Lagrangian/Stationary-action principle

[Lambda96]

Homework Statement

Find a stationary solution

Relevant Equations

Lagrangian $L=x\dot{x}(2\alpha - \dot{x})$

Hi

the text is unfortunately a bit longer, but I have included it anyway to provide all the information.

I am not sure if I understood the task b correctly, but I understood it as follows: using the equation $H(x,\dot{x},t)$, solve the differential equation for $x(t)$.

Once you have found $x(t)$, you have to use it to find the statin solutions, i.e. form the first derivative of $x(t)$ and set it to 0.

I then first formed $H(x,\dot{x},t)$ and got the following $H(x,\dot{x},t)=-x\dot{x}^2$

And $H(x,\dot{x},t)=c$ which leads to $c=-x\dot{x}^2$

After that, I did the separation of variables,

$$\sqrt{x}dx=-\sqrt{c}dt$$

Then I integrated both sides and solved for $x(t)$ and got the following.

$$x(t)=\sqrt[3]{\frac{9}{4}(a-\sqrt{c}*t)^2}$$

$a$ is the constant of integration.

Now, to get the stationary solution, I have to form $\frac{dx(t)}{dt}=0$.

Have I done everything right up to this point, or have I completely misunderstood the task?

 

[vanhees71]

I guess they meant to find the solution of the Euler-Lagrange equations, which makes the "action functional" stationary. I've not checked your calculation in detail, but it looks reasonable.

 

[Lambda96]

Thank you vanhees71 for your help

I then determined the following form with the help of the initial condition $x(t_i)=0$.

$$x(t)=\frac{9c}{4}^{1/3}(t_i-t)^{1/3}$$

In task c you had to derive the Euler Lagrange equation and put $x(t)$ in there, and with the above result I get 0.

Unfortunately, I can't get any further with task d.

I would have simply calculated the above functional with $S[x]= \int_{t_i}^{t_f} L(x,\dot{x})dt$ with $x(t)$, $\dot{x}(t)$ which I got in task b but Q does not appear at all when I solve the integral?

 

[vanhees71]

I don't understand (d).

Note that the piece
$$\tilde{L}=2 \alpha x \dot{x}=\frac{\mathrm{d}}{\mathrm{d} t} (\alpha x^2).$$
Check, what this term contributes to the equations of motion!

 

[Lambda96]

Thank you vanhees71 for your help

I had already thought of that, that I form the antiderivatives of $S=2x\dot{x}\alpha-x\dot{x}^2$ and just put $x(t)$ there, unfortunately the expression $x\dot{x}^2$ gives me problems, because I can't find the antiderivatives for this expression, even with wolframalpha I can't find any.

 

[vanhees71]

I wanted to hint you at something else. Think about a Lagrangian of the form
$$L=\frac{\mathrm{d}}{\mathrm{d} t} f(x,t)=\dot{x} \partial_x f(x,t) + \partial_t f(x,t).$$
What are the Euler-Lagrange equations for such a Lagrangian?