Stationary-action principle

The term "least action" is often used by physicists even though the principle has no general minimality requirement. Historically the principle was known as "least action" and Feynman adopted this name over "Hamilton's principle" when he adapted it for quantum mechanics.

The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity, as well as classical electrodynamics and quantum field theory. In these cases, a different action must be minimized or maximized. For relativity, it is the Einstein–Hilbert action. For quantum field theory, it involves the path integral formulation.

The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics.

The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746.

General statement

The action, denoted $S$ , of a physical system is defined as the integral of the Lagrangian L between two instants of time t₁ and t₂ – technically a functional of the N generalized coordinates q = (q₁, q₂, ... , q_N) which are functions of time and define the configuration of the system:

$q : R \to R^{N}$ $S [q, t_{1}, t_{2}] = \int_{t_{1}}^{t_{2}} L (q (t), \dot{q} (t), t) d t$

where the dot denotes the time derivative, and t is time.

Mathematically the principle is

$δ S = 0,$

where δ (lowercase Greek delta) means a small change. In words this reads:

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. It is a minimum principle for sufficiently short, finite segments in the path of a finite-dimensional system.

In applications the statement and definition of action are taken together in "Hamilton's principle", written in modern form as:

$δ \int_{t_{1}}^{t_{2}} L (q, \dot{q}, t) d t = 0.$

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).

2 more chapters are free with a Lably account.

History

The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path was the shortest length and least time.

Building on the early work of Pierre Louis Maupertuis, Leonhard Euler, and Joseph Louis Lagrange defining versions of principle of least action, William Rowan Hamilton and in tandem Carl Gustav Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation.

In 1915 David Hilbert applied the variational principle to derive Albert Einstein's equations of general relativity.

In 1933, the physicist Paul Dirac demonstrated how this principle can be used in quantum calculations by discerning the quantum mechanical underpinning of the principle in the quantum interference of amplitudes. Subsequently Julian Schwinger and Richard Feynman independently applied this principle in quantum electrodynamics.

Disputes about possible teleological aspects

The mathematical equivalence of the differential equations 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

$F = m a$

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 and Stanley Mandelstam, the teleological approach... presupposes that the variational principles themselves have mathematical characteristics which they de facto do not possess 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.

General statement

History

Disputes about possible teleological aspects

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