Skip to main content Accessibility help
×
  1. Home
  2. Browse subjects
  3. Engineering
  4. Engineering: general interest
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

9746 results in Engineering: general interest

Page 103 of 488

  • 1 - Fundamental concepts

  • Patrick Hamill, San José State University, California
  • Book:
    A Student's Guide to Lagrangians and Hamiltonians
    Published online:
    05 June 2014
    Print publication:
    21 November 2013, pp 3-43

  • Summary

    This book is about Lagrangians and Hamiltonians. To state it more formally, this book is about the variational approach to analytical mechanics. You may not have been exposed to the calculus of variations, or may have forgotten what you once knew about it, so I am not assuming that you know what I mean by, “the variational approach to analytical mechanics.” But I think that by the time you have worked through the first two chapters, you will have a good grasp of the concept.

    We being with a review of introductory concepts and an overview of background material. Some of the concepts presented in this chapter will be familiar from your introductory and intermediate mechanics courses. However, you will also encounter several new concepts that will be useful in developing an understanding of advanced analytical mechanics.

    Kinematics

    A particle is a material body having mass but no spatial extent. Geometrically, it is a point. The position of a particle is usually specified by the vector r from the origin of a coordinate system to the particle. We can assume the coordinate system is inertial and for the sake of familiarity you may suppose the coordinate system is Cartesian. See Figure 1.1.

  • 2 - The calculus of variations

  • Patrick Hamill, San José State University, California
  • Book:
    A Student's Guide to Lagrangians and Hamiltonians
    Published online:
    05 June 2014
    Print publication:
    21 November 2013, pp 44-69

  • Summary

    Introduction

    The calculus of variations is a branch of mathematics which considers extremal problems; it yields techniques for determining when a particular definite integral will be a maximum or a minimum (or, more generally, the conditions for the integral to be “stationary”). The calculus of variations answers questions such as the following.

    1. • What is the path that gives the shortest distance between two points in a plane? (A straight line.)

    2. • What is the path that gives the shortest distance between two points on a sphere? (A geodesic or “great circle.”)

    3. • What is the shape of the curve of given length that encloses the greatest area? (A circle.)

    4. • What is the shape of the region of space that encloses the greatest volume for a given surface area? (A sphere.)

    The technique of the calculus of variations is to formulate the problem in terms of a definite integral, then to determine the conditions under which the integral will be maximized (or minimized). For example, consider two points (P1 and P2)inthe x–y plane. These can be connected by an infinite number of paths, each described by a function of the form y = y(x). Suppose we wanted to determine the equation y = y(x) for the curve giving the shortest path between P1 and P2.

Page 103 of 488