Solve Ball Thrown Up in Air: Stationary Action Min

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In summary, the forum poster has correctly used the Euler-Lagrange equation and the action equation to find the equation of motion and calculate the value of the action for a ball thrown up in the air. To show that the stationary value of the action is a global minimum, they can use the principle of least action and consider a different path for the ball, showing that the parabolic path is indeed the one that minimizes the action.
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Homework Statement



For a ball thrown up in the air show that the stationary value of the action is always a global minimum

Homework Equations



euler lagrange equation and action equation(sorry i don't know syntax here)

The Attempt at a Solution


well basically here i used the euler lagrange equation where L=T-V and by using the principle of stationary action figured out that a=g...by solving the differential equation i found a equaiton of motion for y(t). Thus i sub this into the action equation of S=integral(L)dt and integrate it getting the answer mg^2t^3/6+g^2T^3/2 . and this is where I'm stuck ...i don't know how to show whether this is a global minimum or nor not and also if even my initial steps are correct. If someone could please point me in the right direction it would be great. Thanks.
 
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Thank you for sharing your solution so far. It seems that you have correctly used the Euler-Lagrange equation to find the equation of motion for the ball and the action equation to calculate the value of the action. To determine whether this value is a global minimum, we can use the concept of the principle of least action.

The principle of least action states that the actual path taken by a particle between two points in space and time is the one that minimizes the action. In this case, the ball is thrown up in the air and comes back down to the same point, so the two points are the same. This means that the actual path taken by the ball is the one that minimizes the action.

To show that the stationary value of the action is a global minimum, we need to show that any other path taken by the ball will result in a greater value for the action. In other words, we need to show that any other path will require more energy to be expended by the ball, which is not possible since the ball is moving under the influence of a conservative force (gravity).

To prove this, you can consider a different path for the ball, such as one that is not a parabola. You can then use the same steps you used before to find the equation of motion and calculate the action for this path. You will find that the action for this path is greater than the action for the parabolic path, thus showing that the parabolic path is indeed the path that minimizes the action.

I hope this helps guide you in the right direction. Keep up the good work!
 

1. What is the "Solve Ball Thrown Up in Air: Stationary Action Min" problem?

The "Solve Ball Thrown Up in Air: Stationary Action Min" problem is a physics problem that involves finding the minimum action needed for a ball thrown up in the air to reach its highest point and return to the ground. It is typically solved using the principle of least action, which states that the actual path taken by a system between two points is the one that minimizes the action, or the integral of the system's Lagrangian over time.

2. What is the principle of least action?

The principle of least action is a fundamental principle in classical mechanics that states that the actual path taken by a system between two points is the one that minimizes the action, or the integral of the system's Lagrangian over time. This principle is used to solve various physics problems, including the "Solve Ball Thrown Up in Air: Stationary Action Min" problem.

3. How is the "Solve Ball Thrown Up in Air: Stationary Action Min" problem solved?

The "Solve Ball Thrown Up in Air: Stationary Action Min" problem is typically solved using the principle of least action. This involves setting up the Lagrangian of the system, which takes into account the kinetic and potential energies of the ball, and then using the Euler-Lagrange equations to find the path that minimizes the action. This path will be the one that the ball follows in order to reach its highest point and return to the ground.

4. What are the key variables and assumptions in the "Solve Ball Thrown Up in Air: Stationary Action Min" problem?

The key variables in the "Solve Ball Thrown Up in Air: Stationary Action Min" problem are the initial velocity of the ball, the height at which it is thrown, and the acceleration due to gravity. The assumptions made in this problem include neglecting air resistance and assuming a flat and uniform gravitational field.

5. What are some real-world applications of the "Solve Ball Thrown Up in Air: Stationary Action Min" problem?

The "Solve Ball Thrown Up in Air: Stationary Action Min" problem has various real-world applications, including calculating the trajectory of a rocket launch, predicting the motion of a thrown ball in sports, and understanding the behavior of celestial bodies in space. It is also used in engineering and design to optimize the motion of objects and systems.

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