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Variational methods in general relativity
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<h2 id="lagrangian-methods">Lagrangian methods</h2> <p class="note">Main article: <a href="/facts/Einstein%25E2%2580%2593Hilbert_action/DGbTkCe9">Einstein–Hilbert action</a></p> <p>The <a href="/facts/Equations_of_motion/25msr1vE">equations of motion</a> in physical theories can often be derived from an object called the <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a>. In <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>, this object is usually of the form, '<a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a> − <a href="/facts/Potential_energy/I03KLbmn">potential energy</a>'. In general, the Lagrangian is that function which when integrated over produces the Action functional. </p><p><a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a> gave an early and classic formulation of the equations in Einstein's general relativity.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> This used the functional now called the <a href="/facts/Einstein-Hilbert_action/DGbTkCe9">Einstein-Hilbert action</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Palatini_action/jyEhSaHM">Palatini action</a></li> <li><a href="/facts/Plebanski_action/xjsVTu4R">Plebanski action</a></li> <li><a href="/facts/MacDowell%25E2%2580%2593Mansouri_action/JFPFEK9M">MacDowell–Mansouri action</a></li> <li>Freidel–Starodubtsev action</li> <li><a href="/facts/Mathematics_of_general_relativity/5RktLkom">Mathematics of general relativity</a></li> <li><a href="/facts/Fermat%2527s_and_energy_variation_principles_in_field_theory/rboC9ufh">Fermat's and energy variation principles in field theory</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Poisson, Eric (2004). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511606601. ISBN 978-0-521-83091-1. <a href="978-0-521-83091-1" target="_blank">978-0-521-83091-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hilbert, D. (1915). "Die Grundlagen der Physik . (Erste Mitteilung.)". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 1915: 395–408. <a href="https://eudml.org/doc/58946" target="_blank">https://eudml.org/doc/58946</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>