Double tangent bundle - Revision history

en>Yobot: /* Canonical tensor fields on the tangent bundle */WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T10:01:27Z

Canonical tensor fields on the tangent bundle: WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 12:01, 5 May 2014
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	'''Variational integrators''' are [[Numerical ordinary differential equations\|numerical integrators]] for [[Hamiltonian system]]s derived from the [[Euler-Lagrange equations]] of a discretized [[Hamilton's principle]]. Variational integrators are momentum-preserving and [[Symplectic integrator\|symplectic]].

	~~==Derivation of a Simple Variational Integrator==~~

	~~Consider~~ a ~~mechanical system with~~ a ~~single particle degree~~ of ~~freedom described by~~ the ~~Lagrangian~~		If you present photography effectively, it helps you look much more properly at the globe around you. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. This plugin allows a blogger get more Facebook fans on the related fan page. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. <br><br>When you loved this post and you would want to receive details about [http://2am.eu/wordpressdropboxbackup300514 backup plugin] kindly visit our own website. Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. The first thing you need to do is to choose the right web hosting plan. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Word - Press is the most popular open source content management system (CMS) in the world today. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Internet is not only the source for information, it is also one of the source for passive income. Visit our website to learn more about how you can benefit. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site.

	~~: <math>L(t,q,v) = \frac{1}{2} m v^2 - V(q)</math>,~~

	~~where <math>m</math> is~~ the ~~mass of~~ the ~~particle,~~ and ~~<math>V</math> is a potential~~. ~~To construct a variational integrator for this system, we begin by forming~~ the ~~'''discrete Lagrangian'~~''. ~~The discrete Lagrangian approximates the action for the system over a short time interval:~~

	: <~~math~~>L_d\left(t_0, t_1, q_0, q_1\right) = \frac{t_1 - t_0}{2} \left[ L\left(t_0, q_0, \frac{q_1-q_0}{t_1-t_0}\right) + L\left(t_1, q_1, \frac{q_1-q_0}{t_1-t_0}\right) \right] \approx \int_{t_0}^{t_1} dt\, L(t, q(t), v(t)) <~~/math~~>.

	~~Here we have chosen to approximate the time integral using the trapezoid method~~, and we use a ~~linear approximation~~ to the ~~trajectory~~,

	~~: <math>q(t) \approx \frac{q_1~~ - ~~q_0}{t_1-t_0} \left( t - t_0 \right) + q_0~~<~~/math~~>

	~~between~~ <~~math~~>~~t_0<~~/~~math> and <math>t_1<~~/~~math>~~, ~~resulting in~~ a ~~constant velocity <math>v \approx \left(q_1 - q_0 \right)/\left(t_1 - t_0 \right)</math>~~. ~~Different choices for the approximation~~ to the ~~trajectory and~~ the ~~time integral give different variational integrators~~. The ~~order of accuracy of the integrator~~ is ~~controlled by the accuracy of our approximation~~ to the ~~action; since~~

	~~: <math>L_d\left( t_0, t_1, q_0, q_1 \~~right~~) = \int_{t_0}^{t_1} dt\, L(t,q(t),v(t)) + \mathcal{O}\left(t_1~~ - ~~t_0\right)^3</math>,~~

	~~our integrator will be second~~-~~order accurate.~~

	~~Evolution equations for~~ the ~~discrete system can be derived from a stationary-action principle~~. ~~The discrete action over an extended time interval is a sum of discrete Lagrangians over many sub-intervals:~~

	: <~~math~~>~~S_d = L_d\left(t_0, t_1, q_0, q_1 \right) + L_d\left( t_1, t_2, q_1, q_2 \right) + \ldots~~<~~/math~~>.

	~~The principle of stationary action states~~ that the ~~action is stationary~~ with ~~respect to variations~~ of ~~coordinates that leave~~ the ~~endpoints~~ of the ~~trajectory fixed~~. ~~So, varying the coordinate <math>q_1</math>, we have~~

	~~: <math>\frac{\partial S_d}{\partial q_1} = 0 = \frac{\partial}{\partial q_1} L_d\left(t_0, t_1, q_0, q_1 \right) + \frac{\partial}{\partial q_1} L_d\left( t_1, t_2, q_1, q_2 \right)</math>.~~

	~~Given an initial condition <math>(q_0, q_1)</math>, and a sequence~~ of ~~times <math>(t_0,t_1,t_2)</math> this provides a relation that can be solved for <math>q_2</math>~~. ~~The solution~~ is

	~~: <math>q_2 = q_1 + \frac{t_2 - t_1}{t_1 - t_0} \left~~( ~~q_1 - q_0 \right) - \frac{\left( t_2 - t_0 \right) \left( t_2 - t_1 \right)}{2m} \frac{d}{dq_1}V\left(q_1\right~~)~~</math>.~~

	~~We can write this~~ in ~~a simpler form if we define~~ the ~~discrete momenta,~~

	~~: <math>p_0 \equiv~~ -~~\frac{\partial}{\partial q_0} L_d\left( t_0, t_1, q_0, q_1 \right)</math>~~

	and

	~~: <math>p_1 \equiv \frac{\partial}{\partial q_1} L_d\left( t_0, t_1, q_0, q_1 \right)</math>~~.


	~~Given an initial condition~~ <~~math~~>~~(q_0,p_0)~~<~~/math~~>~~, the stationary action condition~~ is ~~equivalent to solving~~ the ~~first of these equations~~ for ~~<math>q_1</math>~~, ~~and then determining <math>p_1</math> using~~ the ~~second equation~~. This ~~evolution scheme gives~~

	~~: <math>q_1 = q_0 + \frac{t_1 - t_0}{m} p_0 - \frac{\left(t_1 - t_0\right)^2}{2m} \frac{d}{dq_0} V\left( q_0 \right)</math>~~

	~~and~~

	~~: <math>p_1 = m \frac{q_1 - q_0}{t_1 - t_0} - \frac{t_1 - t_0}{2} \frac{d}{dq_1} V\left(q_1\right)</math>~~.

	This is ~~a [[leapfrog integration]] scheme for the system; two steps~~ of ~~this evolution are equivalent to~~ the ~~formula above~~ for ~~<math>q_2</math>~~

	~~==References==~~
	* E. ~~Hairer, C. Lubich, and G. Wanner. ''Geometric Numerical Integration''. Springer, 2002.~~
	* J. Marsden and M. ~~West. ''Discrete mechanics and variational integrators''. Acta Numerica, 2001, pp. 357–514.~~




	~~[[Category:Numerical differential equations]]~~

en>Dthomsen8: clean up, typos fixed: , → , using AWB

2013-06-01T15:13:01Z

clean up, typos fixed: , → , using AWB

← Older revision		Revision as of 17:13, 1 June 2013
Line 1:		Line 1:
	~~by Nas, is very fitting and~~ the ~~film agrees with it. It is very easy to customize plugins according to the needs~~ of a ~~particular business~~. ~~The Word~~ - ~~Press Dashboard~~ : ~~an administrative management tool that supports FTP content upload~~ 2~~. Word~~ - ~~Press also provides protection against spamming~~, ~~as security~~ is a ~~measure issue~~. It'~~s as simple as hiring~~ a ~~Wordpress plugin developer or learning how to create what is needed.~~ <br><br>~~As you know today Word - Press has turn out~~ to be a ~~tremendously popular open source publishing~~ and ~~blogging display place~~. ~~The higher your blog ranks on search engines,~~ the ~~more likely people will find your online marketing site. You are able~~ to ~~set them within your theme options~~ and ~~so they aid~~ the ~~search engine to get a suitable title and description for~~ the ~~pages that get indexed~~ by ~~Google. It primarily lays emphasis on improving~~ the ~~search engine results~~ of ~~your website whenever a related query is typed in~~ the ~~search box~~. ~~This~~ can be ~~done by using~~ a ~~popular layout format and your unique Word~~ - ~~Press design can be achieved in other elements~~ of the ~~blog~~. <br><br>~~If you adored this article therefore you would like to receive more info pertaining to [http~~:/~~/www~~.~~shortlinki.com~~/~~wordpress_dropbox_backup_9267297 backup plugin] kindly visit our own page. Usually~~, ~~Wordpress owners selling the ad space on monthly basis~~ and this ~~means~~ a ~~residual income source~~. ~~After sending these details, your Word~~ - ~~Press blog will be setup within a few days~~. ~~Use~~ this ~~section to change many formatting elements. These frequent updates have created menace~~ in the ~~task of optimization~~. ~~After that~~ the ~~developer adds~~ the ~~unordered list~~ for ~~navigations.~~ <br><br>~~If all else fails~~, ~~please leave a comment on this post with~~ the ~~issue~~(s) ~~you're having~~ and ~~help will be on the way. The SEOPressor Word~~ - ~~Press SEO Plugin works by analysing each page and post against your chosen keyword~~ (~~or keyword phrase~~) and giving a score, with instructions on how to improve it. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. This ~~includes enriching~~ the ~~content with proper key words, tactfully defining~~ the ~~tags and URL.~~ <br><br>~~A sitemap is useful for enabling web spiders and also on rare occasions clients~~, ~~too~~, ~~to more easily~~ and ~~navigate your website~~. ~~I don~~'~~t want that for my visitors and I~~'~~m quite sure they don~~'~~t either~~. ~~The days of spending a lot of time~~ and ~~money to have a website built are long gone~~. ~~Working with a Word - Press blog~~ and ~~the appropriate cost-free Word - Press theme~~, ~~you can get a professional internet site up and published in no time at all~~. ~~You can check out the statistics of page of views for your web pages using free tools that are available on the internet~~.		'''Variational integrators''' are [[Numerical ordinary differential equations\|numerical integrators]] for [[Hamiltonian system]]s derived from the [[Euler-Lagrange equations]] of a discretized [[Hamilton's principle]]. Variational integrators are momentum-preserving and [[Symplectic integrator\|symplectic]].

			==Derivation of a Simple Variational Integrator==

			Consider a mechanical system with a single particle degree of freedom described by the Lagrangian

			: <math>L(t,q,v) = \frac{1}{2} m v^2 - V(q)</math>,

			where <math>m</math> is the mass of the particle, and <math>V</math> is a potential. To construct a variational integrator for this system, we begin by forming the '''discrete Lagrangian'''. The discrete Lagrangian approximates the action for the system over a short time interval:

			: <math>L_d\left(t_0, t_1, q_0, q_1\right) = \frac{t_1 - t_0}{2} \left[ L\left(t_0, q_0, \frac{q_1-q_0}{t_1-t_0}\right) + L\left(t_1, q_1, \frac{q_1-q_0}{t_1-t_0}\right) \right] \approx \int_{t_0}^{t_1} dt\, L(t, q(t), v(t)) </math>.

			Here we have chosen to approximate the time integral using the trapezoid method, and we use a linear approximation to the trajectory,

			: <math>q(t) \approx \frac{q_1 - q_0}{t_1-t_0} \left( t - t_0 \right) + q_0</math>

			between <math>t_0</math> and <math>t_1</math>, resulting in a constant velocity <math>v \approx \left(q_1 - q_0 \right)/\left(t_1 - t_0 \right)</math>. Different choices for the approximation to the trajectory and the time integral give different variational integrators. The order of accuracy of the integrator is controlled by the accuracy of our approximation to the action; since

			: <math>L_d\left( t_0, t_1, q_0, q_1 \right) = \int_{t_0}^{t_1} dt\, L(t,q(t),v(t)) + \mathcal{O}\left(t_1 - t_0\right)^3</math>,

			our integrator will be second-order accurate.

			Evolution equations for the discrete system can be derived from a stationary-action principle. The discrete action over an extended time interval is a sum of discrete Lagrangians over many sub-intervals:

			: <math>S_d = L_d\left(t_0, t_1, q_0, q_1 \right) + L_d\left( t_1, t_2, q_1, q_2 \right) + \ldots</math>.

			The principle of stationary action states that the action is stationary with respect to variations of coordinates that leave the endpoints of the trajectory fixed. So, varying the coordinate <math>q_1</math>, we have

			: <math>\frac{\partial S_d}{\partial q_1} = 0 = \frac{\partial}{\partial q_1} L_d\left(t_0, t_1, q_0, q_1 \right) + \frac{\partial}{\partial q_1} L_d\left( t_1, t_2, q_1, q_2 \right)</math>.

			Given an initial condition <math>(q_0, q_1)</math>, and a sequence of times <math>(t_0,t_1,t_2)</math> this provides a relation that can be solved for <math>q_2</math>. The solution is

			: <math>q_2 = q_1 + \frac{t_2 - t_1}{t_1 - t_0} \left( q_1 - q_0 \right) - \frac{\left( t_2 - t_0 \right) \left( t_2 - t_1 \right)}{2m} \frac{d}{dq_1}V\left(q_1\right)</math>.

			We can write this in a simpler form if we define the discrete momenta,

			: <math>p_0 \equiv -\frac{\partial}{\partial q_0} L_d\left( t_0, t_1, q_0, q_1 \right)</math>

			and

			: <math>p_1 \equiv \frac{\partial}{\partial q_1} L_d\left( t_0, t_1, q_0, q_1 \right)</math>.


			Given an initial condition <math>(q_0,p_0)</math>, the stationary action condition is equivalent to solving the first of these equations for <math>q_1</math>, and then determining <math>p_1</math> using the second equation. This evolution scheme gives

			: <math>q_1 = q_0 + \frac{t_1 - t_0}{m} p_0 - \frac{\left(t_1 - t_0\right)^2}{2m} \frac{d}{dq_0} V\left( q_0 \right)</math>

			and

			: <math>p_1 = m \frac{q_1 - q_0}{t_1 - t_0} - \frac{t_1 - t_0}{2} \frac{d}{dq_1} V\left(q_1\right)</math>.

			This is a [[leapfrog integration]] scheme for the system; two steps of this evolution are equivalent to the formula above for <math>q_2</math>

			==References==
			* E. Hairer, C. Lubich, and G. Wanner. ''Geometric Numerical Integration''. Springer, 2002.
			* J. Marsden and M. West. ''Discrete mechanics and variational integrators''. Acta Numerica, 2001, pp. 357–514.




			[[Category:Numerical differential equations]]

en>Josetomas2008: Bibliography

2012-04-21T00:15:04Z

Bibliography

New page

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