What is Least action: Definition and 59 Discussions

This article discusses the history of the principle of least action. For the application, please refer to action (physics).The stationary action principle – also known as the principle of least action – is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are stationary points (a.k.a. critical points) of the system's action functional. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal (even locally) on the trajectories.The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). In relativity, a different action must be minimized or maximized.
The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics. 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.The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.
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 was the shortest length and least time.Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746. However, Leonhard Euler discussed the principle in 1744, and evidence shows that Gottfried Leibniz preceded both by 39 years.

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  1. H

    I Noether's second theorem: two questions

    A technical subject, well above my level it seems (I'm still learning about quantum physics and special relativity), but one about which I absolutely must get some clear ideas as soon as possible. From what I 'understand', Noether's second theorem applies to infinite-dimensional symmetry...
  2. O

    Modifying Euler-Lagrange equation to multivariable function

    I'm confused on how to derive the multidimensional generalization for a multivariable function. Everything makes sense here except the line, $$ \frac{\delta S}{\delta \psi} = \frac{\partial L}{\partial \psi} - \frac{d}{dx} \frac{\partial L}{\partial(\frac{\partial \psi}{\partial x})} -...
  3. R

    A Aligning effect in uniform field

    I would like to discuss the nature of the following effect. At whatever angle and with whatever initial speed the particle fly into a uniform potential field, over time the directions of the instantaneous velocity and field strength converge. The kinematics and dynamics here are trivial, but I...
  4. Delta2

    I The principle of least action

    From what I know Newton's 2nd law in classical mechanics can be derived from the principle of least action. Also from what I know, two of the Maxwell's equations (those that contain the time derivatives, i.e. Maxwell-Faraday law and Maxwell-Ampere law) also can be derived from the principle of...
  5. Haorong Wu

    I Variational operator in the least action principle

    Hello. Since I learned the least action principle several years ago, I cannot figure out the difference between the variational operator ##\delta## in ##\delta S=0## and the differential operator ##d## in, say ##dS##. Everytime I encountered the variational operator, I just treated it as a...
  6. hyksos

    A Derive the Principle of Least Action from the Path Integral?

    Several weeks ago I had considered the question as to how one can start from the Schroedinger Equation, and after several transformations, derive F=ma as a limiting case. At some point in my investigations of this derivation, it occurred to me that this is simply too much work. While in...
  7. JD_PM

    Euler-Lagrange equations and the principle of least action

    Summary:: I am missing something in my integration by parts Consider the infinitesimal variation of the fields ##\phi_a (x)## $$\phi_a \rightarrow \phi_a + \delta \phi_a$$ The infinitesimal variation vanishes at the boundary of the region considered (ie. ##\delta \phi (x) = 0## at the...
  8. N

    Laplace's equation vs Principle of Least Action?

    There seems to be a similarity between the solutions of laplace's equation and the principle of least action. e.g. the solution of a one dimensional laplace equation is a straight and the curve that minimizes the action is also a straight line. Was one derived from the other? Newbie here. Id...
  9. dRic2

    Jacobi's principle of least action

    I'm studying a chapter on the parametric form of the canonical equation so basically the author says that time is no more an independent variable but it is expressed as a function of an other variable called ##\tau##. In this way the canonical integral is reduced to the one I've written in...
  10. Andy_K

    B Quantum Wave & Principle of Least Action

    Dear All, I would like to better understand how the Principle of Least Action applies in observations / measurements in quantum physics. Does the wave function of a particle correspond directly to the principle of least action, as in, the positions with higher probability of detecting the...
  11. S

    B Double Slit experiment and the path of least Action

    Using the principle of least action can you figure out which path the photon took, or which slit it went thru given some initial condition. Or is this not possible and why.
  12. LarryS

    Principle of Least Action for optics?

    Fermat's Principle states that light always travels the path of least time. In Classical Physics, other than the above, is there a separate "Principle of Lease Action" for light? Thanks in advance.
  13. C

    I Does the Principle of Least Action Have a Physical Meaning?

    I have found that some people say “yes, definitely”, and other days “no, definitely not”. Those who say “no” seem to regard PLA as merely a neat way of packaging the equations. Those who say “yes” seem to regard PLA as somehow fundamental. (There have actually been two recent books on this...
  14. BookWei

    I Is the Action Always a Minimum in the Principle of Least Action?

    Hello, When we applying the principle of least action, we require ##\delta S=0##, which corresponding to the action S being an extremum. I am just wondering why do we say that the action is a minimum instead of a maximum for a physical path? Can I use the path integral to explain this problem...
  15. Ben Geoffrey

    I Definite Integrals and the Principle of Least Action: Exploring the Connection

    This is with regard to my doubt in the derivation of the principle of least of action in Goldstein Is there any theorem in math about definite integrals like this ∫a+cb+df(x)dx = f(a)c-f(b)d The relevant portion of the derivation is given in the image.
  16. J

    I Feynman's Thesis, 3 Principles

    Hi, I've just started studying Feynman's thesis and am in need of some discussion regarding the three principles he put forward on the development of his 'Principle of least action in quantum mechanics'. The three principles are 1) The acceleration of a point charge is due to the sum of its...
  17. P

    Euler-Lagrange Equations for geodesics

    Homework Statement The Lagrange Function corresponding to a geodesic is $$\mathcal{L}(x^\mu,\dot{x}^\nu)=\frac{1}{2}g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$ Calculate the Euler-Lagrange equations Homework Equations The Euler Lagrange equations are $$\frac{\mathrm{d}}{\mathrm{d}s}...
  18. G

    I Classical/QM justification of principle of least action

    Hi. Is the principle of least (better: stationary) action only an axiom in classical mechanics, or can it be derived from a more profound (classical) principle? As far as I know, it can be derived from the path integral formulation of QM. Is this a more profound justification for the principle...
  19. S

    The principle of least Action proof of minimum

    Homework Statement Reading Feynman The Principle of Least Action out of The Feynman Lectures on Physics, Vol 2. Link to text http://www.feynmanlectures.caltech.edu/II_19.html So I'm having a problem proving that, section 19-2 5th paragraf, that "Now the mean square of something that deviates...
  20. StrawberrySaturn

    The principle of least action and diffraction

    When reading through one of the feynman lectures (http://www.feynmanlectures.caltech.edu/II_19.html) there was a paragraph that said: "In the case of light we also discussed the question: How does the particle find the right path? From the differential point of view, it is easy to understand...
  21. Y

    A Least Action Principle Applied to Vector Field ##A_{\mu}##

    hello, everyone. When a vector field ##A_{\mu}## has the Lagrangian of the form as ##L=Const.{\times}F^{\mu\nu}F_{\mu\nu}##, where...
  22. C

    What if the Principle of Least Action were different?

    What would the world be like if the Principle of Least Action were different? Let's say that it minimized a different quantity than KE - PE. EF Taylor et al argue that if the quantity minimized were KE + PE, physical systems would accelerate apart from one another. Here's their short articlet...
  23. S

    Principle of least action in field theory

    In page 15, Peskin and Schroeder states that The principle of least action states that when a system evolves from one given configuration to another between times ##t_1## and ##t_2##, it does so along the path in configuration space for which ##S## is an extremum. What is the definition of...
  24. amjad-sh

    Principle of least action ...some confusion

    Principle of least action states that the particle choose the path which optimizes its action.Where the action is defined by S=t1∫t2(L)dt and L is the Lagrangian of the system.This leads to δS=0 and it is a condition to optimize S. I will summarize what confuses me about this principle in a...
  25. Last-cloud

    Obtain Equation Using Hamilton's Principle

    I want to obtain equation using Hamilton principle but I just couldn't figure it out; i have The kinetic energy : \begin{equation} E_{k}=\dfrac{1}{2}m_{z} \displaystyle\int\limits_{0}^{L}\ \left[ \left( \dfrac{\partial w(x,t)}{\partial t}\right)^{2}+\left( \dfrac{\partial v(x,t)}{\partial...
  26. V

    Total derivative in action of the field theory

    When applying the least action I see that a term is considered total derivative. Two points are not clear to me. We say that first $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu \phi)}\delta \phi) d^4x= \int d(\frac {\partial L}{\partial(\partial_\mu \phi)}\delta \phi)= (\frac...
  27. adoion

    Exploring the Principle of Least Action in Physics

    hi, please if somebody could explain why anybody would consider the "action" and is there any proof that the minimal action actually gives the correct route of a problem?
  28. I

    Interesting book explaining the principle of least action in detail

    Hi, I'm doing further maths and I would like to study maths at university. I have been asked to read a number of books to put on my personal statement, and as I am finding it difficult coming to terms with the fact that mechanics is taught as maths and not physics, a maths don at Oxford...
  29. C

    Relationship between Principle of Least Action and Continuity Equation

    Is there a profound relationship between Principle of Least Action and Continuity Equation? Can we derive one from another?
  30. E

    Principle of least action used to predict particle movement?

    I am having a tough time understanding the principle of least action and I would really appreciate it if someone would clear it up for me. And from what I understand from it, it can be used to predict particle motion?
  31. M

    Valid curves with least action

    Greetings, I'm simulating the principle of least action for simple object motion and reading from Feynman Vol. 2, Chpt 19 -- The Principle of Least Action. He states (with my paraphrasing) that the true path of a trajectory is the one for which the integral over all points of kinetic energy...
  32. B

    Principle of Least Action via Finite-Difference Method

    I have to be honest, the principle of least action seems to me more of a religious claim one takes on complete faith, though of course I'm hoping this is just because I don't understand it. I tried to explain this to a friend suffering through a mechanics class & was literally pushed to say 'one...
  33. anorlunda

    Principle of least action as a law?

    We say that the laws of nature (e.g. Newton's Laws, relativity, ...) must be confirmed and must be checked for invariance (e.g Lorentz, gauge, ...). Yet many of these laws may be directly derived from the principle of least action. Why do we not consider the principle of least action a law...
  34. nomadreid

    Confused about the principle of least action

    To make the confusions both concise and explicit, I will put down some incorrect calculations, and ask for corrections Take the Lagrangian KE - PE = T - V Action = S=∫ L dt (with given limits) Principle of least action: δS= 0: S(t1)-S(t2) =0 if t1-t2 is small (using the (.) as function...
  35. A

    Lagrangian and principle of least action

    So the integral of the lagrangian over time must be stationary according to hamiltons principle. One can show that this leads to the euler lagrange equations, one for each pair of coordinates (qi,qi'). But my book has now started on defining a generalized lagrangian where lagrangian...
  36. D

    Exploring Nature's Preference for Least Action

    Hamilton's principle says that the action for the true path that a system follows will be stationary. As I understand it, the action is almost always least. Is there a reason why nature prefers least action rather over greatest action?
  37. T

    Equation of Motion: Deriving from Principle of Least Action

    I'm going through Landau/Lifgarbagez's book II of theoretical physics. In it they have a derivation of the equation of motion from the principle of least action, however I don't understand one step. Homework Statement Derive the equation of motion: \frac{d^2x^i}{ds^2}+\Gamma^i_{kj}...
  38. Q

    Formulating the correct least action expression

    I am trying to understand how, for a given physical system it is possible to come up with the correct expression for the lagrangian and thence the least action formulation from which the governing equations can be derived by application of the euler lagrange equation. From what I have seen so...
  39. M

    Is the principle of least action a tautology ?

    In classical mechanics, if we consider a force field (uniform or non-uniform) in which the acceleration \vec{a}_{\scriptscriptstyle \mathrm A} of a particle A is constant, then \vec{a}_{\scriptscriptstyle \mathrm A} - \, \vec{a}_{\scriptscriptstyle \mathrm A} = 0{\vphantom{\delta...
  40. M

    On the principle of least action in classical mechanics

    The principle of least action applicable in an uniform field can be obtained as follows: Particle A \vec{a}_A = \vec{a}_A \int \vec{a}_A \cdot d\vec{r}_A = \int \vec{a}_A \cdot d\vec{r}_A \int \vec{a}_A \cdot d\vec{r}_A = \Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 \int \vec{a}_A...
  41. M

    Principle of Least Action help me clear it up

    I've been reading Landau and Lifgarbagez's Mechanics and have some issues I need clearing up, so I hope folk here can help :) It is stated that the Principle of Least Action is not always valid for the entire path of a system, but only for a sufficiently small segment...what does this mean...
  42. M

    Principle of Least Action - not always valid?

    It is stated in Landau & Lifgarbagez Mechanics that the Principle of Least Action is not always valid for the entire path of a system in phase space, but only for a sufficiently small segment of the path. Can anyone expand on this? How can we be sure that when we derive Lagrange's Equation...
  43. T

    Principle of Least Action Derivation

    There is one step I am having trouble understanding in the derivation of the principle of least action which leads to the Euler-Lagrange equations. When you have the variation of the action: \delta S = \int_{t_1}^{t_2} [ L(q+\delta q, \dot q + \delta \dot q, t ) - L(q, \dot q, t ) ] \...
  44. A

    Principle of the least action

    Hi! I'm new of this forum and I'm searching a way to understand why the action is E-U, but in this moment i don't know how to do...ther's someone who can help me? Thank you
  45. P

    Least action principle for a free relativistic particle (Landau)

    Reading the Landau's "The classical theory of fields" (chapter 2, section 9 ) I have some doubts in explaining the steps in derivig the formula for the variation of the action for the relativistic free particle...
  46. J

    Woes With the Principle of Least Action.

    I am now attempting to figure out how to calculate trajectories using the highly coveted "principle of least action". I apologize beforehand if this is more of a mathematical problem than a problem that needs to be placed under classical mechanics. I also apologize if I can't do the Latex...
  47. N

    Why is the action stationary for the true path in mechanics?

    What is the physically intuitive reason behind the fact that the action is stationary for the true path a particle takes? I understand that a path that satisfies the euler-lagrange equation minimizes (or maximizes or "saddle-points") the result of the action functional. I also understand...
  48. R

    Principle of Least Action - Straight Worldline on a Geodesic

    What does it mean to say that something moves on a straight wordline in terms of the principle of least action? I know it generally means that action is minimum or stationary but since I only really know some physics from a conceptual standpoint and not a mathematical one I don't really know...
  49. N

    Principle of Least Action in Electrostatics

    In Feynman's lecture on the principle of least action, he show that you can describe electrostatics by saying that a certain integral is a minimum or maximum. He states the integral, and goes on to show how it can be used to approximate the capacitance of a coaxial line. I've been able to...
  50. S

    Principle of Least Action OR Hamilton's Principle

    Are the principle of least action(http://astro.berkeley.edu/~converse/Lagrange/Kepler%27sFirstLaw.htm) and the hamilton principle 'exactly' the same? As far as I know, yes. How do I go from one to the other