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Action principles

Action principles are fundamental to physics, from classical mechanics through quantum mechanics , particle physics , and general relativity . [ 1 ] Action principles start with...

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Action principles are fundamental to physics, from classical mechanics through quantum mechanics, particle physics, and general relativity.[1] 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 increased and expanded 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.[2]:162 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: F = ma. This approach to mechanics focuses on a single point in space and time, attempting to answer the question: "What happens next?".[3] 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?[3] The Newtonian and action-principle forms are equivalent, and either one can solve the same problems, but selecting the appropriate form may make solutions much easier.

The energy function in the action principles is not the total energy (conserved in an isolated system), 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.[4]:125

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.[5]:xxv Force requires an inertial frame of reference;[6]:65 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.[7]

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.[8] These diagrammatic conventions reiterate the different strong points of each method.

Diagrammatic aid for forces
Diagrammatic aid for action principle

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:[8]

action S=∫t1t2(KE(t)−PE(t))dt,{\displaystyle S=\int _{t_{1}}^{t_{2}}{\bigl (}{\text{KE}}(t)-{\text{PE}}(t){\bigr )}\,dt,}

where the form of the kinetic energy (KE) and potential energy (PE) 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 = ⁠action/ħ⁠). 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 constantħ (the classical limit), only the stationary action path survives the interference.

El símbolo δ se utiliza para indicar las variaciones de trayectoria , de modo que un principio de acción aparece matemáticamente como

(δS)do=0,{\displaystyle (\delta S)_{C}=0,}

lo que significa que en el punto estacionario , la variación de la acción S con algunas restricciones fijas C es cero. [ 9 ] : 38 Para los principios de acción, el punto estacionario puede ser un mínimo o un punto de silla , pero no un máximo. [ 10 ] Las órbitas planetarias elípticas proporcionan un ejemplo simple de dos caminos con igual acción : uno en cada dirección alrededor de la órbita; ninguno puede ser el mínimo o "menor acción". [ 2 ] : 175 La variación de camino implícita por δ no es lo mismo que un diferencial como dt . La integral de acción depende de las coordenadas de los objetos, y estas coordenadas dependen del camino tomado. Por lo tanto, la integral de acción es un funcional , una función de una función. 

Principios de conservación

Un resultado importante de la geometría, conocido como el teorema de Noether, establece que cualquier cantidad conservada en un lagrangiano implica una simetría continua y viceversa. [ 11 ] Por ejemplo, un lagrangiano independiente del tiempo corresponde a un sistema con energía conservada; la independencia de la traslación espacial implica la conservación del momento; la invariancia de la rotación angular implica la conservación del momento angular. [ 12 ] : 489 Estos ejemplos son simetrías globales, donde la independencia es en sí misma independiente del espacio o del tiempo; las simetrías locales más generales que tienen una dependencia funcional del espacio o del tiempo conducen a la teoría de gauge . [ 13 ] La conservación observada del isospín fue utilizada por Yang Chen-Ning y Robert Mills en 1953 para construir una teoría de gauge para mesones , lo que condujo algunas décadas más tarde a la teoría moderna de la física de partículas . [ 14 ] : 202

Principios distintos

Los principios de acción se aplican a una amplia variedad de problemas físicos, incluyendo toda la física fundamental. Las únicas excepciones importantes son los casos que involucran fricción o cuando solo se dan la posición y las velocidades iniciales. [ 3 ] Los diferentes principios de acción tienen distintos significados para las variaciones; cada aplicación específica de un principio de acción requiere un lagrangiano específico que describa la física. Un nombre común para cualquiera o todos estos principios es "el principio de mínima acción". Para una discusión sobre los nombres y el origen histórico de estos principios, véase nombres de principios de acción .

Puntos finales fijos con energía conservada

Dwyane Wade lanzando tiros libres, ilustrando el tipo de limitaciones físicas adecuadas para la aplicación del principio de mínima acción de Maupertuis.

Cuando la energía total y los puntos finales son fijos, se aplica el principio de mínima acción de Maupertuis . Por ejemplo, para anotar puntos en baloncesto, el balón debe salir de la mano del jugador y pasar por el aro, pero el tiempo de vuelo no está restringido. [ 3 ] El principio de mínima acción de Maupertuis se expresa matemáticamente como la condición estacionaria. (δW)mi=0{\displaystyle (\delta W)_{E}=0} sobre la acción abreviadaW[q] =definición ∫q1q2pag⋅dq,{\displaystyle W[\mathbf {q} ]\ {\stackrel {\text{def}}{=}}\ \int _{q_{1}}^{q_{2}}\mathbf {p} \cdot \mathbf {dq} ,} (a veces escrito S 0 ), donde p = ( p 1 , p 2 ,…, p N ) son los momentos de las partículas o los momentos conjugados de las coordenadas generalizadas , definidos por la ecuación pagk =definición ∂L∂q˙k,{\displaystyle p_{k}\ {\stackrel {\text{def}}{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}},} donde L ( q , q̇ , t ) es el lagrangiano . Algunos libros de texto escriben [ 15 ] : 76 [ 9 ] : 356 ( δW ) E = 0 como Δ S 0 , para enfatizar que la variación utilizada en esta forma del principio de acción difiere de la variación de Hamilton . Aquí la energía total E es fija durante la variación, pero no el tiempo, lo contrario de las restricciones en el principio de Hamilton. [ 16 ] En consecuencia, la misma trayectoria y los mismos puntos finales toman tiempos y energías diferentes en las dos formas. Las soluciones en el caso de esta forma del principio de Maupertuis son órbitas : funciones que relacionan coordenadas entre sí en las que el tiempo es simplemente un índice o un parámetro. [ 16 ]

Potenciales independientes del tiempo; sin fuerzas

Para un sistema invariante en el tiempo, la acciónS{\displaystyle S}se relaciona simplemente con la acción abreviada W en la trayectoria estacionaria como [ 9 ] : 434ΔS=ΔW−miΔt{\displaystyle \Delta S=\Delta WE\Delta t} para la energía E y la diferencia de tiempo Δ t = t 2 − t 1 . Para un cuerpo rígido sin fuerza neta, las acciones son idénticas y los principios variacionales se vuelven equivalentes al principio de Fermat del tiempo mínimo: [ 9 ] : 360δ(t2−t1)=0.{\displaystyle \delta (t_{2}-t_{1})=0.}

Eventos fijos

Para trazar una ruta hacia la Luna, es necesario tener en cuenta el movimiento de la Luna durante el viaje.

Cuando el problema de física da los dos puntos extremos como una posición y un tiempo, es decir como eventos , se aplica el principio de acción de Hamilton . Por ejemplo, imagina planificar un viaje a la Luna. Durante tu viaje, la Luna continuará su órbita alrededor de la Tierra: es un objetivo en movimiento. El principio de Hamilton para objetos en posiciones q ( t ) se escribe matemáticamente como (δS)Δt=0,dónde S[q] =dmiF ∫t1t2L(q(t),q˙(t),t)dt.{\displaystyle (\delta {\mathcal {S}})_{\Delta t}=0,\quad {\text{donde}}\ {\mathcal {S}}[\mathbf {q} ]\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt.} La restricción Δ t = t 2 − t 1 significa que solo consideramos trayectorias que toman el mismo tiempo, además de conectar los mismos dos puntos q ( t 1 ) y q ( t 2 ) . El lagrangianoL=T−V{\displaystyle L=TV}es la diferencia entre la energía cinética y la energía potencial en cada punto de la trayectoria. [ 17 ] : 62 La solución de las ecuaciones resultantes da la línea de universo q ( t ) . [ 3 ] Partiendo del principio de Hamilton, se puede derivar la ecuación diferencial local de Euler-Lagrange para sistemas de energía fija. La acciónS{\displaystyle S}En el principio de Hamilton se encuentra la transformación de Legendre de la acción en el principio de Maupertuis. [ 18 ]

Teoría clásica de campos

Los conceptos y muchos de los métodos útiles para la mecánica de partículas también se aplican a campos continuos. La integral de acción se extiende sobre una densidad lagrangiana, pero los conceptos son tan similares que a menudo la densidad se denomina simplemente lagrangiana. [ 19 ] : 15

Principios de acción cuántica

Para la mecánica cuántica, los principios de acción presentan ventajas significativas: solo se necesita un postulado mecánico, si se utiliza un lagrangiano covariante en la acción, el resultado es relativísticamente correcto y se transicionan claramente a equivalentes clásicos. [ 2 ] : 128

Both Richard Feynman and Julian Schwinger developed quantum action principles based on early work by Paul Dirac. Feynman's integral method was not a variational principle but reduces to the classical least action principle; it led to his Feynman diagrams. Schwinger's differential approach relates infinitesimal amplitude changes to infinitesimal action changes.[2]:138

Feynman's action principle

When quantum effects are important, new action principles are needed. Instead of a particle following a path, quantum mechanics defines a probability amplitude ψ(xk,t) at one point xk and time t related to a probability amplitude at a different point later in time: ψ(xk+1,t+ε)=1A∫eiℏS(xk+1,xk)ψ(xk,t)dxk,{\displaystyle \psi (x_{k+1},t+\varepsilon )={\frac {1}{A}}\int e^{{\frac {i}{\hbar }}S(x_{k+1},x_{k})}\psi (x_{k},t)\,dx_{k},} where S(xk + 1,xk) is the classical action.[20] Instead of a single path with stationary action, all possible paths add (the integral over xk), weighted by a complex probability amplitude eiS⁄ħ. The phase of the amplitude is given by the action divided by the Planck constant or quantum of action: ⁠S/ħ⁠. When the action of a particle is much larger than ħ, ⁠S/ħ⁠ ≫ 1, the phase changes rapidly along the path: the amplitude averages to a small number.[8] Thus the Planck constant sets the boundary between classical and quantum mechanics.[21]

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 π 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.[22]

Schwinger's action principle

Schwinger's approach relates variations in the transition amplitudes (qf|qi) to variations in an action matrix element:

δ(qrf|qri)=i(qrf|δS|qri),{\displaystyle \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=∫titfLdt.{\displaystyle 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.[2]:138

Optico-mechanical analogy

Surfaces of constant action shown as wavefronts perpendicular to paths for the case of light

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.

[Maupertuis] … thus pointed to that remarkable analogy between optical and mechanical phenomena which was observed much earlier by John Bernoulli and which was later fully developed in Hamilton's ingenious optico-mechanical theory. This analogy played a fundamental role in the development of modern wave-mechanics.

Applications

Action principles are applied to derive differential equations like the Euler–Lagrange equations[9]:44 or as direct applications to physical problems.

Classical mechanics

Action principles can be directly applied to many problems in classical mechanics, such as the shape of elastic rods under load,[23]:9 the shape of a liquid between two vertical plates (a capillary),[23]:22 or the motion of a pendulum when its support is in motion.[23]:39

Chemistry

Quantum action principles are used in the quantum theory of atoms in molecules (QTAIM), a way of decomposing the computed electron density of molecules in to atoms as a way of gaining insight into chemical bonding.[24]

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.[25]:186 His action, now known as the Einstein–Hilbert action,

S=12κ∫R−gd4x,{\displaystyle S={\frac {1}{2\kappa }}\int R{\sqrt {-g}}\,d^{4}x,}

contained a relativistically invariant volume element √−gd4x and the Ricci scalar curvatureR. The scale factor κ{\displaystyle \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,[26][27][28]fluid mechanics,[29] the theory of relativity, quantum mechanics,[30]particle physics, and string theory.[31]

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.[32]Hero of Alexandria later showed that this path has the shortest length and least time.[33]

Building on the early work of Pierre Louis Maupertuis, Leonhard Euler, and Joseph-Louis Lagrange defining versions of principle of least action,[34]:580William Rowan Hamilton and in tandem Carl Gustav Jacob Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation.[35]:201

In 1915, David Hilbert applied the variational principle to derive Albert Einstein's equations of general relativity.[36]

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.[37] Subsequently Julian Schwinger and Richard Feynman independently applied this principle in quantum electrodynamics.[38][39]

References

  1. ↑Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia of Physics, John Rigden, editor, Simon & Schuster Macmillan, 1996, volume 2, page 840.
  2. 12345Yourgrau, Wolfgang; Mandelstam, Stanley (1979). Variational principles in dynamics and quantum theory. Dover books on physics and chemistry (Republ. of the 3rd ed., publ. in 1968 ed.). New York, NY: Dover Publ. ISBN 978-0-486-63773-0.
  3. 12345Hanc, Jozef; Taylor, Edwin F.; Tuleja, Slavomir (2005-07-01). "Variational mechanics in one and two dimensions". American Journal of Physics. 73 (7): 603–610. Bibcode:2005AmJPh..73..603H. doi:10.1119/1.1848516. ISSN 0002-9505.
  4. ↑Coopersmith, Jennifer (2017). The lazy universe : an introduction to the principle of least action. Oxford; New York, NY : Oxford University Press. ISBN 978-0-19-874304-0.
  5. 12Lanczos, Cornelius (1986). The variational principles of mechanics (4 ed.). New York: Dover. ISBN 978-0-486-65067-8.
  6. ↑Kleppner, Daniel; Kolenkow, Robert J. (2014). "Chapter 3: Forces and equations of motion". An Introduction to Mechanics (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0521198110.
  7. ↑Moore, Thomas A. (2004-04-01). "Getting the most action out of least action: A proposal". American Journal of Physics. 72 (4): 522–527. Bibcode:2004AmJPh..72..522M. doi:10.1119/1.1646133. ISSN 0002-9505.
  8. 123"Ch. 19: The Principle of Least Action". The Feynman Lectures on Physics. Vol. II. Retrieved 2023-11-03.
  9. 12345Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2008). Classical mechanics (3rd ed., [Nachdr.] ed.). San Francisco, Munich: Addison Wesley. ISBN 978-0-201-65702-9.
  10. ↑Gray, C. G.; Taylor, Edwin F. (May 2007). "When action is not least". American Journal of Physics. 75 (5): 434–458. Bibcode:2007AmJPh..75..434G. doi:10.1119/1.2710480. ISSN 0002-9505.
  11. ↑Hill, E. L. (1951-07-01). "Hamilton's Principle and the Conservation Theorems of Mathematical Physics". Reviews of Modern Physics. 23 (3): 253–260. Bibcode:1951RvMP...23..253H. doi:10.1103/RevModPhys.23.253. ISSN 0034-6861.
  12. ↑Penrose, Roger (2005). The road to reality: a complete guide to the laws of the universe. New York: Alfred A. Knopf. ISBN 978-0-679-45443-4.
  13. ↑Brading, Katherine (March 2002). "Which Symmetry? Noether, Weyl, and Conservation of Electric Charge". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 33 (1): 3–22. Bibcode:2002SHPMP..33....3B. doi:10.1016/S1355-2198(01)00033-8.
  14. ↑Baggott, J. E. (2013). The quantum story: a history in 40 moments (Impression: 3 ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-956684-6.
  15. ↑Hand, Louis N.; Finch, Janet D. (2008). Analytical mechanics (7. print ed.). Cambridge: Cambridge Univ. Press. ISBN 978-0-521-57572-0.
  16. 12Gray, Chris G. (2009-12-09). "Principle of least action". Scholarpedia. 4 (12): 8291. Bibcode:2009SchpJ...4.8291G. doi:10.4249/scholarpedia.8291.
  17. ↑Kibble, T. W. B.; Berkshire, F. H. (2004). Classical Mechanics (5th ed.). Imperial College Press. ISBN 9781860944352.
  18. ↑ Gray, CG; Karl, G.; Novikov, VA (2004-02-01). "Progreso en principios variacionales clásicos y cuánticos" . Reports on Progress in Physics . 67 (2): 159– 208. arXiv : physics/0312071 . Bibcode : 2004RPPh...67..159G . doi : 10.1088/0034-4885/67/2/R02 . ISSN 0034-4885 . 
  19. ↑ Peskin, Michael E. (31 de enero de 2018). Introducción a la teoría cuántica de campos . Boca Raton: CRC Press. doi : 10.1201/9780429503559 . ISBN 978-0-429-50355-9.
  20. ↑ Feynman, RP (1948-04-01). "Enfoque espacio-temporal de la mecánica cuántica no relativista" . Reviews of Modern Physics . 20 (2): 367– 387. Bibcode : 1948RvMP...20..367F . doi : 10.1103/RevModPhys.20.367 . ISSN 0034-6861 . 
  21. ↑ Curtis, Lorenzo J. (2011-09-01). "Una perspectiva del siglo XXI como introducción a la física" . European Journal of Physics . 32 (5): 1259– 1274. Bibcode : 2011EJPh...32.1259C . doi : 10.1088/0143-0807/32/5/014 . ISSN 0143-0807 . 
  22. ↑ Ogborn, Jon; Taylor, Edwin F. (24 de diciembre de 2004). "La física cuántica explica las leyes del movimiento de Newton" (PDF) . Educación en Física . 40 (1): 26–34 . doi : 10.1088/0031-9120/40/1/001 . ISSN 0031-9120 . 
  23. 1 2 3 Dittrich, Walter (2021). El desarrollo del principio de acción: una historia didáctica desde Euler-Lagrange hasta Schwinger . SpringerBriefs in Physics. Cham: Springer International Publishing. doi : 10.1007/978-3-030-69105-9 . ISBN 978-3-030-69104-2.
  24. ↑ Bader, Richard FW (junio de 2005). "La base mecánico-cuántica de la química conceptual" . Monatshefte für Chemie – Chemical Monthly . 136 (6): 819– 854. doi : 10.1007/s00706-005-0307-x . ISSN 0026-9247 . 
  25. ↑Rojo, Alberto; Bloch, Anthony, eds. (2018). "Relativity and Least Action". The Principle of Least Action: History and Physics. Cambridge: Cambridge University Press. pp. 162–188. doi:10.1017/9781139021029.007. ISBN 978-0-521-86902-7.
  26. ↑García-Morales, Vladimir; Pellicer, Julio; Manzanares, José A. (2008). "Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators". Annals of Physics. 323 (8): 1844–1858. arXiv:cond-mat/0602186. Bibcode:2008AnPhy.323.1844G. doi:10.1016/j.aop.2008.04.007. S2CID 118464686.
  27. ↑Gay-Balmaz, François; Yoshimura, Hiroaki (2018). "From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective". Entropy. 21 (1): 8. arXiv:1904.03738. Bibcode:2018Entrp..21....8G. doi:10.3390/e21010008. ISSN 1099-4300. PMC 7514189. PMID 33266724.
  28. ↑Biot, Maurice Anthony (1975). "A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics". Bulletin de la Classe des sciences. 61 (1): 6–30. doi:10.3406/barb.1975.57878. ISSN 0001-4141.
  29. ↑Gray, Chris (2009). "Principle of least action". Scholarpedia. 4 (12): 8291. Bibcode:2009SchpJ...4.8291G. doi:10.4249/scholarpedia.8291.
  30. ↑Feynman, Richard Phillips (1942), The Principle of Least Action in Quantum Mechanics (thesis), Bibcode:1942PhDT.........5F.
  31. ↑"Principle of Least Action – damtp"(PDF). Archived from the original(PDF) on 2015-10-10. Retrieved 2016-07-18.
  32. ↑Helzberger, Max (1966). "Optics from Euclid to Huygens". Applied Optics. 5 (9): 1383–1893. Bibcode:1966ApOpt...5.1383H. doi:10.1364/AO.5.001383. PMID 20057555. In Catoptrics the law of reflection is stated, namely that incoming and outgoing rays form the same angle with the surface normal.
  33. ↑Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 167–168. ISBN 0-19-501496-0.
  34. ↑Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 167–168. ISBN 0-19-501496-0.
  35. ↑Nakane, Michiyo, and Craig G. Fraser. "The Early History of Hamilton-Jacobi Dynamics 1834–1837." Centaurus 44.3-4 (2002): 161–227.
  36. ↑Mehra, Jagdish (1987). "Einstein, Hilbert, and the Theory of Gravitation". In Mehra, Jagdish (ed.). The physicist's conception of nature (Reprint ed.). Dordrecht: Reidel. ISBN 978-90-277-2536-3.
  37. ↑Dirac, Paul A. M. (1933). "The Lagrangian in Quantum Mechanics"(PDF). Physikalische Zeitschrift der Sowjetunion. 3 (1): 64–72.
  38. ↑R. Feynman, Quantum Mechanics and Path Integrals, McGraw-Hill (1965), ISBN 0-07-020650-3.
  39. ↑J. S. Schwinger, Quantum Kinematics and Dynamics, W. A. Benjamin (1970), ISBN 0-7382-0303-3.