Solving Problem 3.5 from Hartle's "Gravity

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In summary, the problem involves finding the action of a system and using the Lagrangian equation to solve for the second derivative of the function. The resulting exponential behaviour is expected and the initial conditions are used to determine the constants of integration in the general solution.
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M-Speezy
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I was working problem 3.5 out of Hartle's "Gravity" and have come to a bit of confusion. The problem states that the action for some system is given by the integral from 0 to T of d(x(t)/dt squared plus x(t) squared. My next step was to take the Lagrangian and set it equal to zero. Doing this, I found that the second derivative of x was equal to negative x, which is suggestive of exponential behavior. I then assumed the form e^x and solved for the initial conditions listed (x(0)=0, and x(T)=1). Is this all sound? Something about it seems off, but I'm not sure where it is that I may have taken a misstep.

Thanks!
 
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  • #2
Yes, your steps are sound. Exponential behaviour is expected when you have a second derivative equal to a negative multiple of the function itself. However, you need to be careful when solving for the initial conditions. Since you have assumed that the form of the solution is exponential, you can only solve for the constants of integration when you have the initial conditions. That is, you should use the initial conditions to determine the constants of integration and then find the general solution.
 

1. How do I approach solving Problem 3.5 from Hartle's "Gravity"?

In order to solve Problem 3.5 from Hartle's "Gravity", it is important to first understand the concepts and equations related to the problem. Make sure to read the problem carefully and identify all the given information. Then, use the appropriate equations and mathematical techniques to solve the problem step by step.

2. What is the significance of Problem 3.5 in Hartle's "Gravity"?

Problem 3.5 in Hartle's "Gravity" is significant because it allows for the application of important concepts and equations related to Newtonian gravity. It also introduces the concept of escape velocity and its relationship to the mass and radius of a planet.

3. How can I check my solution to Problem 3.5 from Hartle's "Gravity"?

You can check your solution to Problem 3.5 from Hartle's "Gravity" by plugging your answers back into the original equations and seeing if they hold true. You can also compare your solution to the expected answer, if provided, or consult with a classmate or instructor for verification.

4. What are some common mistakes to avoid when solving Problem 3.5 from Hartle's "Gravity"?

Some common mistakes to avoid when solving Problem 3.5 from Hartle's "Gravity" include using incorrect equations or not correctly identifying all the given information. It is also important to check your units and perform all calculations accurately.

5. Can I apply the concepts learned from solving Problem 3.5 to other problems in "Gravity" by Hartle?

Yes, the concepts and equations used to solve Problem 3.5 from Hartle's "Gravity" can be applied to other problems in the textbook. It is important to understand the fundamentals and build upon them when tackling more complex problems.

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