Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Feynman's formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations can be derived as conditions of stationary action.
Many problem in of physics can be represented and solved in the form of an action principle, such as finding the quickest way to run down the beach for reaching a drowning person. Water running downhill seeks the steepest descent, the quickest way down, and water running into a basin distributes itself so that its surface is as low as possible. Light finds the quickest trajectory through an optical system (Fermat's principle of Least Time). The path of a body in a graviational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.
Symmetries in a physical situation can better be treated with the action principle, together with the Euler-Lagrange equations which are derived from the action principle. For example, Emmy Noether's theorem which states that with every continuous symmetry in a physical situation there corresponds a conservation law. This deep connection, however, requires that the action principle is assumed.
In classical mechanics (non-relativistic, non-quantum mechanics), the correct choice of the action can be proven from Newton's laws of motion. Conversely, the action principle proofs Newton's equation of motion given the correct choice of action. So in classical mechanics the action principle is equivalent to Newton's equation of motion. The use of the action principle often is simpler than the direct application of Newton's equation of motion. The action principle is a scalar theory, with derivations and applications that employ elementary calculus.
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2 Action principle in classical mechanics 3 Euler-Lagrange equations for the action integral 4 See also 5 Literature 6 External Links |
The Principle of Least Action was first formulated by Maupertuis [1] in 1746 and further developed (from 1748 onwards) by the mathematicians Euler, Lagrange, and Hamilton.
Maupertuis arrived at this principle from a feeling that the very perfection of the universe demands a certain economy in nature and is opposed to any needless expenditure of energy. Natural motions must be such as to make some quantity a minimum. It was only necessary to find that quantity, and this he proceeded to do. It was the product of the duration (time) of movement within a system by the "vis viva" or twice what we now call the kinetic energy of the system.
Euler (in "Reflexions sur quelques loix generales de la nature..", 1748) adopts the least-action principle, calling the quantity "effort". His expression corresponds to what we would now call potential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.
Newton's laws of motion can be stated in various alternative ways. One of them is the Lagrangian formalism, also called Lagrangian mechanics. If we denote the trajectory of a particle as a function of time as , with a velocity , than the Lagrangian is a function depending these quantities and possibly also explicitely on time:
History
Action principle in classical mechanics
The action integral is the integral of the Lagrangian over time between a given starting point at time and a given end point at time
In Lagrangian mechanics, the trajectory of an object is derived by finding the path for which the action integral is stationary (a minimum or a saddle point). The action integral is a functional (a function depending on a function, in this case ).
For a system with conservative forces (forces that can be described in terms of a potential, like the gravitational force and not like friction forces), the choice of a Lagrangian as the kinetic energy minus the potential energy results in the correct laws of Newtionain mechanics
(Note that the sum of kinetic and potential energy is the total energy of the system).
The stationary point of an integral along a path is equivalent to a set of
differental-equations, called the Euler-Lagrange equations. This can be seen as follows where we restrict ourselves to one coordinate only. The extention to more coordinates is straightforward.
Suppose we have an action integral of an integrand which depends on
coordinates and ,
its derivatives with respect to :
The difference between the integrals along curve one and along curve two is:
Where we have replaced for , since this must hold for every coordinate.
This set of equations is called the Euler-Lagrange Equations for the variational problem.
An important simple consequence of these equations is that
if does not explicitly contain coordinate ,
i.e.
Trivial examples help to appreciate the use of the action principle via the Euler-Lagrangian equations. A free particle (mass and velocity ) in Euclidean space moves in a straight line. Using the Euler-Lagrange equations, this can be shown in polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy in orthonormal coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time ).
In polar coordinates the kinetic energy and hence the Lagrangian becomes
For an annotated bibliography, see Edwin F. Taylor
[[1]
who lists, among other things, the following books
Euler-Lagrange equations for the action integral
Consider a second curve which starts and ends at the
same points as the first curve, and assume that the distance
between the two curves is small everywhere:
is small.
At the begin and endpoint we have .
where we have used the first order expansion of in
and .
Now use partial integration on the last term
and use the conditions to find:
reaches a stationary point (an extremum), i.e.
for each .
Note that this is the only requirement: the extremum could either be a minimum, saddle-point or formally even a maximum.
for each if and only if
Such a coordinate is called a cyclic coordinate,
and is called the
conjugate momentum, which is conserved.
For example if doesnot depend on time, the associated constant of motion
(the conjugate momentum) is called the energy. If we use spherical coordinates
and doesnot depend on ,
the conjugate momentum is the conserved angular momentum.Example: Free particle in polar coordinates
The radial and components of the Euler-Lagrangian equations become, respectively
The solution of these two equations is given by
for a set of constants determined by initial conditions.
Thus, indeed, the solution is a straight line given in polar coordinates.
The formalisms above are valid in classical mechanics in a very restrictive sense of the term. More generally, an action is a functional from the configuration space to the real numbers and in general, it needn't even necessarily be an integral because nonlocal actions are possible. The configuration space needn't even necessarily be a functional space because we could have things like noncommutative geometry.See also
Literature
External Links