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| In [[general relativity]] and gravitation the '''Palatini variation''' is nowadays thought of as variation of a [[Lagrangian]] with respect to the connection.
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| In fact, as is well known, the [[Einstein-Hilbert Lagrangian|Einstein-Hilbert action]] for general relativity was first formulated purely in terms of the [[Metric tensor (general relativity)|spacetime metric]] <math>\scriptstyle {g_{\mu\nu}}</math>. In the Palatini variational method one takes as independent field variables not only the ten components <math>\scriptstyle {g_{\mu\nu}}</math> but also the forty components of the [[affine connection]] <math>\scriptstyle {\Gamma^{\alpha}_{\,\beta\mu}}</math>, assuming, a priori, no dependence of the <math>\scriptstyle {\Gamma^{\alpha}_{\,\beta\mu}}</math> from the <math>\scriptstyle {g_{\mu\nu}}</math> and their derivatives.
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| The reason that Palatini variation is considered important is that it means that the use of the [[Affine connection|Christoffel connection]] in general relativity does not have to be added as a separate assumption, the information is already in the Lagrangian. For theories of gravitation which have more complex lagrangians than the [[Einstein-Hilbert Lagrangian]] of general relativity Palatini variation sometimes gives more complex connections and sometimes tensorial equations.
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| [[Attilio Palatini]] (1889-1949) was an Italian mathematician who received his doctorate from the [[University of Padova]], where he studied under [[Tullio Levi-Civita|Levi-Civita]] and [[Gregorio Ricci-Curbastro|Ricci-Curbastro]].
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| The history of the subject, and Palatini's connection with it, is not straightfoward, see the references. In fact, it seems that what the textbooks now call Palatini formalism was actually invented in 1925 by Einstein and as the years passed people tended to mix up the [[Palatini identity]] and the Palatini formalism. | |
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| ==See also==
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| *[[Palatini identity]]
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| *[[Self-dual Palatini action]]
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| *[[Tetradic Palatini action]]
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| ==References==
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| * A. Palatini (1919) ''Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton'', Rend. Circ. Mat. Palermo '''43''', 203-212 [English translation by R.Hojman and C. Mukku in [[Peter Bergmann|P.G. Bergmann]] and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)]
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| * M. Tsamparlis, ''On the Palatini method of Variation'', J. Math. Phys. '''19''', 555 (1977).
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| * M. Ferraris, [[Mauro Francaviglia|M. Francaviglia]], C. Reina, ''Variational Formulation of General Relativity from 1915 to 1925 'Palatini's Method' Discovered by Einstein in 1925'', Gen. Rel. Grav. '''14''' (1982) 243-254.
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| [[Category:Lagrangian mechanics]]
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| [[Category:General relativity]]
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I'm Tammie and I live in Gipf-Oberfrick.
I'm interested in Graduate School, Gaming and Russian art. I like to travel and watching The Vampire Diaries.
Also visit my web blog - sexanzeigen [Discover More]