Principle Of Least Action
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Action principles lie at the heart of fundamental physics, from classical mechanics through
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
,
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. Action principles start with an energy function called a Lagrangian describing the physical system. The accumulated value of this energy function between two states of the system is called the action. Action principles apply the calculus of variation to the action. The action depends on the energy function, and the energy function depends on the position, motion, and interactions in the system: variation of the action allows the derivation of the equations of motion without vectors or forces. Several distinct action principles differ in the constraints on their initial and final conditions. The names of action principles have evolved over time and differ in details of the endpoints of the paths and the nature of the variation. Quantum action principles generalize and justify the older classical principles by showing they are a direct result of quantum interference patterns. Action principles are the basis for Feynman's version of quantum mechanics, general relativity and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. The action principles have applications as broad as physics, including many problems in classical mechanics but especially in modern problems of quantum mechanics and general relativity. These applications built up over two centuries as the power of the method and its further mathematical development rose. This article introduces the action principle concepts and summarizes other articles with more details on concepts and specific principles.


Common concepts

Action principles are "
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
" approaches rather than the " differential" approach of Newtonian mechanics. The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.


Energy, not force

Introductory study of mechanics, the science of interacting objects, typically begins with Newton's laws based on the concept of
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
, defined by the acceleration it causes when applied to
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
: F = ma. This approach to mechanics focuses on a single point in space and time, attempting to answer the question: "What happens next?". Mechanics based on action principles begin with the concept of action, an energy tradeoff between
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
and
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
, defined by the physics of the problem. These approaches answer questions relating starting and ending points: Which trajectory will place a basketball in the hoop? If we launch a rocket to the Moon today, how can it land there in 5 days? The Newtonian and action-principle forms are equivalent, and either one can solve the same problems, but selecting the appropriate form will make solutions much easier. The energy function in the action principles is not the total energy ( conserved in an isolated system), but the Lagrangian, the difference between kinetic and potential energy. The kinetic energy combines the energy of motion for all the objects in the system; the potential energy depends upon the instantaneous position of the objects and drives the motion of the objects. The motion of the objects places them in new positions with new potential energy values, giving a new value for the Lagrangian. Using energy rather than force gives immediate advantages as a basis for mechanics. Force mechanics involves 3-dimensional vector calculus, with 3 space and 3 momentum coordinates for each object in the scenario; energy is a scalar magnitude combining information from all objects, giving an immediate simplification in many cases. The components of force vary with coordinate systems; the energy value is the same in all coordinate systems. Force requires an inertial frame of reference; once velocities approach the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
,
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
profoundly affects mechanics based on forces. In action principles, relativity merely requires a different Lagrangian: the principle itself is independent of coordinate systems.


Paths, not points

The explanatory diagrams in force-based mechanics usually focus on a single point, like the center of momentum, and show vectors of forces and velocities. The explanatory diagrams of action-based mechanics have two points with actual and possible paths connecting them. These diagrammatic conventions reiterate the different strong points of each method. Depending on the action principle, the two points connected by paths in a diagram may represent two particle positions at different times, or the two points may represent values in a configuration space or in a phase space. The mathematical technology and terminology of action principles can be learned by thinking in terms of physical space, then applied in the more powerful and general abstract spaces.


Action along a path

Action principles assign a number—the action—to each possible path between two points. This number is computed by adding an energy value for each small section of the path multiplied by the time spent in that section: : action S = \int_^ \left( \text(t) - \text(t)\right) \,dt, where the form of the kinetic (\text) and potential (\text) energy expressions depend upon the physics problem, and their value at each point on the path depends upon relative coordinates corresponding to that point. The energy function is called a Lagrangian; in simple problems it is the kinetic energy minus the potential energy of the system.


Path variation

In classical mechanics, a system moving between two points takes one particular path; other similar paths are not taken. Each conceivable path corresponds to a value of the action. An action principle predicts or explains that the particular path taken has a stationary value for the system's action: similar paths near the one taken have very similar action value. This variation in the action value is key to the action principles. In quantum mechanics, every possible path contributes an amplitude to the system's behavior, with the phase of each amplitude determined by the action for that path (phase = action/ \hbar). The classical path emerges because: * Only near the path of stationary action do neighboring paths have similar phases, leading to constructive interference, * Neighboring paths have rapidly varying actions with the phase that interfere with other paths, When the scale of the problem is much larger than the Planck constant \hbar (the classical limit), only the stationary action path survives the interference. The symbol \delta is used to indicate the path variations so an action principle appears mathematically as : (\delta S)_C = 0, meaning that at the stationary point, the variation of the action S with some fixed constraints C is zero. For action principles, the stationary point may be a minimum or a saddle point, but not a maximum. Elliptical planetary orbits provide a simple example of two paths with equal action one in each direction around the orbit; neither can be the minimum or "least action". The path variation implied by \delta is not the same as a differential like dt. The action integral depends on the coordinates of the objects, and these coordinates depend upon the path taken. Thus the action integral is a functional, a function of a function.


Conservation principles

An important result from geometry known as Noether's theorem states that any conserved quantities in a Lagrangian imply a continuous symmetry and conversely. For examples, a Lagrangian independent of time corresponds to a system with conserved energy; spatial translation independence implies momentum conservation; angular rotation invariance implies angular momentum conservation. These examples are global symmetries, where the independence is itself independent of space or time; more general ''local'' symmetries having a functional dependence on space or time lead to gauge theory. The observed conservation of isospin was used by Yang Chen-Ning and Robert Mills in 1953 to construct a gauge theory for
meson In particle physics, a meson () is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, the ...
s, leading some decades later to modern particle physics theory.


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 (\delta W)_E = 0 on the abbreviated action W mathbf \stackrel\ \int_^ \mathbf \cdot \mathbf, (sometimes written S_0), where \mathbf = (p_1, p_2, \ldots, p_N) are the particle momenta or the conjugate momenta of generalized coordinates, defined by the equation p_k\ \stackrel\ \frac, where L(\mathbf, \dot, t) is the Lagrangian. Some textbooks write (\delta W)_E = 0 as \Delta S_0, 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 In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...
: 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 S relates simply to the abbreviated action W on the stationary path as \Delta S = \Delta W - E\Delta t for energy E and time difference \Delta t = t_2 - t_1. 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: \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's a moving target. Hamilton's principle for objects at positions \mathbf(t) is written mathematically as (\delta \mathcal)_ = 0,\ \mathrm\ \mathcal mathbf \stackrel\ \int_^ L(\mathbf(t), \dot(t), t) \,dt. The constraint \Delta t = t_2 - t_1 means that we only consider paths taking the same time, as well as connecting the same two points \mathbf(t_1) and \mathbf(t_2). The Lagrangian 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 The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
\mathbf(t). Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy. The action S in Hamilton's principle is the Legendre transformation of the action in Maupertuis' 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 In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
. 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 \psi(x_k, t) at one point x_k and time t related to a probability amplitude at a different point later in time: \psi(x_, t + \varepsilon) = \frac \int e^ \psi(x_k, t) \,dx_k, where S(x_, x_k) is the classical action. Instead of single path with stationary action, all possible paths add (the integral over x_k), weighted by a complex probability amplitude e^. The phase of the amplitude is given by the action divided by the Planck constant or quantum of action: S/\hbar. When the action of a particle is much larger than \hbar, S/\hbar \gg 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 \pi 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_\text, q_\text) to variations in an action matrix element: : \delta(q_, q_) = i(q_, \delta S, q_), where the action operator is : S = \int_^ L \,dt. 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.


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 Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, e.g. the shape of elastic rods under load, the shape of a liquid between two vertical plates (a
capillary A capillary is a small blood vessel, from 5 to 10 micrometres in diameter, and is part of the microcirculation system. Capillaries are microvessels and the smallest blood vessels in the body. They are composed of only the tunica intima (the inn ...
), 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 General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, 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, : S = \frac \int R \sqrt \, \mathrm^4x, contained a relativistically invariant volume element \sqrt \, \mathrm^4x and the Ricci scalar curvature R. The scale factor \kappa is the Einstein gravitational constant.


Other applications

The action principle is so central in modern physics and
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
that it is widely applied including in
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
,
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
, the
theory of relativity The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical ph ...
,
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
,
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, and string theory.


History

The action principle is preceded by earlier ideas in
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
. In
ancient Greece Ancient Greece () was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity (), that comprised a loose collection of culturally and linguistically r ...
,
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, and
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia William Rowan Hamilton and in tandem Carl Gustav Jacob Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation.Nakane, Michiyo, and Craig G. Fraser. "The Early History of Hamilton-Jacobi Dynamics 1834–1837." Centaurus 44.3-4 (2002): 161–227. In 1915, David Hilbert applied the variational principle to derive
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's equations of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. 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.J. S. Schwinger, Quantum Kinematics and Dynamics, W. A. Benjamin (1970), .


References

{{reflist, 30em Dynamics (mechanics) Classical mechanics