Action principles

Distinct principles

Action principles apply to a wide variety of physical problems, including all of fundamental physics. The only major exceptions are cases involving friction or when only the initial position and velocities are given. Different action principles have different meaning for the variations; each specific application of an action principle requires a specific Lagrangian describing the physics. A common name for any or all of these principles is "the principle of least action". For a discussion of the names and historical origin of these principles see action principle names.

Fixed endpoints with conserved energy

When total energy and the endpoints are fixed, Maupertuis's least action principle applies. For example, to score points in basketball the ball must leave the shooters hand and go through the hoop, but the time of the flight is not constrained. Maupertuis's least action principle is written mathematically as the stationary condition $${\displaystyle (\delta W{)}_E=0}$$ on the abbreviated action $${\displaystyle W[\mathbf{q}]\stackrel{\text{def}}{=}{\int }_{q_1}^{q_2}\mathbf{p}\cdot \mathbf{d}\mathbf{q},}$$ (sometimes written S₀), where p = (p₁,p₂,…,p_N) are the particle momenta or the conjugate momenta of generalized coordinates, defined by the equation $${\displaystyle p_k\stackrel{\text{def}}{=}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_k},}$$ where L(q,q̇,t) is the Lagrangian. Some textbooks write (δW)_E = 0 as ΔS₀, to emphasize that the variation used in this form of the action principle differs from Hamilton's variation. Here the total energy E is fixed during the variation, but not the time, the reverse of the constraints on Hamilton's principle. Consequently, the same path and end points take different times and energies in the two forms. The solutions in the case of this form of Maupertuis's principle are orbits: functions relating coordinates to each other in which time is simply an index or a parameter.

Time-independent potentials; no forces

For time-invariant system, the action ${\displaystyle S}$ relates simply to the abbreviated action W on the stationary path as $${\displaystyle \mathrm{\Delta }S=\mathrm{\Delta }W-E\mathrm{\Delta }t}$$ for energy E and time difference Δt = t₂ − t₁. For a rigid body with no net force, the actions are identical, and the variational principles become equivalent to Fermat's principle of least time: $${\displaystyle \delta (t_2-t_1)=0.}$$

Fixed events

When the physics problem gives the two endpoints as a position and a time, that is as events, Hamilton's action principle applies. For example, imagine planning a trip to the Moon. During your voyage the Moon will continue its orbit around the Earth: it is a moving target. Hamilton's principle for objects at positions q(t) is written mathematically as $${\displaystyle (\delta \mathcal{S}{)}_{\mathrm{\Delta }t}=0,\text{where}\mathcal{S}[\mathbf{q}]\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{t_1}^{t_2}L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)dt.}$$ The constraint Δt = t₂ − t₁ means that we only consider paths taking the same time, as well as connecting the same two points q(t₁) and q(t₂). The Lagrangian ${\displaystyle L=T-V}$ is the difference between kinetic energy and potential energy at each point on the path. Solution of the resulting equations gives the world line q(t). Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy. The action ${\displaystyle S}$ in Hamilton's principle is the Legendre transformation of the action in Maupertuis's principle.

Classical field theory

The concepts and many of the methods useful for particle mechanics also apply to continuous fields. The action integral runs over a Lagrangian density, but the concepts are so close that the density is often simply called the Lagrangian.

Quantum action principles

For quantum mechanics, the action principles have significant advantages: only one mechanical postulate is needed, if a covariant Lagrangian is used in the action, the result is relativistically correct, and they transition clearly to classical equivalents.

Both Richard Feynman and Julian Schwinger developed quantum action principles based on early work by Paul Dirac. Feynman's integral method was not a variational principle but reduces to the classical least action principle; it led to his Feynman diagrams. Schwinger's differential approach relates infinitesimal amplitude changes to infinitesimal action changes.

Feynman's action principle

When quantum effects are important, new action principles are needed. Instead of a particle following a path, quantum mechanics defines a probability amplitude ψ(x_k,t) at one point x_k and time t related to a probability amplitude at a different point later in time: $${\displaystyle \psi (x_{k+1},t+\epsilon )=\frac{1}{A}\int e^{\frac{i}{\hslash }S(x_{k+1},x_k)}\psi (x_k,t)d{x}_k,}$$ where S(x_{k + 1},x_k) is the classical action. Instead of a single path with stationary action, all possible paths add (the integral over x_k), weighted by a complex probability amplitude e^iS⁄ħ. The phase of the amplitude is given by the action divided by the Planck constant or quantum of action: ⁠S/ħ⁠. When the action of a particle is much larger than ħ, ⁠S/ħ⁠ ≫ 1, the phase changes rapidly along the path: the amplitude averages to a small number. Thus the Planck constant sets the boundary between classical and quantum mechanics.

All of the paths contribute in the quantum action principle. At the end point, where the paths meet, the paths with similar phases add, and those with phases differing by π subtract. Close to the path expected from classical physics, phases tend to align; the tendency is stronger for more massive objects that have larger values of action. In the classical limit, one path dominates – the path of stationary action.

Schwinger's action principle

Schwinger's approach relates variations in the transition amplitudes (q_f|q_i) to variations in an action matrix element:

${\displaystyle \delta (q_{r_{\text{f}}}|q_{r_{\text{i}}})=i(q_{r_{\text{f}}}|\delta S|q_{r_{\text{i}}}),}$

where the action operator is

${\displaystyle S={\int }_{t_{\text{i}}}^{t_{\text{f}}}Ldt.}$

The Schwinger form makes analysis of variation of the Lagrangian itself, for example, variation in potential source strength, especially transparent.

Optico-mechanical analogy

For every path, the action integral builds in value from zero at the starting point to its final value at the end. Any nearby path has similar values at similar distances from the starting point. Lines or surfaces of constant partial action value can be drawn across the paths, creating a wave-like view of the action. Analysis like this connects particle-like rays of geometrical optics with the wavefronts of Huygens–Fresnel principle.

[Maupertuis] … thus pointed to that remarkable analogy between optical and mechanical phenomena which was observed much earlier by John Bernoulli and which was later fully developed in Hamilton's ingenious optico-mechanical theory. This analogy played a fundamental role in the development of modern wave-mechanics.

— C. Lanczos

Applications

Action principles are applied to derive differential equations like the Euler–Lagrange equations or as direct applications to physical problems.

Classical mechanics

Action principles can be directly applied to many problems in classical mechanics, such as the shape of elastic rods under load, the shape of a liquid between two vertical plates (a capillary), or the motion of a pendulum when its support is in motion.

Chemistry

Quantum action principles are used in the quantum theory of atoms in molecules (QTAIM), a way of decomposing the computed electron density of molecules in to atoms as a way of gaining insight into chemical bonding.

General relativity

Inspired by Einstein's work on general relativity, the renowned mathematician David Hilbert applied the principle of least action to derive the field equations of general relativity. His action, now known as the Einstein–Hilbert action,

${\displaystyle S=\frac{1}{2\kappa }\int R\sqrt{-g}{d}^{4}x,}$

contained a relativistically invariant volume element √−g d⁴x and the Ricci scalar curvature R. The scale factor ${\displaystyle \kappa }$ is the Einstein gravitational constant.

Other applications

The action principle is so central in modern physics and mathematics that it is widely applied including in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory.

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 has 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 Jacob 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.