The concept of a warp drive has been the stuff of science fiction for the longest time.. But all that is changing with the Alcubierre Drive. In 2000 Alcubierre posted the first paper on the matter and it's been an active area of research ever since. The Alcubierre (or "warp drive") metric is usually discussed in the context of an ADM formalism.

Ch. 1, Problem 8 - Data Reduction and Error Analysis for the Physical Sciences by Philip Bevington:

Justify the second equality in equations (1.8) and (1.14).

Solution:

In both cases this is a straight forward calculation using the distributive property of discrete and continuous (or integral) summation. Observe for equation (1.8),

\[\lim_{N \to \infty} \frac{1}{N} \sum (x_i - \mu)^2\]

\[= \lim_{N \to \infty} \frac{1}{N} \sum ({x_i}^2 - 2 x_i \mu + {\mu}^2)\]

Ch. 1, Problems 1, 2, and 3 - Data Reduction and Error Analysis for the Physical Sciences by Philip Bevington:

1. How many significant figures are there in the following numbers?
   1. 976.45
   2. 84,000
   3. 0.0094
   4. 301.07
   5. 4.000
   6. 10
   7. 5280
   8. 400
2. What is the most significant figure in each of the numbers? What is the least significant?
3. Rround off each of the numbers above to two significant digits.

I've found an excellent discussion on Galilean spacetime from a geometric viewpoint. Everything is written up in a pdf that I found on this site. The mathematical development is set forth in Chapter 1 of the course notes. The notes have apparently formed the basis for a textbook on Geometric Control theory and are no longer being maintained. I'm afraid at some point these notes will disappear so I'm attaching the pdf on the Galilean development here in this blog.

Ch. 1, Problem 2 - Mathematical Methods of Classical Mechanics (V.I. Arnold): Show that if a mechanical system consists of only one point, then its acceleration in an inertial coordinate system is equal to zero ("Newton's first law"). Hint. By examples 1 and 2 the acceleration vector does not depend on \(x, \dot{x}, or t\), and by example 3 the vector \(F\) is invariant with respect to rotation.

Solution:

We start with \[F(x, \dot{x}, t) = m\ddot{x}\] associated with the motion of one point.