need to add contraint)

\begin{align} \frac{d}{dt}\frac{\partial L}{\partial\dot{r}}-\frac{\partial L}{\partial r}+\lambda\frac{\partial f}{\partial r} & =0\tag{2}\\ \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}-\frac{\partial L} {\partial\theta}+\lambda\frac{\partial f}{\partial\theta} & =0 \tag{3} \end{align} But \begin{align*} \frac{d}{dt}\frac{\partial L}{\partial\dot{r}} & =m\ddot{r}\\ \frac{\partial L}{\partial\dot{\theta}} & =mr^{2}\dot{\theta}\\ \frac{d}{dt}\left( \frac{\partial L}{\partial\dot{\theta}}\right) & =m\left( 2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta}\right) \\ \frac{\partial L}{\partial r} & =mr\dot{\theta}^{2}-mg\sin\theta\\ \frac{\partial L}{\partial\theta} & =-mgr\cos\theta\\ \frac{\partial f}{\partial r} & =1\\ \frac{\partial f}{\partial\theta} & =0 \end{align*} Hence (2) becomes \begin{equation} m\ddot{r}-mr\dot{\theta}^{2}+mg\sin\theta+\lambda=0 \tag{4} \end{equation} And (3) becomes \begin{align} m\left( 2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta}\right) +mgr\cos\theta & =0\nonumber\\ r\ddot{\theta}+2\dot{r}\dot{\theta}+g\cos\theta & =0 \tag{5} \end{align} We now need to solve (1,4,5) for $\lambda$. Now we have to apply the constrain that $r=R$ in the above to be able to solve (4,5) equations. Therefore, (4,5) becomes \begin{align} -mR\dot{\theta}^{2}+mg\cos\theta+\lambda & =0\tag{4A}\\ R\ddot{\theta}+g\cos\theta & =0 \tag{5A} \end{align} Where (4A,5A) were obtained from (4,5) by replacing $r=R$ and $\dot{r}=0$ and $\ddot{r}=0$ since we are using that $r=R$ which is constant (the radius).

From (5A) we see that this can be integrated giving \begin{equation} R\dot{\theta}^{2}+2g\sin\theta+c=0 \tag{6} \end{equation} Where $c$ is constant. Since if we differentiate the above with time, we obtain \begin{align*} 2R\dot{\theta}\ddot{\theta}+2g\dot{\theta}\cos\theta & =0\\ R\ddot{\theta}+g\cos\theta & =0 \end{align*} Which is the same as (5A). Therefore from (6) we find $\dot{\theta}^{2}$ to use in (4A). Hence from (6) $$ \dot{\theta}^{2}=-2\frac{g}{R}\sin\theta+c $$ To find $c$ we use initial conditions. At $t=0$, $\theta=90^{0}$ and $\dot{\theta}\left( 0\right) =0$ hence $$ c=2\frac{g}{R} $$ Therefore \begin{align*} \dot{\theta}^{2} & =-2\frac{g}{R}\sin\theta+2\frac{g}{R}\\ & =2\frac{g}{R}\left( 1-\sin\theta\right) \end{align*} Plugging the above into (4A) in order to find $\lambda$ gives \begin{align*} -mR\left( 2\frac{g}{R}\left( 1-\sin\theta\right) \right) +mg\sin \theta+\lambda & =0\\ \lambda & =m\left( 2g\left( 1-\sin\theta\right) \right) -mg\sin\theta\\ \lambda & =2mg-2mg\sin\theta-mg\sin\theta\\ & =mg\left( 2-3\sin\theta\right) \end{align*} Now that we found $\lambda\,,$we can find the constraint force in the radial direction \begin{align*} N & =\lambda\frac{\partial f}{\partial r}\\ & =mg\left( 2-3\sin\theta\right) \end{align*} The particle will leave when $N=0$ which will happen when \begin{align*} 2-3\sin\theta & =0\\ \theta & =\sin^{-1}\left( \frac{2}{3}\right) \\ & =41.8^{0} \end{align*} Therefore, the angle from the vertical is $$ 90-41.8=48.2^{0} $$