Lagrangian/Stationary-action principle

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In summary: Thank you vanhees71 for your help 👍In summary, the person was trying to find the statin solutions using the equation ##H(x,\dot{x},t)## and the statin solutions were found to be ##x(t)=\sqrt[3]{\frac{9}{4}(a-\sqrt{c}*t)^2}##.
  • #1
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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.

Bildschirmfoto 2022-11-29 um 19.17.39.png

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?
 
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  • #2
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.
 
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  • #3
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.

Bildschirmfoto 2022-11-30 um 20.01.15.png

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?
 
  • #4
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!
 
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  • #5
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.
 
  • #6
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?
 
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What is the Lagrangian/Stationary-action principle?

The Lagrangian/Stationary-action principle is a fundamental principle in physics that states that the path taken by a physical system between two points in time is the one that minimizes the action, which is the integral of the Lagrangian over time. In simpler terms, it is a mathematical tool used to find the most likely path that a system will take between two points.

How is the Lagrangian/Stationary-action principle used in physics?

The Lagrangian/Stationary-action principle is used in various fields of physics, such as classical mechanics, quantum mechanics, and electromagnetism. It is used to derive equations of motion, predict the behavior of particles and fields, and solve problems in different systems.

What is the difference between the Lagrangian and the action?

The Lagrangian is a function that describes the dynamics of a system, while the action is a quantity that represents the total energy of the system over a period of time. The action is the integral of the Lagrangian over time, and it is used in the Lagrangian/Stationary-action principle to find the path of a system.

What are the advantages of using the Lagrangian/Stationary-action principle?

The Lagrangian/Stationary-action principle has several advantages, such as providing a more elegant and concise way of describing the dynamics of a system compared to other methods. It also allows for a more straightforward and systematic approach to solving complex physical problems and can be applied to a wide range of systems and phenomena.

Are there any limitations to the Lagrangian/Stationary-action principle?

While the Lagrangian/Stationary-action principle is a powerful tool in physics, it also has its limitations. It may not be applicable to systems with non-conservative forces or systems with infinite degrees of freedom. It also requires knowledge of the Lagrangian function, which may be challenging to determine in some cases.

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