Action principles


Action principles are fundamental to physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. 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.
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" 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? 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, 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, 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. 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:
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. 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 , only the stationary action path survives the interference.
The symbol is used to indicate the path variations so an action principle appears mathematically as
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". The path variation implied by is not the same as a differential like. 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 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. 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
on the abbreviated action
, where are the particle momenta or the conjugate momenta of generalized coordinates, defined by the equation
where is the Lagrangian. Some textbooks write as, to emphasize that the variation used in this form of the action principle differs from Hamilton's variation. Here the total energy 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 relates simply to the abbreviated action on the stationary path as
for energy and time difference. 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: