nLab principle of extremal action

Idea

The principle of extremal action in physics is a historical precursor of the modern understanding that trajectories of particles and configurations of fields realized in classical mechanics are characterized as being the variational extrema or critical loci of a functional on the space of all possible configurations, called the action functional. For typical examples of this principle – some of which were known to the ancient Greeks and which were then formalized in some generality in the 18th century – these extrema are indeed minima, as with, for instance, the propagation of a ray of light through optical media in optics. More generally one speaks of the principle of extremal action or the principle of stationary action. But in fact mostly today one just speaks of the Euler-Lagrange equations of motion, which are the equations that characterize these extremal trajectories.

With the advent of quantum mechanics, the principle of extremal action found an explanation in more fundamental terms as the classical limit of quantum dynamics. A simple, though somewhat heuristic, way to see this is via path integral quantization: whereas propagation in quantum mechanics is given by the path integral by which every trajectory $\phi$ between two prescribed configurations contributes with a probability amplitude given by the exponentiated value $\exp(i S(\phi))$ of the action functional $S$ on this trajectory, at least under some conditions the main contribution to this integral is from trajectories close to the critical points of the action functional.

Examples

• Einstein-Hilbert action$\implies$ Einstein equation of motion for gravity/general relativity

References

Historical and expository accounts of the “principle of extremal action” or “action principle”, for short:

• Agamenon R. E. Oliveira, History of Two Fundamental Principles of Physics: Least Action and Conservation of Energy, Advances in Historical Studies 3 2 (2014) [doi:10.4236/ahs.2014.32008]

• Alberto Rojo, Anthony Bloch: The Principle of Least Action – History and Physics, Cambridge University Press (2018) [ISBN:9780521869027]

• Walter Dittrich, The Development of the Action Principle – A Didactic History from Euler-Lagrange to Schwinger, SpringerBriefs in Physics, Springer (2021) [doi:10.1007/978-3-030-69105-9]

• Douglas Cline, Variational Principles in Classical Mechanics, University of Rochester (2021) [pdf, online version]

See also:

• Wikipedia, Principle of least action

• geometry of physics– Equations of motion