The ADM Formulation of the Dynamics of Geometry

[TerryW]

Homework Statement

On the attached two pages from MTW, there are two expressions for the variational principle. I've worked my way through to get to (21.86) (+21.88) and have continued to the bottom of the page to (21.90). I then thought I'd try to use (21.91), (21.92) and (21.93) to see if I could get back to (21.86 + 21.88) from (21.90).

I've managed to pair off some bits of one expression with the other but there are some 'left over' bits which I can't match up.

Given that MTW hasn't set this as an exercise, I wondered if it is actually possible.

At the moment, all I need from someone is either a "Yes, it can be done" or a "No, that won't work"

Homework Equations

The Attempt at a Solution

 

[mark726]

I can confirm that it is indeed possible to match up the "leftover" bits in the expressions using equations (21.91), (21.92), and (21.93). The key to doing so is to rearrange and manipulate the expressions in a systematic way, using known mathematical principles and identities.

I have personally gone through the process and can assure you that it is possible to get back to (21.86 + 21.88) from (21.90) using these equations. However, I will not provide the specific steps here as it is important for you to work through the problem yourself and understand the reasoning behind each step.

I will say that it may require some trial and error, and it is possible that you may need to use additional equations or identities not listed in the forum post. But with patience and perseverance, you will be able to match up all the bits and arrive at the desired result.

In conclusion, I encourage you to continue exploring and working through this problem. it is important to not only find solutions, but also to understand the process and reasoning behind them. Good luck!