The stationary-action principle – also known as the principle of least action – is a
variational principle that, when applied to the action of a
mechanical system, yields the
equations of motion for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are stationary points of the system's action functional.[1]
The term "least action" is often used[1] by physicists even though the principle has no general minimality requirement.[2] Historically the principle was known as "least action" and Feynman adopted this name over "Hamilton's principle" when he adapted it for quantum mechanics.[3]
The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics.[4]
where δ (lowercase Greek
delta) means a small change. In words this reads:[13]
The path taken by the system between times t1 and t2 and configurations q1 and q2 is the one for which the action is stationary (i.e., not changing) to first order.
Stationary action is not always a minimum, despite the historical name of least action.[16][1]: 19–6 It is a minimum principle for sufficiently short, finite segments in the path of a finite-dimensional system.[2]
In applications the statement and definition of action are taken together in "
Hamilton's principle", written in modern form as:[17]
The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the
configuration space, i.e. the curve q(t), parameterized by time (see also
parametric equation for this concept).
The mathematical equivalence of the
differentialequations of motion and their
integral
counterpart has important philosophical implications. The differential equations are statements about quantities localized to a single point in space or single moment of time. For example,
Newton's second law
states that the instantaneous force F applied to a mass m produces an acceleration a at the same instant. By contrast, the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) an extended region of space. Moreover, in the usual formulation of
classical action principles, the initial and final states of the system are fixed, e.g.,
Given that the particle begins at position x1 at time t1 and ends at position x2 at time t2, the physical trajectory that connects these two endpoints is an
extremum of the action integral.
In particular, the fixing of the final state has been interpreted as giving the action principle a
teleological character which has been controversial historically. However, according to
Wolfgang Yourgrau [
de] and
Stanley Mandelstam, the teleological approach... presupposes that the variational principles themselves have mathematical characteristics which they de facto do not possess[26] In addition, some critics maintain this apparent
teleology occurs because of the way in which the question was asked. By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities) we are making some inferences about the initial conditions from the final conditions, and it is this "backward" inference that can be seen as a teleological explanation.
^
abStehle, Philip M. (1993).
"Least-action principle". In Parker, S. P. (ed.). McGraw-Hill Encyclopaedia of Physics (2nd ed.). New York: McGraw-Hill. p. 670.
ISBN0-07-051400-3.
^Biot, Maurice Anthony (1975). "A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics". Bulletin de la Classe des sciences. 61 (1): 6–30.
doi:
10.3406/barb.1975.57878.
ISSN0001-4141.
^Goodman, Bernard (1993).
"Action". In Parker, S. P. (ed.). McGraw-Hill Encyclopaedia of Physics (2nd ed.). New York: McGraw-Hill. p. 22.
ISBN0-07-051400-3.
^Helzberger, Max (1966). "Optics from Euclid to Huygens". Applied Optics. 5 (9): 1383–93.
Bibcode:
1966ApOpt...5.1383H.
doi:
10.1364/AO.5.001383.
PMID20057555. In Catoptrics the law of reflection is stated, namely that incoming and outgoing rays form the same angle with the surface normal.
^Nakane, Michiyo, and Craig G. Fraser. "The Early History of Hamilton‐Jacobi Dynamics 1834–1837." Centaurus 44.3‐4 (2002): 161-227.
^Mehra, Jagdish (1987). "Einstein, Hilbert, and the Theory of Gravitation". In Mehra, Jagdish (ed.). The physicist's conception of nature (Reprint ed.). Dordrecht: Reidel.
ISBN978-90-277-2536-3.
^R. Feynman, Quantum Mechanics and Path Integrals, McGraw-Hill (1965),
ISBN0-07-020650-3
^J. S. Schwinger, Quantum Kinematics and Dynamics, W. A. Benjamin (1970),
ISBN0-7382-0303-3
^Stöltzner, Michael (1994). "Action Principles and Teleology". In H. Atmanspacher; G. J. Dalenoort (eds.). Inside Versus Outside. Springer Series in Synergetics. Vol. 63. Berlin: Springer. pp. 33–62.
doi:
10.1007/978-3-642-48647-0_3.
ISBN978-3-642-48649-4.
Further reading
For an annotated bibliography, see Edwin F. Taylor who
lists, among other things, the following books
Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia of Physics (Simon & Schuster Macmillan, 1996), Volume 2,
ISBN0-02-897359-3,
OCLC35269891, pages 840–842.
Dare A. Wells, Lagrangian Dynamics, Schaum's Outline Series (McGraw-Hill, 1967)
ISBN0-07-069258-0, A 350-page comprehensive "outline" of the subject.
Robert Weinstock, Calculus of Variations, with Applications to Physics and Engineering (Dover Publications, 1974).
ISBN0-486-63069-2. An oldie but goodie, with the formalism carefully defined before use in physics and engineering.
Wolfgang Yourgrau and
Stanley Mandelstam,
Variational Principles in Dynamics and Quantum Theory (Dover Publications, 1979). A nice treatment that does not avoid the philosophical implications of the theory and lauds the Feynman treatment of quantum mechanics that reduces to the principle of least action in the limit of large mass.
Georgiev, Georgi Yordanov (2012). "A Quantitative Measure, Mechanism and Attractor for Self-Organization in Networked Complex Systems". Self-Organizing Systems. Lecture Notes in Computer Science. Vol. 7166. pp. 90–5.
doi:
10.1007/978-3-642-28583-7_9.
ISBN978-3-642-28582-0.
S2CID377417.
Georgiev, Georgi; Georgiev, Iskren (2002). "The Least Action and the Metric of an Organized System". Open Systems & Information Dynamics. 9 (4): 371–380.
arXiv:1004.3518.
doi:
10.1023/a:1021858318296.
S2CID43644348.