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Solutions to Problems in Goldstein,
Classical Mechanics, Second Edition
Homer Reid
April 21, 2002
Chapter 7
Problem 7.2
Obtain the Lorentz transformation in which the velocity is at an innitesimal angle
d counterclockwise from the z axis, by means of a si
Solutions to Problems in Goldstein,
Classical Mechanics, Second Edition
Homer Reid
June 17, 2002
Chapter 8
Problem 8.4
The Lagrangian for a system can be written as
L = ax2 + b
y
+ cxy + f y 2 xz + g y k
x
x2 + y 2 ,
where a, b, c, f, g, and k are constan
Classical Mechanics  Homework Assignment 8
Alejandro Gmez Espinosa
o
November 20, 2012
Goldstein, Ch.8, 14 The Lagrangian for a system can be written as
L = ax2 + b
y
+ cxy + f y 2 xz + g y 2 k
x
x2 + y 2
where a, b, c, f, g and k are constants. What is
Math 711: Lecture of October 10, 2007
We now want to make precise the assertion at the end of the preceding lecture to the
eect that, under mild conditions on the local ring R, if one system of parameters of R
generates a tightly closed ideal then R is F
Math 711: Lecture of October 31, 2007
Discussion: local cohomology. Let y1 , . . . , yd be a sequence of elements of a Noetherian ring S and let N be an Smodule, which need not be nitely generated. Let J be
an ideal whose radical is the same as the radic
Math 711: Lecture of October 17, 2007
We next want to prove the Theorem stated at the end of the Lecture Notes from October
12. Recall that A = K[x1 , . . . , xn ] and that is conite in a xed p base for K.
First note that it is clear that K[x1 , . . . ,
Math 711: Lecture of October 29, 2007
We want to study at local homomorphism (R, m, K) (S, n, L) to obtain information about S from corresponding information about the base R and the closed ber S/mS.
We recall one fact of this type from p. 3 of the Lectur
Math 711: Lecture of October 22, 2007
By the singular locus Sing (R) in a Noetherian ring R we mean the set
cfw_P Spec (R) : RP is not regular.
We know that if R is excellent, then Sing (R) is a Zariski closed set, i.e., it has the form
V(I) for some idea
Math 711: Lecture of October 12, 2007
Capturing the contracted expansion from an integral extension
Using the result of the rst problem in Problem Set #1, we can now prove that tight
closure has one of the good properties, namely property (3) on p. 15 of
Math 711: Lecture of November 5, 2007
The following result is one we have already established in the Fnite case. We can now
extend it to include rings essentially of nite type over an excellent semilocal ring.
Theorem. Let R be a reduced ring of prime ch
Math 711: Lecture of October 8, 2007

Properties of regular sequences
In the sequel we shall need to make use of certain standard facts about regular sequences
on a module: for convenience, we collect these facts here. Many of the proofs can be made
simp
Math 711: Lecture of October 5, 2007
More on mapping cones and Koszul complexes
Let : B A be a map of complexes that is injective. We shall write d for the
dierential on A and for the dierential on B . Then we may form a quotient complex
Q such that Qn =
Math 711: Lecture of November 7, 2007
Our current theory of test elements permits the extension of many results proved under
other hypotheses, such as the condition that the ring under consideration be a homomorphic image of a CohenMacaulay ring, to the
Math 711: Lecture of October 19, 2007
Our next goal is to prove the two results stated at the end of the Lecture Notes of
October 17.
Proof of the theorem on preserving the eld property for a nite purely inseparable extension. Recall that L is a nite pure
Math 711: Lecture of October 26, 2007
It still remains to prove the nal assertion of the Theorem from p. 3 of the Lecture Notes
of October 22: that if R is Fnite and weakly Fregular, then R is strongly Fregular.
Before doing so, we want to note some co