[help]how to obtain gravity eqution from gravity field Action S?

In summary, the conversation is about obtaining the gravity equation from the gravity field action, specifically through the use of Hamilton's principle and the calculus of variations. The Lagrangian and Euler-Lagrange equations play a key role in this derivation, and the equation for two particles of mass m and M is also mentioned. The speaker also requests any additional papers or resources that could aid in their understanding of this topic.
  • #1
zhushualbert
4
0
could you please tell me how to obtain gravity equation from gravity field Action S as in detail as possible, especially the derivation of calculus of variations?
or you could point out some papers for me,
and thanks very very very very very very very very very much!
p.s. as in detail as possible! thanks!
 
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  • #2
The action is given by the integral S = ∫ L(q,v) dt, whre L is a function of coordinates and velocities called the 'Lagrangian'. Hamilton's principle states that this integral is stationary for the actual motion of the system (or minimum for a sufficiently small segment of the motion). Using the calculus of variations, one can show that this leads to the differential equation

(d/dt)(∂L/∂v) = ∂L/∂q (Euler-Lagrange)

(for each coordinate q and and it's time derivative v).

For two particles of mass m and M, the lagrangian is

[tex] \mathcal{L} = \frac{mv_1^2}{2} + \frac{Mv_2^2}{2} + \frac{GmM}{|r_1 - r_2|} [/tex]

Use this in the Euler-Lagrange equations to get the differential equations of motion.
 
  • #3
I kown what you said, what's important for me is the derivation from the gravity field action(Einstain-hiber action) to gravity field equation(einstain eqution).
Thank you all the same!
 

What is a gravity field Action S?

A gravity field Action S is a mathematical representation of the gravitational force in a specific area or region. It takes into account the mass and distance of objects to calculate the strength of the gravitational force at any given point.

Why is it important to obtain the gravity equation from a gravity field Action S?

The gravity equation allows scientists to accurately measure and predict the effects of gravity in a particular area. This can be helpful in understanding the behavior of objects in space, as well as in designing structures and vehicles that can withstand the force of gravity.

What factors are considered when obtaining the gravity equation from a gravity field Action S?

The gravity equation takes into account the mass of objects and the distance between them. It also considers the gravitational constant, which is a universal value that represents the strength of gravity.

How is the gravity equation derived from a gravity field Action S?

The gravity equation is derived using mathematical principles, such as calculus and Newton's law of universal gravitation. By plugging in the relevant values for mass, distance, and the gravitational constant, the equation can be calculated.

Can the gravity equation from a gravity field Action S be used for all objects and situations?

While the gravity equation is a fundamental formula for calculating the force of gravity, it may not be applicable in all situations. For example, it does not take into account the effects of relativity or quantum mechanics, which may be necessary for extremely large or small objects. Additionally, the equation may need to be modified for non-uniform or complex gravitational fields.

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