algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
In variational calculus the Euler-Lagrange equations of a nonlinear functional arising from transgression of a local Lagrangian density characterize the extrema of that functional, hence its critical locus (the “shell”).
This originates from and is mainly used in physics, specifically in Lagrangian field theory, where the functional in question is the action functional of a physical system, and where its critical points encode the physically realized field histories by the principle of extremal action.
See at A first idea of quantum field theory this def.
The Euler-Lagrange equations of the Einstein-Hilbert action are Einstein's equations of gravity.
Named after Leonhard Euler and Joseph-Louis de Lagrange.
Textbook account in the context of gauge theory:
See also:
Wikipedia, Euler-Lagrange equation.
Robert Bryant, Phillip Griffiths, Daniel Grossman, Exterior differential systems and Euler-Lagrange partial differential equations, Chicago Lectures in Mathematics 2003, 205+xiv pp. math.DG/0207039 bookpage
Discussion in the convenient context of smooth sets:
Higher prequantum geometry II: The principle of extremal action – comonadically
Grigorios Giotopoulos, Hisham Sati, §5 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]
Last revised on April 14, 2024 at 15:42:25. See the history of this page for a list of all contributions to it.