Action (physics)
Action | |
---|---|
Common symbols | S |
SI unit | joule-second |
Other units | J⋅Hz−1 |
In SI base units | kg⋅m2⋅s−1 |
Dimension |
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the
In the simple case of a single particle moving with a constant velocity (thereby undergoing
More formally, action is a
Introduction
Introductory physics often begins with Newton's laws of motion, relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages.[1]
Simple example
For a trajectory of a baseball moving in the air on Earth the action is defined between two points in time, and as the kinetic energy minus the potential energy, integrated over time.[4]
The action balances kinetic against potential energy.[4] The kinetic energy of a baseball of mass is where is the velocity of the ball; the potential energy is where is the gravitational constant. Then the action between and is
The action value depends upon the trajectory taken by the baseball through and . This makes the action an input to the powerful
Planck's quantum of action
The Planck constant, written as or when including a factor of , is called the quantum of action.
The energy of light quanta, , increases with frequency , but the product of the energy and time for a vibration of a light wave—the action of the quanta—is the constant .[9]
History
Definitions
Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action. Action has the
Several different definitions of "the action" are in common use in physics.[12][13] The action is usually an integral over time. However, when the action pertains to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.
The action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:[12]
Action (functional)
Most commonly, the term is used for a functional which takes a
Abbreviated action (functional)
In addition to the action functional, there is another functional called the abbreviated action. In the abbreviated action, the input function is the path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path.
The abbreviated action (sometime written as ) is defined as the integral of the generalized momenta,
Hamilton's characteristic function
When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:[12]: 225
This can be integrated to give
which is just the abbreviated action.[16]: 434
Action of a generalized coordinate
A variable Jk in the action-angle coordinates, called the "action" of the generalized coordinate qk, is defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:[16]: 454
The corresponding canonical variable conjugate to Jk is its "angle" wk, for reasons described more fully under action-angle coordinates. The integration is only over a single variable qk and, therefore, unlike the integrated dot product in the abbreviated action integral above. The Jk variable equals the change in Sk(qk) as qk is varied around the closed path. For several physical systems of interest, Jk is either a constant or varies very slowly; hence, the variable Jk is often used in perturbation calculations and in determining adiabatic invariants. For example, they are used in the calculation of planetary and satellite orbits.[16]: 477
Single relativistic particle
When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper time is
If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t1 to t2, then the action becomes
Physical laws are frequently expressed as
Action is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or more generally, is
The action principle provides deep insights into physics, and is an important concept in modern theoretical physics. Various action principles and related concepts are summarized below.
Maupertuis's principle
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). Maupertuis's principle uses the abbreviated action between two generalized points on a path.
Hamilton's principal function
Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.
Hamilton's principle applies not only to the
Hamilton–Jacobi equation
Hamilton's principal function is obtained from the action functional by fixing the initial time and the initial endpoint while allowing the upper time limit and the second endpoint to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.
Euler–Lagrange equations
In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations.
Classical fields
The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field. Maxwell's equations can be derived as conditions of stationary action.
The
Conservation laws
Implications of symmetries in a physical situation can be found with the action principle, together with the
Path integral formulation of quantum field theory
In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, which gives the probability amplitudes of the various outcomes.
Although equivalent in classical mechanics with
Modern extensions
The action principle can be generalized still further. For example, the action need not be an integral, because
See also
- Calculus of variations
- Functional derivative
- Functional integral
- Hamiltonian mechanics
- Lagrangian
- Lagrangian mechanics
- Measure (physics)
- Noether's theorem
- Path integral formulation
- Principle of least action
- Principle of maximum entropy
- Some actions:
References
- ^ ISSN 0031-921X.
- S2CID 250809103.
- ISSN 0002-9505.
- ^ a b c d e "The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action". www.feynmanlectures.caltech.edu. Retrieved 2023-11-03.
- ISBN 0-07-051400-3.
- ISBN 0-07-051400-3.
- ^ "Max Planck Nobel Lecture". Archived from the original on 2023-07-14. Retrieved 2023-07-14.
- S2CID 34765637.
- JSTOR 27825934.
- ISBN 0-19-501496-0.
- ISBN 978-0-486-63773-0.
- ^ ISBN 978-0-521-57572-0
- ISBN 0-89573-752-3(VHC Inc.)
- ^ ISBN 0-679-77631-1
- ISBN 0-07-084018-0
- ^ ISBN 978-0-201-65702-9.
- ^ L. D. Landau and E. M. Lifshitz (1971). The Classical Theory of Fields. Addison-Wesley. Sec. 8. p. 24–25.
- ^ ISBN 978-0-13-146100-0
Further reading
- The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
- Dare A. Wells, Lagrangian Dynamics, Schaum's Outline Series (McGraw-Hill, 1967) ISBN 0-07-069258-0, A 350-page comprehensive "outline" of the subject.
External links
- Principle of least action interactive Interactive explanation/webpage