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1080:, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the
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In the following exposition, we assume that the manifolds are real manifolds, so that cotangent vectors acting on tangent vectors produce real numbers.
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is defined on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of the
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164:
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1312:{\displaystyle \left\{q^{i},q^{j}\right\}=0\qquad \left\{p_{i},p_{j}\right\}=0\qquad \left\{q^{i},p_{j}\right\}=\delta _{ij}}
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which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the
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A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the
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1704:, by the duality between the tangent and cotangent spaces. That is, define a function
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up to a total differential. A change of coordinates that preserves this form is a
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Sets of coordinates on phase space which can be used to describe a physical system
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2062:{\displaystyle X_{q}=\sum _{i}X^{i}(q){\frac {\partial }{\partial q^{i}}}}
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are defined as the momentum functions corresponding to the vectors
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2648:(3rd ed.). San Francisco: Addison Wesley. pp. 347–349.
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s denoting the coordinates on the underlying manifold and the
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formalism. The canonical coordinates satisfy the fundamental
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together form a coordinate system on the cotangent bundle
317:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
2434:, a different set of coordinates are used, called the
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Canonical coordinates are defined as a special set of
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1426:, or from another set of canonical coordinates by a
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coordinates are referred to as "conjugate momenta".
1091:
2634:
2195:{\displaystyle P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}}
171:. Unsourced material may be challenged and removed.
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2324:{\displaystyle p_{i}=P_{\partial /\partial q^{i}}}
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1322:A typical example of canonical coordinates is for
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2490:{\displaystyle \left(q^{i},{\dot {q}}^{i}\right)}
2114:. The conjugate momentum then has the expression
1639:{\displaystyle \sum _{i}p_{i}\,\mathrm {d} q^{i}}
2689:
1700:) can be thought of as a function acting on the
98:but its sources remain unclear because it lacks
1414:Canonical coordinates can be obtained from the
1006:
1751:{\displaystyle P_{X}:T^{*}Q\to \mathbb {R} }
1068:As Hamiltonian mechanics are generalized by
1053:. A closely related concept also appears in
64:Learn how and when to remove these messages
2425:
2181:
1088:(the mathematical notion of phase space).
1013:
999:
1744:
1620:
249:Learn how and when to remove this message
231:Learn how and when to remove this message
129:Learn how and when to remove this message
2263:{\displaystyle \partial /\partial q^{i}}
2103:{\displaystyle \partial /\partial q^{i}}
1543:{\displaystyle \left(x^{i},p_{j}\right)}
1493:{\displaystyle \left(q^{i},p_{j}\right)}
1450:. They are usually written as a set of
14:
2690:
1667:
695:Newton's law of universal gravitation
1927:, the tangent space to the manifold
169:adding citations to reliable sources
140:
70:
29:
2673:, (1978) Benjamin-Cummings, London
2418:; these coordinates are called the
1821:{\displaystyle P_{X}(q,p)=p(X_{q})}
676:Mechanics of planar particle motion
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2438:. These are commonly denoted as
1579:in the cotangent bundle at point
1092:Definition in classical mechanics
45:This article has multiple issues.
1831:holds for all cotangent vectors
1653:; these are a special case of a
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1434:Definition on cotangent bundles
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1063:canonical commutation relations
156:needs additional citations for
53:or discuss these issues on the
2557:{\displaystyle {\dot {q}}^{i}}
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602:Koopman–von Neumann mechanics
2587:Linear discriminant analysis
2110:are the coordinate frame on
670:Non-inertial reference frame
7:
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597:Appell's equation of motion
467:Inertial frame of reference
10:
2739:
1863:{\displaystyle T_{q}^{*}Q}
1590:to be written in the form
2642:; Safko, John L. (2002).
2622:Canonical quantum gravity
2612:Complementarity (physics)
2574:Hamilton–Jacobi equations
1074:canonical transformations
1059:Stone–von Neumann theorem
2671:Foundations of Mechanics
1651:canonical transformation
1428:canonical transformation
1384:. Hence in general, the
1380:to be the components of
760:Rotating reference frame
592:Hamilton–Jacobi equation
84:This article includes a
2597:Symplectic vector field
2436:generalized coordinates
2426:Generalized coordinates
1424:Legendre transformation
1416:generalized coordinates
1078:contact transformations
1047:Hamiltonian formulation
701:Newton's laws of motion
561:Newton's laws of motion
180:"Canonical coordinates"
113:more precise citations.
2617:Canonical quantization
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2411:{\displaystyle T^{*}Q}
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728:Simple harmonic motion
641:Euler's laws of motion
435:D'Alembert's principle
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2708:Hamiltonian mechanics
2698:Differential topology
2640:Poole, Charles P. Jr.
2559:
2519:
2517:{\displaystyle q^{i}}
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2420:canonical coordinates
2413:
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2381:{\displaystyle p_{j}}
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2354:{\displaystyle q^{i}}
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2225:{\displaystyle p_{i}}
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1955:{\displaystyle P_{X}}
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1890:{\displaystyle X_{q}}
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1404:{\displaystyle p_{i}}
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1373:{\displaystyle p_{i}}
1351:Cartesian coordinates
1344:
1342:{\displaystyle q^{i}}
1314:
1162:that are used in the
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1151:{\displaystyle p_{i}}
1126:
1124:{\displaystyle q^{i}}
1102:canonical coordinates
1035:canonical coordinates
582:Hamiltonian mechanics
400:Statistical mechanics
319:
2713:Lagrangian mechanics
2566:generalized velocity
2532:
2526:generalized position
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2432:Lagrangian mechanics
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805:Angular acceleration
797:Rotational frequency
577:Lagrangian mechanics
570:Analytical mechanics
326:Second law of motion
279:
165:improve this article
2703:Symplectic geometry
2645:Classical Mechanics
2592:Symplectic manifold
1977:, the vector field
1856:
1659:symplectic manifold
1098:classical mechanics
1076:are generalized by
1070:symplectic geometry
1051:classical mechanics
1031:classical mechanics
657:Harmonic oscillator
635:Equations of motion
270:Classical mechanics
264:Part of a series on
2718:Coordinate systems
2667:Jerrold E. Marsden
2636:Goldstein, Herbert
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2361:together with the
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1985:may be written as
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1668:Formal development
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1588:canonical one-form
1573:conjugate momentum
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973:Physics portal
587:Routhian mechanics
462:Frame of reference
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86:list of references
18:Conjugate momentum
2602:Symplectomorphism
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2473:
2152:
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2008:
1975:local coordinates
1966:corresponding to
1964:momentum function
1672:Given a manifold
1655:symplectomorphism
1600:
1583:in the manifold.
1055:quantum mechanics
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770:Centrifugal force
765:Centripetal force
721:Euler's equations
706:Relative velocity
482:Moment of inertia
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16:(Redirected from
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2607:Kinetic momentum
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1935:. The function
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1702:cotangent bundle
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1104:are coordinates
1082:cotangent bundle
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876:Johann Bernoulli
871:Daniel Bernoulli
792:Tangential speed
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647:Fictitious force
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494:Mechanical power
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425:Angular momentum
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221:November 2018
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185:
182: –
181:
177:
176:Find sources:
170:
166:
160:
159:
154:This article
152:
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143:
142:
133:
130:
122:
119:November 2018
112:
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2071:
1972:
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1696:
1678:vector field
1671:
1663:
1648:
1585:
1580:
1575:, which are
1572:
1565:
1558:
1551:
1437:
1413:
1321:
1101:
1095:
1067:
1037:are sets of
1034:
1024:
815: /
811: /
809:displacement
807: /
668: /
630:Displacement
568:
559:
553:Formulations
540:Virtual work
480: /
420:Acceleration
413:Fundamentals
245:
227:
218:
208:
201:
194:
187:
175:
163:Please help
158:verification
155:
125:
116:
105:Please help
97:
61:
54:
48:
47:Please help
44:
2570:Hamiltonian
2524:called the
1440:coordinates
1170:relations:
1164:Hamiltonian
1160:phase space
1043:phase space
1039:coordinates
1027:mathematics
951:von Neumann
618:Core topics
111:introducing
2692:Categories
2629:References
2568:. When a
2205:where the
2072:where the
1761:such that
1420:Lagrangian
1057:; see the
886:d'Alembert
866:Maupertuis
829:Scientists
711:Rigid body
385:Kinematics
191:newspapers
50:improve it
2543:˙
2471:˙
2401:∗
2307:∂
2299:∂
2248:∂
2240:∂
2154:∑
2088:∂
2080:∂
2044:∂
2040:∂
2010:∑
1981:at point
1931:at point
1853:∗
1741:→
1733:∗
1602:∑
1550:with the
1298:δ
931:Liouville
813:frequency
733:Vibration
450:potential
375:Continuum
370:Celestial
347:Textbooks
56:talk page
2580:See also
1870:. Here,
1448:manifold
1382:momentum
1086:manifold
986:Category
911:Hamilton
896:Lagrange
891:Clairaut
856:Horrocks
817:velocity
787:Pendulum
775:reactive
747:Rotation
716:dynamics
666:Inertial
652:Friction
535:Velocity
510:Momentum
390:Kinetics
380:Dynamics
358:Branches
342:Timeline
1691:of the
1689:section
1577:1-forms
1442:on the
1418:of the
946:Koopman
906:Poisson
901:Laplace
846:Huygens
841:Galileo
686: (
625:Damping
478:Inertia
472:Impulse
445:kinetic
395:Statics
365:Applied
337:History
205:scholar
107:improve
2677:
2652:
1353:, and
984:
936:Appell
921:Cauchy
916:Jacobi
861:Halley
851:Newton
836:Kepler
688:linear
684:Motion
530:Torque
505:Moment
440:Energy
430:Couple
207:
200:
193:
186:
178:
2497:with
1569:'
1562:'
1557:s or
1555:'
1446:of a
1084:of a
941:Gibbs
926:Routh
881:Euler
520:Speed
515:Space
457:Force
212:JSTOR
198:books
92:, or
2675:ISBN
2665:and
2650:ISBN
2564:the
2528:and
2334:The
1676:, a
1131:and
1072:and
1061:and
1029:and
525:Time
488:Mass
184:news
2430:In
1973:In
1835:in
1687:(a
1683:on
1500:or
1158:in
1096:In
1049:of
1041:on
1025:In
167:by
2694::
2669:,
2638:;
2576:.
2422:.
2270::
2112:TQ
1970:.
1697:TQ
1661:.
1430:.
1100:,
1033:,
96:,
88:,
59:.
2684:.
2658:.
2550:i
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2510:i
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2484:)
2478:i
2468:q
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2397:T
2374:j
2370:p
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2303:/
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2291:=
2286:i
2282:p
2256:i
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2244:/
2218:i
2214:p
2188:i
2184:p
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2176:q
2173:(
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2158:i
2150:=
2147:)
2144:p
2141:,
2138:q
2135:(
2130:X
2126:P
2096:i
2092:q
2084:/
2052:i
2048:q
2035:)
2032:q
2029:(
2024:i
2020:X
2014:i
2006:=
2001:q
1997:X
1983:q
1979:X
1968:X
1948:X
1944:P
1933:q
1929:Q
1915:Q
1910:q
1906:T
1883:q
1879:X
1858:Q
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1844:T
1833:p
1816:)
1811:q
1807:X
1803:(
1800:p
1797:=
1794:)
1791:p
1788:,
1785:q
1782:(
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1773:P
1745:R
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1729:T
1725::
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1527:p
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1459:(
1397:i
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1335:i
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1302:i
1294:=
1290:}
1284:j
1280:p
1276:,
1271:i
1267:q
1262:{
1257:0
1254:=
1250:}
1244:j
1240:p
1236:,
1231:i
1227:p
1222:{
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1214:=
1210:}
1204:j
1200:q
1196:,
1191:i
1187:q
1182:{
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1140:p
1117:i
1113:q
1014:e
1007:t
1000:v
690:)
309:t
306:d
300:p
296:d
290:=
285:F
252:)
246:(
234:)
228:(
223:)
219:(
209:·
202:·
195:·
188:·
161:.
132:)
126:(
121:)
117:(
103:.
66:)
62:(
20:)
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