Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-27T16:29:36.063Z Has data issue: false hasContentIssue false

Chapter 14 - Hamilton's equations

Published online by Cambridge University Press:  05 December 2012

Lawrence Sklar
Affiliation:
University of Michigan, Ann Arbor
Get access

Summary

The Hamiltonian formalism

The nineteenth century saw several new proposals for fundamental principles to be placed at the foundations of dynamics. In 1829 Gauss offered the “Principle of Least Constraint.” Let a system of particles be connected together by constraints. Suppose each particle n starts at An and after a time interval ends up at Bn. Let Cn be the position it would have ended up at at the end of the time interval had the constraints not been imposed. Then, for the actual motion undergone, the sum of the mass of each particle times the square of the distance from Bn to Cn will be minimal over the class of all motions compatible with the constraints binding the particles together.

In 1894 Hertz offered the “Principle of Least Curvature.” Generalizations of the ordinary Euclidean notions of distance along a path and curvature of a path are constructed. It is then shown that, using these definitions, a system of particles that is isolated, that is not subject to some external force, will evolve in such a way that the motions of the individual particles will generate a path in a multi-dimensional space for the point representing the system as a whole that has, at each point, minimal curvature in the sense defined by Hertz. This principle provides a useful opening for applying methods of differential geometry in dynamics, for in a curved space the paths of least curvature, the geodesics, are objects of intensive study in differential geometry. Hertz's principle then tells us that, with the proper definitions of distance and curvature for our space, one can view the dynamical trajectory of a system as a geodesic in the representing space.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Hamilton's equations
  • Lawrence Sklar, University of Michigan, Ann Arbor
  • Book: Philosophy and the Foundations of Dynamics
  • Online publication: 05 December 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139034340.014
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Hamilton's equations
  • Lawrence Sklar, University of Michigan, Ann Arbor
  • Book: Philosophy and the Foundations of Dynamics
  • Online publication: 05 December 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139034340.014
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Hamilton's equations
  • Lawrence Sklar, University of Michigan, Ann Arbor
  • Book: Philosophy and the Foundations of Dynamics
  • Online publication: 05 December 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139034340.014
Available formats
×