nLab Euler-Lagrange equation

1. Idea

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.

2. Details

See at A first idea of quantum field theory this def.

3. Examples

• The Euler-Lagrange equations of the Einstein-Hilbert action are Einstein's equations of gravity.

• Maxwell's equations

• Yang-Mills equations

4. Related concepts

• Euler-Lagrange form

• equations of motion

  • Newton's laws of motion

  • Hamilton's equations, de Donder-Weyl-Hamilton equation

  • Einstein's equations

• Schwinger-Dyson equation, Ward identity

• variational calculus

• topos of laws of motion

5. References

Named after Leonhard Euler and Joseph-Louis de Lagrange.

Textbook account in the context of gauge theory:

• Marc Henneaux, Claudio Teitelboim, §1.1.1 in: Quantization of Gauge Systems, Princeton University Press (1992) [ISBN:9780691037691, jstor:j.ctv10crg0r]

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]