Action principles

Action principles are "integral" 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, defined by the acceleration it causes when applied to mass: . 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 and potential energy, 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? Using energy rather than force gives immediate advantages as a basis for mechanics. Force mechanics involves three-dimensional vector calculus, with three space and three 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. once velocities approach the speed of light, special relativity 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. : action S = \int_{t_1}^{t_2} \bigl( \text{KE}(t) - \text{PE}(t)\bigr) \,dt, where the form of the kinetic energy () and potential 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 = ). 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 (the classical limit), only the stationary action path survives the interference. The symbol 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 with some fixed constraints 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". 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 mesons, 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. 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. Solution of the resulting equations gives the world line . 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. Instead of a single path with stationary action, all possible paths add (the integral over ), weighted by a complex probability amplitude . The phase of the amplitude is given by the action divided by the Planck constant or quantum of action: . When the action of a particle is much larger than , , the phase changes rapidly along the path: the amplitude averages to a small number. 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 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 to variations in an action matrix element: : \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 : S = \int_{t_\text{i}}^{t_\text{f}} 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 the shape of a liquid between two vertical plates (a capillary), 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, : S = \frac{1}{2\kappa} \int R \sqrt{-g} \, d^4x, contained a relativistically invariant volume element and the Ricci scalar curvature . The scale factor \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. == References ==