arXiv:1401.2660v1 [math.HO] 12 Jan 2014
The straight line, the catenary, the
brachistochrone, the circle, and Fermat
Raul Rojas
Freie Universitt Berlin
a
January 2014
Abstract
This paper shows that the well-known curve optimization problems which
lead to
Introduction to Lagrangian Mechanics
(1) Newtons method is just as fundamental as Lagrangian, and
in some cases, has wider range of application. However it is
cumbersome to use for problem solving.
(2) Many mechanics problems are about trajectory under co
PHY509: SOLUTION HW #3. P1. With f = -ku3 plugged into the EOM, we have d2 u mk + 1 - 2 u = 0. 2 d l (a) For mk/l2 < 1, the EOM is u + 2 u = 0 with 2 = 1 - mk/l2 and the solution is sinosoidal, u() = A cos[( - 0 )]. (b) Since the energy conservation is ob
PHY509: HOMEWORK 3. (due 09/26/05) P1. Solve trajectory from a central force f (r) = k/r3 . Use the equation of motion m d2 u + u = 2 2f 2 d l u You may consult Landau and Lifshitz, Prob.1 after 15. (a) For mk/l2 < 1, what is the general form of solution
1
CHAPTER 13
LAGRANGIAN MECHANICS
13.1 Introduction
The usual way of using newtonian mechanics to solve a problem in dynamics is first of
all to draw a large, clear diagram of the system, using a ruler and a compass. Then mark
in the forces on the various
PHY509: HOMEWORK 2. (due 09/16/05) P1. We re-derive the trajectory r() of the elliptic planetary orbit in the inverse-square law force field. This method is simpler than that discussed in class. You may look up the derivation in the Goldstein 3rd Ed. pp.
PHY509: HOMEWORK 1. (due 09/09/05)
m F m (A)
1.
A dumbbell consists of two point masses,
F 2R
m, connected by a massless rod of length 2R. A constant force F is applied as shown during a very short time interval t. In (A), the force acts on the center of
Qualication Exam: Classical Mechanics
Name:
, QEID#81245504:
September, 2014
Qualication Exam
1
QEID#81245504
2
Undergraduate level
Problem 1. 1983-Fall-CM-U-1.
ID:CM-U-2
1. Consider a particle of mass m moving in a plane under the inuence of a sphericall
TMME, vol5, nos.2&3, p.169
The Brachistochrone Problem: Mathematics for a Broad Audience
via a Large Context Problem
Jeff Babb1 & James Currie2
Department of Mathematics and Statistics,
University of Winnipeg,Canada
Abstract
Large context problems (LCP) a
THERE ONCE WAS A
CLASSICAL THEORY
Introductory Classical Mechanics,
with Problems and Solutions
David Morin
Of which quantum disciples were leery.
They said, Why spend so long
On a theory thats wrong?
Well, it works for your everyday query!
Copyright 2004
THE BRACHISTOCHRONE PROBLEM.
Imagine a metal bead with a wire threaded through a hole in it, so that
the bead can slide with no friction along the wire.1 How can one choose
the shape of the wire so that the time of descent under gravity (from rest)
is sma
Page 1 of 67
8.01 IAP Mechanics ReView Course
January 2009
Practice Problems
This document contains a large number of practice problems on the material in
8.01. Practice makes perfect, especially in physics, and doing problems and
understanding them is pr
Classical Mechanics and Electrodynamics
Lecture notes FYS 3120
Jon Magne Leinaas
Department of Physics, University of Oslo
December 2009
2
Preface
These notes are prepared for the physics course FYS 3120, Classical Mechanics and Electrodynamics,
at the De
14
Calculus of Variations
and Applications1
This chapter is a little more classic than the others. It introduces calculus of variations, an elegant eld not often covered in modern math curricula. A knowledge of
multivariable calculus will suce, but it hel
PHY509: HOMEWORK 6. (due 10/21/05)
1 1. (a) Show that a Lagrangian L = 2 mx2 - q + qv A with
electrostatic potential and vector potential A is consistent with the Newton's equation ma = F = q(E + v B).
q
B
(b) At any given time, a vector potential A(r, t)
Chapter 6
The Lagrangian Method
Copyright 2007 by David Morin, [email protected] (draft version)
In this chapter, were going to learn about a whole new way of looking at things. Consider
the system of a mass on the end of a spring. We can analyze