Olation - Revision history

181.118.153.138 at 22:04, 22 August 2014

2014-08-22T22:04:26Z

← Older revision		Revision as of 00:04, 23 August 2014
Line 1:		Line 1:
	~~'''Non-autonomous mechanics''' describe non-relativistic mechanical systems subject to time-dependent transformations. In particular~~, ~~this is the case of mechanical systems whose [[Lagrangian]]s~~ and ~~[[Hamiltonian mechanics\|Hamiltonian]]s depend on the time~~. ~~The configuration space of non-autonomous mechanics is a [[fiber bundle]] <math>Q\to \mathbb R</math> over~~ the ~~time axis <math>\mathbb R</math> coordinated by <math>(t,q^i)</math>.~~		Corporate Normal Manager Fidel Fruits from Powassan, has hobbies including hunting, top eleven and rowing. Finds the entire world an enjoyable place having spent 5 days at Komodo National Park.<br><br>Visit my blog - [http://on.fb.me/1x3Loy6 top eleven hack]

	This bundle is trivial, but its different trivializations <math>Q=\mathbb R\times M</math> correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a [[connection (mathematics)\|connection]]
	<math>\Gamma</math> on <math>Q\to\mathbb R</math> which takes a form <math>\Gamma^i =0</math> with respect to this trivialization. The corresponding covariant differential <math>(q^i_t-\Gamma^i)\partial_i</math>
	~~determines the relative velocity with respect to a reference frame <math>\Gamma</math>.~~

	As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a [[covariant classical field theory]] (in particular [[covariant Hamiltonian field theory]]) on <math>X=\mathbb R</math>. Accordingly, the velocity phase space of non-autonomous mechanics is the [[jet bundle\|jet manifold]] <math>J^1Q</math> of <math>Q\to \mathbb R</math> provided with the coordinates <math>(t,q^i,q^i_t)</math>. Its momentum phase space is the vertical cotangent bundle <math>VQ</math> of <math>Q\to \mathbb R</math> coordinated by <math>(t,q^i,p_i)</math> and endowed with the canonical [[Poisson manifold\|Poisson structure]]. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form <math>p_idq^i-H(t,q^i,p_i)dt</math>.

	~~One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle~~ <~~math~~>TQ<~~/math~~> ~~of <math>Q</math> coordinated by <math>(t,q^i,p,p_i)</math> and provided with the canonical [[symplectic manifold\|symplectic form]]; its [[Hamiltonian mechanics\|Hamiltonian]] is <math>p-H</math>.~~

	~~== References ==~~
	* De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989).
	* Echeverria Enriquez, A., Munoz Lecanda, M., Roman Roy, N., Geometrical setting of time-dependent regular systems. Alternative models, Rev. Math. Phys. '''3''' (1991) 301.
	* Carinena, J., Fernandez-Nunez, J., Geometric theory of time-dependent singular Lagrangians, Fortschr. Phys., '''41''' (1993) 517.
	* Mangiarotti, L., [[Gennadi Sardanashvily\|Sardanashvily, G.]], Gauge Mechanics (World Scientific, 1998) ISBN 981-02-3603-4.
	* Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily\|Sardanashvily, G.]], Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 ([http://~~xxx~~.~~lanl~~.~~gov~~/~~abs/0911.0411 arXiv: 0911.0411]).~~

	~~==See also==~~
	* [[Analytical mechanics]]
	* [[Non-autonomous system (mathematics)]]
	* [[Hamiltonian mechanics]]
	* [[Symplectic manifold]]
	* [[Covariant Hamiltonian field theory]]
	* [[Free motion equation]]
	* [[Relativistic system (mathematics)]]

	~~[[Category:Theoretical physics]]~~
	~~[[Category:Classical mechanics]]~~
	~~[[Category:Hamiltonian mechanics]]~~
	~~[[Category:Symplectic geometry]]~~


	~~{{physics-stub}}~~
	~~{{applied-math-stub}}~~

129.241.83.162 at 14:13, 8 May 2013

2013-05-08T14:13:26Z

← Older revision		Revision as of 16:13, 8 May 2013
Line 1:		Line 1:
			'''Non-autonomous mechanics''' describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose [[Lagrangian]]s and [[Hamiltonian mechanics\|Hamiltonian]]s depend on the time. The configuration space of non-autonomous mechanics is a [[fiber bundle]] <math>Q\to \mathbb R</math> over the time axis <math>\mathbb R</math> coordinated by <math>(t,q^i)</math>.

			This bundle is trivial, but its different trivializations <math>Q=\mathbb R\times M</math> correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a [[connection (mathematics)\|connection]]
			<math>\Gamma</math> on <math>Q\to\mathbb R</math> which takes a form <math>\Gamma^i =0</math> with respect to this trivialization. The corresponding covariant differential <math>(q^i_t-\Gamma^i)\partial_i</math>
			determines the relative velocity with respect to a reference frame <math>\Gamma</math>.

	~~They're always ready to help~~, ~~and they're always making changes to the site to make sure you won't have troubles~~ in ~~the first place. It is very easy to customize plugins according to the needs of a~~ particular ~~business. Wordpress Content management systems, being customer friendly~~, can be ~~used extensively to write and manage websites and blogs. If you adored this short article and you would like to obtain additional information regarding~~ [~~http:~~/~~/shortener~~.~~us/backup_plugin_3448647 backup plugin] kindly see~~ the ~~web~~-~~page. If you are using videos on your site then this~~ is the ~~plugin~~ to ~~use. After activating, you will find their website link and get~~ the ~~activation code from their website.~~ <br><br>~~The Internet is a vast open market where businesses and consumers congregate. If you are a positive thinker businessman then today you have to put your business online~~. ~~This~~ is the ~~reason for the increased risk~~ of ~~Down Syndrome babies in women over age 35. So if you want~~ to ~~create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Aided~~ by ~~the completely foolproof j - Query color selector~~, ~~you're able to change the colors of factors of your theme a the click on the screen~~, ~~with very little previous web site design experience.~~ <br>~~<br>This gives a clearer picture that online shoppers are familiar~~ with the ~~WP ecommerce system~~. ~~By using Word~~ - ~~Press MLM websites or blogs~~, ~~you and your companies presence on the internet can be created swiftly and simply. If Gandhi was empowered with a blogging system~~, ~~every event in his life would have been minutely documented so that it could be recounted to the future generations~~. ~~The animation~~ can ~~be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as~~ an ~~alternative gives some viewing enjoyment for~~ the ~~visitor~~ of ~~the internet site. " Thus working~~ with ~~a Word - Press powered web application, making any changes in~~ the ~~website design or website content~~ is ~~really easy and self explanatory.~~ <br><br>~~Numerous bloggers are utilizing Word - Press and with good reason~~. ~~php file~~ in ~~the Word - Press root folder and look for this line~~ (~~line 73 in our example~~):. ~~Normally~~, ~~the Word~~ - ~~Press developers make a thorough research on your website goals and then ingrain the most suitable graphical design elements to your website~~. ~~A whole lot worse~~, ~~your site will likely be useless as well as your merchandise won~~'~~t sell if no one has the endurance to wait for the web pages to load~~. ~~Now all you have to do is log into your Word~~ - ~~Press site making use of the very same username and password that you initially had in your previous site~~. ~~<br><br>There is no denying that Magento is an ideal platform for building ecommerce websites~~, ~~as it comes with an astounding number~~ of ~~options that can help your online business do extremely well~~. ~~Automated deal feed integration option to populate your blog with relevant deals~~. It'~~s not a secret that a lion share~~ of ~~activity on the internet is takes place on the Facebook~~. ~~) Remote Login~~: ~~With the process of PSD to Wordpress conversion comes the advantage of flexibility~~. ~~Get started today so that people searching for your type of business will be directed to you~~.		As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a [[covariant classical field theory]] (in particular [[covariant Hamiltonian field theory]]) on <math>X=\mathbb R</math>. Accordingly, the velocity phase space of non-autonomous mechanics is the [[jet bundle\|jet manifold]] <math>J^1Q</math> of <math>Q\to \mathbb R</math> provided with the coordinates <math>(t,q^i,q^i_t)</math>. Its momentum phase space is the vertical cotangent bundle <math>VQ</math> of <math>Q\to \mathbb R</math> coordinated by <math>(t,q^i,p_i)</math> and endowed with the canonical [[Poisson manifold\|Poisson structure]]. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form <math>p_idq^i-H(t,q^i,p_i)dt</math>.

			One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle <math>TQ</math> of <math>Q</math> coordinated by <math>(t,q^i,p,p_i)</math> and provided with the canonical [[symplectic manifold\|symplectic form]]; its [[Hamiltonian mechanics\|Hamiltonian]] is <math>p-H</math>.

			== References ==
			* De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989).
			* Echeverria Enriquez, A., Munoz Lecanda, M., Roman Roy, N., Geometrical setting of time-dependent regular systems. Alternative models, Rev. Math. Phys. '''3''' (1991) 301.
			* Carinena, J., Fernandez-Nunez, J., Geometric theory of time-dependent singular Lagrangians, Fortschr. Phys., '''41''' (1993) 517.
			* Mangiarotti, L., [[Gennadi Sardanashvily\|Sardanashvily, G.]], Gauge Mechanics (World Scientific, 1998) ISBN 981-02-3603-4.
			* Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily\|Sardanashvily, G.]], Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 ([http://xxx.lanl.gov/abs/0911.0411 arXiv: 0911.0411]).

			==See also==
			* [[Analytical mechanics]]
			* [[Non-autonomous system (mathematics)]]
			* [[Hamiltonian mechanics]]
			* [[Symplectic manifold]]
			* [[Covariant Hamiltonian field theory]]
			* [[Free motion equation]]
			* [[Relativistic system (mathematics)]]

			[[Category:Theoretical physics]]
			[[Category:Classical mechanics]]
			[[Category:Hamiltonian mechanics]]
			[[Category:Symplectic geometry]]


			{{physics-stub}}
			{{applied-math-stub}}

en>MystBot: r2.7.1) (Robot: Adding it:Olazione

2011-10-14T13:56:19Z

r2.7.1) (Robot: Adding it:Olazione

New page

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