Solve Mercury's Perihelion Shift with Euler-Lagrange Equation

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In summary, the conversation discusses the process of calculating S in order to determine the perihelion shift of Mercury. The Euler Lagrange equation, which is a form of the Hamilton-Jacobi equation, is used for this calculation. The individual is unsure of how to begin the calculation and is seeking help, specifically with the substitution process in the equation E+Gm/r(1-ro/2r)=0.
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LoopQG
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Homework Statement



I need to calculate S in order to calculate the perihelion shift of Mercury. I have found the Euler Lagrange equation to be:

[tex] G^{\nu\beta}(\partial_{\beta}S)(\partial_{\nu}S)-m^{2}=0 [/tex]

Which is a form of the Hamilton-Jacobi equation.

Which my professor tells me is correct. I am just not sure on how to start calculating S. I have been given [tex] G^{00},G^{11},G^{22},G^{33} [/tex] Do I just plug those into solve for S? I think I am just confused on how to start I think. Any help appreciated.
 
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space dynamics

I am working out a problem on orbital motion: Thus finding the minimum kinetic energy required for a comet to change the new orbit to a parabola after colliding with another planet of a bigger mass.

My idea is first setting the condition for a parabolic orbit: thus e=1
and using the equation E+Gm/r(1-ro/2r)=0
I am expand these equation further by making a few substitutions can i get help in continuing this? Thanks
 

FAQ: Solve Mercury's Perihelion Shift with Euler-Lagrange Equation

What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical formula used in classical mechanics to describe the motion of a system. It is derived from the principle of least action, which states that the actual path taken by a system is the one that minimizes the action, a quantity that combines the system's kinetic and potential energies.

How does the Euler-Lagrange equation relate to Mercury's perihelion shift?

The Euler-Lagrange equation can be used to calculate the motion of any system, including the orbit of a planet like Mercury. By applying this equation to Mercury's orbit, scientists were able to accurately predict and explain the observed perihelion shift, which is the gradual rotation of Mercury's orbit around the sun over time.

Why is Mercury's perihelion shift important to study?

Mercury's perihelion shift is important to study because it was one of the first pieces of evidence that supported Einstein's theory of general relativity. The observed shift could not be explained by Newtonian mechanics, but the application of the Euler-Lagrange equation in conjunction with general relativity accurately predicts and explains this phenomenon.

What other applications does the Euler-Lagrange equation have?

The Euler-Lagrange equation has many applications in physics, including in the study of fluid mechanics, electromagnetism, and quantum mechanics. It is also used in engineering and economics to optimize systems and minimize energy consumption.

Is the Euler-Lagrange equation the only way to solve Mercury's perihelion shift?

No, there are other methods and equations that can be used to solve Mercury's perihelion shift, such as the Schwarzschild solution of Einstein's field equations. However, the Euler-Lagrange equation is a useful and widely applicable tool for solving problems in classical mechanics and has been successfully applied to explain Mercury's perihelion shift.

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