Referentni sistem — разлика између измена

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Верзија на датум 30. мај 2023. у 09:14

U fizici, referentni sistem (ili referentni okvir) se sastoji od apstraktnog koordinatnog sistema i seta fizičkih referentnih tačaka koje jedinstveno fiksiraju (lociraju i orijentišu) koordinatni sistem i standardizuju merenja unutar tog okvira.

Za n dimenzija, n + 1 referentne tačke su dovoljne da u potpunosti definišu referentni okvir. Korištenjem pravougaonih (kartezijanskih) koordinata, referentni okvir može se definisati s referentnom tačkom u koordinatnom početku i referentnom točkom na jednom jedinstvenom razmaku duž svake od n koordinatnih osa.

U Ajnštajnovoj relativnosti, referentni okviri se koriste za specifikaciju odnosa između pokretnog posmatrača i pojave, ili pojava pod posmatranjem. U tom kontekstu, fraza često postaje „opservacioni okvir reference” (ili „referentni okvir posmatranja”), što podrazumeva da je posmatrač u okviru u stanju mirovanja, iako nije nužno lociran u koordinatnom početku. Relativistički referentni okvir uključuje (ili podrazumeva) koordinatno vreme, koje se ne izjednačava kroz različite okvire koji se kreću relativno jedan u odnosu na drugi. Situacija se stoga razlikuje od Galilejske relativnosti, gde su sva moguća koordinatna vremena esencijalno jednaka.

Različiti aspekti „referentnog okvira”

Potreba da se razlikuju različita značenja „referentnog okvira“ dovela je do različitih pojmova. Na primer, ponekad je tip koordinatnog sistema priključen kao modifikator, kao u kartezijskom referentnom okviru. Ponekad se naglašava stanje kretanja, kao u rotirajućem referentnom okviru. Ponekad je naglašen način na koji se transformiše u okvire koji se smatraju povezanima kao u Galilejevom referentnom okviru.^[1] Ponekad se okviri razlikuju po obimu njihovih opservacija, kao u makroskopskim i mikroskopskim referentnim okvirima.^[2]

U ovom članku se termin referentni okvir posmatranja koristi kada je naglasak na stanju kretanja, a ne na izboru koordinata ili karakteru posmatranja ili posmatračkog aparata. U tom smislu, referentni okvir posmatranja omogućava proučavanje uticaja kretanja na čitavu porodicu koordinatnih sistema koji bi mogli biti priključeni na ovaj okvir. S druge strane, koordinatni sistem se može koristiti u mnoge svrhe u kojima stanje kretanja nije glavni predmet interesa. Na primer, može se primeniti dati koordinatni sistem da bi se iskoristila simetrija sistema. Gledano sa još šire perspektive, formulacija mnogih problema iz fizike koristi generalizovane koordinate, normalne modove ili sopstvene vektore, koji su samo posredno povezani sa prostorom i vremenom. Stoga je korisno da se razdvoje različiti aspekti referentnog okvira radi diskusije u nastavku. Referentni okviri posmatranja, koordinatni sistemi i opservaciona oprema se uzimaju kao nezavisni koncepti.

Okvir posmatranja (kao što je inercijalni okvir ili neinercijalni referentni okvir) fizički je koncept povezan sa stanjem kretanja.
Koordinatni sistem je matematički koncept, koji se sastoji od izbora jezika korištenog za opisivanje opažanja.^[3] Shodno tome, posmatrač u posmatračkom okviru može izabrati da koristi bilo koji koordinatni sistem (kartezijanski, polarni, krivolinijski, generalizovani ...) da bi opisao zapažanja sagledana iz tog referentnog okvira. Promena izbora ovog koordinatnog sistema ne menja posmatračevo stanje kretanja, i ne podrazumeva promenu u referentnom okviru posmatrača. Ovo gledište se može naći i drugde.^[4] Neosporno je da su neki koordinatni sistemi bolji izbor za neka zapažanja od drugih.
Izbor merene veličine i opservacionog aparata je zasebno pitanje od posmatračevog stanja kretanja i izbora koordinatnog sistema.

Brading i Kastelani su diskusiju odveli izvan jednostavnih sistemsko-vremenskih koordinatnih sistema.^[5] Proširenje na koordinatne sisteme koristeći generalizovane koordinate u osnovi je Hamiltonskih i Lagranžovih formulacija^[6] kvantne teorije polja, klasične relativističke mehanike i kvantne gravitacije.^[7]^[8]^[9]^[10]^[11]

Reference

^ Fields, Douglas E. (пролеће 2020). „Lecture25: Galilean and Special Relativity” (PDF). PHYC 2310: Calculus Based Physics III. University of New Mexico. стр. 8. Приступљено 7. 11. 2020.
^ The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, Kurt Edmund Oughstun (2006). Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Springer. стр. 165. ISBN 0-387-34599-X. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix. стр. 205. ISBN 1-930832-02-8. .
^ In very general terms, a coordinate system is a set of arcs xⁱ = xⁱ (t) in a complex Lie group; see Lev Semenovich Pontri͡agin (1986). L.S. Pontryagin: Selected Works Vol. 2: Topological Groups (3rd изд.). Gordon and Breach. стр. 429. ISBN 2-88124-133-6. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e₁, e₂,… e_n}; see Edoardo Sernesi; J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press. стр. 95. ISBN 0-412-40680-2. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
^ J X Zheng-Johansson; Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers. стр. 13. ISBN 1-59454-260-0.
^ Katherine Brading; Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press. стр. 417. ISBN 0-521-82137-1. ^{[мртва веза]}
^ Oliver Davis Johns (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. Chapter 16. ISBN 0-19-856726-X.
^ Donald T Greenwood (1997). Classical dynamics (Reprint of 1977 edition by Prentice-Hall изд.). Courier Dover Publications. стр. 313. ISBN 0-486-69690-1.
^ Matthew A. Trump; W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer. стр. 99. ISBN 0-7923-5737-X.
^ A S Kompaneyets (2003). Theoretical Physics (Reprint of the 1962 2nd изд.). Courier Dover Publications. стр. 118. ISBN 0-486-49532-9.
^ M Srednicki (2007). Quantum Field Theory. Cambridge University Press. Chapter 4. ISBN 978-0-521-86449-7.
^ Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press. стр. 98 ff. ISBN 0-521-83733-2.

Literatura

Jean Salençon; Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer. стр. 9. ISBN 3-540-41443-6.
Stewart, James B.; Redlin, Lothar; Watson, Saleem (2008). College Algebra (5th изд.). Brooks Cole. стр. 13—19. ISBN 978-0-495-56521-5.
Moon P, Spencer DE (1988). „Rectangular Coordinates (x, y, z)”. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print изд.). New York: Springer-Verlag. стр. 9—11 (Table 1.01). ISBN 978-0-387-18430-2.
Finney, Ross; George Thomas; Franklin Demana; Bert Waits (јун 1994). Calculus: Graphical, Numerical, Algebraic (Single Variable Version изд.). Addison-Wesley Publishing Co. ISBN 0-201-55478-X.
Margenau, Henry; Murphy, George M. (1956). The Mathematics of Physics and Chemistry. New York City: D. van Nostrand. стр. 178. ISBN 978-0-88275-423-9. LCCN 55010911. OCLC 3017486.
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. стр. 658. ISBN 0-07-043316-X. LCCN 52011515.
Jones, Alfred Clement (1912). An Introduction to Algebraical Geometry. Clarendon.
Hodge, W.V.D.; D. Pedoe (1994) [1947]. Methods of Algebraic Geometry, Volume I (Book II). Cambridge University Press. ISBN 978-0-521-46900-5.
Tang, K. T. (2006). Mathematical Methods for Engineers and Scientists. 2. Springer. ISBN 3-540-30268-9.
Liseikin, Vladimir D. (2007). A Computational Differential Geometry Approach to Grid Generation. Springer. ISBN 978-3-540-34235-9.
Hanspeter Schaub; John L. Junkins (2003). „Rigid body kinematics”. Analytical Mechanics of Space Systems. American Institute of Aeronautics and Astronautics. стр. 71. ISBN 1-56347-563-4.
Voitsekhovskii, M.I.; Ivanov, A.B. (2001) [1994], „Coordinates”, Ур.: Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Woods, Frederick S. (1922). Higher Geometry. Ginn and Co. стр. 1ff.
Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). Geometry of Differential Forms. AMS Bookstore. стр. 12. ISBN 0-8218-1045-6.
Farid M. L. Amirouche (2006). „§2.4: Generalized coordinates”. Fundamentals of multibody dynamics: theory and applications. Springer. стр. 46. ISBN 0-8176-4236-6.
Florian Scheck (2010). „§5.1 Manifolds of generalized coordinates”. Mechanics: From Newton's Laws to Deterministic Chaos (5th изд.). Springer. стр. 286. ISBN 978-3-642-05369-6.
Greenwood, Donald T. (1987). Principles of Dynamics (2nd изд.). Prentice Hall. ISBN 0-13-709981-9.
Ginsberg, Jerry H. (2008). Engineering dynamics (3rd изд.). Cambridge UK: Cambridge University Press. ISBN 978-0-521-88303-0.
Goldstein, Herbert; Poole, Charles; Safko, John (2002). Classical Mechanics (3rd изд.). San Francisco, CA: Addison Wesley. ISBN 0-201-65702-3.
Hand, Louis N.; Finch, Janet D. (1998). Analytical mechanics. Cambridge: Cambridge University Press. ISBN 978-0521575720.
Kibble, T.W.B; Berkshire, F.H. (2004). Classical Mechanics (5th изд.). River Edge NJ: Imperial College Press. ISBN 1860944248.
Landau, L. D.; Lifshitz, E.M. (1976). Mechanics (Third изд.). Oxford. ISBN 978-0750628969.
Torby, Bruce (1984). „Energy Methods”. Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.
Hanspeter Schaub, John L. Junkins (2003). „Rigid body kinematics”. Analytical mechanics of space systems. American Institute of Aeronautics and Astronautics. стр. 71. ISBN 1-56347-563-4.
„LVLH Attitude”. Архивирано из оригинала 2007-07-14. г. Приступљено 2010-10-08.
„Inertial Attitude”. Архивирано из оригинала 2007-07-14. г. Приступљено 2010-10-08.
Hurt, H. H., Jr. (јануар 1965) [1960]. Aerodynamics for Naval Aviators. U.S. Government Printing Office, Washington D.C.: U.S. Navy, Aviation Training Division. стр. 284. NAVWEPS 00-80T-80.

Spoljašnje veze

Медији везани за чланак Referentni sistem на Викимедијиној остави

[1] Fields, Douglas E. (пролеће 2020). „Lecture25: Galilean and Special Relativity” (PDF). PHYC 2310: Calculus Based Physics III. University of New Mexico. стр. 8. Приступљено 7. 11. 2020.

[macroscopic-2] The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where constitutive relations of various time and length scales are used to determine the current and charge densities entering Maxwell's equations. See, for example, Kurt Edmund Oughstun (2006). Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Springer. стр. 165. ISBN 0-387-34599-X. . These distinctions also appear in thermodynamics. See Paul McEvoy (2002). Classical Theory. MicroAnalytix. стр. 205. ISBN 1-930832-02-8. .

[Pontriagin-3] In very general terms, a coordinate system is a set of arcs xⁱ = xⁱ (t) in a complex Lie group; see Lev Semenovich Pontri͡agin (1986). L.S. Pontryagin: Selected Works Vol. 2: Topological Groups (3rd изд.). Gordon and Breach. стр. 429. ISBN 2-88124-133-6. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e₁, e₂,… e_n}; see Edoardo Sernesi; J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press. стр. 95. ISBN 0-412-40680-2. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.

[Johansson-4] J X Zheng-Johansson; Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers. стр. 13. ISBN 1-59454-260-0.

[Brading-5] Katherine Brading; Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press. стр. 417. ISBN 0-521-82137-1. ^{[мртва веза]}

[Johns-6] Oliver Davis Johns (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. Chapter 16. ISBN 0-19-856726-X.

[Greenwood-7] Donald T Greenwood (1997). Classical dynamics (Reprint of 1977 edition by Prentice-Hall изд.). Courier Dover Publications. стр. 313. ISBN 0-486-69690-1.

[Trump-8] Matthew A. Trump; W. C. Schieve (1999). Classical Relativistic Many-Body Dynamics. Springer. стр. 99. ISBN 0-7923-5737-X.

[Kompaneyets-9] A S Kompaneyets (2003). Theoretical Physics (Reprint of the 1962 2nd изд.). Courier Dover Publications. стр. 118. ISBN 0-486-49532-9.

[Srednicki-10] M Srednicki (2007). Quantum Field Theory. Cambridge University Press. Chapter 4. ISBN 978-0-521-86449-7.

[Rovelli-11] Carlo Rovelli (2004). Quantum Gravity. Cambridge University Press. стр. 98 ff. ISBN 0-521-83733-2.

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@@ Ред 8: / Ред 8: @@
 == Različiti aspekti „referentnog okvira” ==
-Potreba da se razlikuju različita značenja „referentnog okvira“ dovela je do različitih pojmova. Na primer, ponekad je tip koordinatnog sistema priključen kao modifikator, kao u ''kartezijskom referentnom okviru''. Ponekad se naglašava stanje kretanja, kao u ''[[Rotating reference frame|rotirajućem referentnom okviru]]''. Ponekad je naglašen način na koji se transformiše u okvire koji se smatraju povezanima kao u ''[[Galilean frame of reference|Galilejevom referentnom okviru]]''. Ponekad se okviri razlikuju po obimu njihovih opservacija, kao u ''makroskopskim'' i ''mikroskopskim referentnim okvirima''.<ref name="macroscopic">The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where [[Constitutive equation|constitutive relations]] of various time and length scales are used to determine the current and charge densities entering [[Maxwell's equations]]. See, for example, {{cite book |title=Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media |author=Kurt Edmund Oughstun |page=165 |url=https://books.google.com/books?id=behRnNRiueAC&pg=PA165&dq=macroscopic+frame++electromagnetism|isbn=0-387-34599-X |year=2006 |publisher=Springer}}. These distinctions also appear in thermodynamics. See {{cite book |title=Classical Theory |author=Paul McEvoy |page=205 |url=https://books.google.com/books?id=dj0wFIxn-PoC&pg=PA206&dq=macroscopic+frame#PPA205,M1 |isbn=1-930832-02-8 |year=2002 |publisher=MicroAnalytix}}.</ref>
+Potreba da se razlikuju različita značenja „referentnog okvira“ dovela je do različitih pojmova. Na primer, ponekad je tip koordinatnog sistema priključen kao modifikator, kao u ''kartezijskom referentnom okviru''. Ponekad se naglašava stanje kretanja, kao u ''[[Rotating reference frame|rotirajućem referentnom okviru]]''. Ponekad je naglašen način na koji se transformiše u okvire koji se smatraju povezanima kao u ''[[Galilean frame of reference|Galilejevom referentnom okviru]]''.<ref>{{cite web | url = https://physics.unm.edu/Courses/Fields/Phys2310/Lectures/lecture25.pdf#page=8 | first = Douglas E. |last = Fields | date = Spring 2020 | title = Lecture25: Galilean and Special Relativity | work= PHYC 2310: Calculus Based Physics III | publisher = [[University of New Mexico]] | page = 8 | access-date=7 November 2020}}</ref>  Ponekad se okviri razlikuju po obimu njihovih opservacija, kao u ''makroskopskim'' i ''mikroskopskim referentnim okvirima''.<ref name="macroscopic">The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where [[Constitutive equation|constitutive relations]] of various time and length scales are used to determine the current and charge densities entering [[Maxwell's equations]]. See, for example, {{cite book |title=Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media |author=Kurt Edmund Oughstun |page=165 |url=https://books.google.com/books?id=behRnNRiueAC&pg=PA165&dq=macroscopic+frame++electromagnetism|isbn=0-387-34599-X |year=2006 |publisher=Springer}}. These distinctions also appear in thermodynamics. See {{cite book |title=Classical Theory |author=Paul McEvoy |page=205 |url=https://books.google.com/books?id=dj0wFIxn-PoC&pg=PA206&dq=macroscopic+frame#PPA205,M1 |isbn=1-930832-02-8 |year=2002 |publisher=MicroAnalytix}}.</ref>
 U ovom članku se termin ''referentni okvir posmatranja'' koristi kada je naglasak na ''stanju kretanja'', a ne na izboru koordinata ili karakteru posmatranja ili posmatračkog aparata. U tom smislu, referentni okvir posmatranja omogućava proučavanje uticaja kretanja na čitavu porodicu koordinatnih sistema koji bi mogli biti priključeni na ovaj okvir. S druge strane, koordinatni sistem se može koristiti u mnoge svrhe u kojima stanje kretanja nije glavni predmet interesa. Na primer, može se primeniti dati koordinatni sistem da bi se iskoristila simetrija sistema. Gledano sa još šire perspektive, formulacija mnogih problema iz fizike koristi ''[[generalized coordinates|generalizovane koordinate]]'', ''[[Normalni mod|normalne modove]]'' ili ''[[Sopstvene vrednosti i sopstveni vektori|sopstvene vektore]]'', koji su samo posredno povezani sa prostorom i vremenom. Stoga je korisno da se razdvoje različiti aspekti referentnog okvira radi diskusije u nastavku. Referentni okviri posmatranja, koordinatni sistemi i opservaciona oprema se uzimaju kao nezavisni koncepti.
@@ Ред 23: / Ред 23: @@
 == Literatura ==
-{{refbegin}}
+{{refbegin|30em}}
-* {{cite book |title=Handbook of Continuum Mechanics: General Concepts, Thermoelasticity |page= 9 |author1=Jean Salençon |author2=Stephen Lyle |url=https://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22|isbn=3-540-41443-6 |year=2001 |publisher=Springer}}</ref>
+* {{cite book |title=Handbook of Continuum Mechanics: General Concepts, Thermoelasticity |page= 9 |author1=Jean Salençon |author2=Stephen Lyle |url=https://books.google.com/books?id=H3xIED8ctfUC&pg=PA9&dq=physical+%22frame+of+reference%22|isbn=3-540-41443-6 |year=2001 |publisher=Springer}}
+* {{cite book |last1=Stewart |first1=James B. |last2= Redlin |first2= Lothar |last3=Watson |first3=Saleem |author-link=James Stewart (mathematician) |title=College Algebra |publisher=[[Brooks Cole]] |year=2008 |edition= 5th |pages=13–19 |isbn=978-0-495-56521-5}}
+* {{cite book |vauthors=Moon P, Spencer DE |year=1988 |chapter=Rectangular Coordinates (x, y, z) |title=Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions |edition=corrected 2nd, 3rd print |publisher=Springer-Verlag |location=New York |pages=9–11 (Table 1.01) |isbn=978-0-387-18430-2}}
+* {{cite book| last=Finney|first=Ross| author2=George Thomas|author3=Franklin Demana|author4=Bert Waits|title=Calculus: Graphical, Numerical, Algebraic|edition=Single Variable Version|date=June 1994|publisher=Addison-Wesley Publishing Co.|isbn=0-201-55478-X|url-access=registration| url=https://archive.org/details/calculusgraphica00ross}}
+* {{cite book |last1=Margenau |first1=Henry |author-link1=Henry Margenau |last2=Murphy |first2=George M. |year=1956 |title=The Mathematics of Physics and Chemistry |url=https://archive.org/details/mathematicsofphy0002marg |url-access=registration |publisher=D. van Nostrand |location=New York City |page=[https://archive.org/details/mathematicsofphy0002marg/page/178 178] |lccn=55010911 |isbn=978-0-88275-423-9 |oclc=3017486}}
+* {{cite book |author= [[Philip M. Morse|Morse PM]], [[Herman Feshbach|Feshbach H]] |year= 1953 |title= Methods of Theoretical Physics, Part I |publisher= McGraw-Hill |location= New York |isbn= 0-07-043316-X |page= 658 |lccn= 52011515}}
+* {{cite book |title=An Introduction to Algebraical Geometry|first=Alfred Clement|last=Jones |publisher=Clarendon|year=1912}}
+* {{cite book | last= Hodge | first= W.V.D. | author-link= W. V. D. Hodge | author2=[[D. Pedoe]] | title= Methods of Algebraic Geometry, Volume&nbsp;I (Book&nbsp;II) | publisher= [[Cambridge University Press]] | year= 1994 | isbn= 978-0-521-46900-5 | orig-year= 1947}}
+* {{cite book |title=Mathematical Methods for Engineers and Scientists |volume=2|first=K. T.|last=Tang|publisher=Springer|year=2006|isbn=3-540-30268-9|page= }}
+* {{cite book |title=A Computational Differential Geometry Approach to Grid Generation |first=Vladimir D.|last=Liseikin|publisher=Springer|year=2007|isbn=978-3-540-34235-9|page= }}
+* {{cite book |title=Analytical Mechanics of Space Systems |author1=Hanspeter Schaub |author2=John L. Junkins |chapter=Rigid body kinematics |page=71 |chapter-url=https://books.google.com/books?id=qXvESNWrfpUC&pg=PA71 |isbn=1-56347-563-4 |year=2003 |publisher=American Institute of Aeronautics and Astronautics}}
+* {{SpringerEOM |title= Coordinates |id= C/c026470 |last= Voitsekhovskii |first=M.I. |last2=Ivanov |first2=A.B.}}
+* {{cite book |title= Higher Geometry |first=Frederick S. |last=Woods |publisher=Ginn and Co.|year=1922 |pages= 1ff |url= https://books.google.com/books?id=3ZULAAAAYAAJ&pg=PA1}}
+* {{cite book |title= Geometry of Differential Forms |last1=Shigeyuki Morita |last2=Teruko Nagase |last3=Katsumi Nomizu |author-link3=Katsumi Nomizu |page=12 |url= https://books.google.com/books?id=5N33Of2RzjsC&pg=PA12 |isbn=0-8218-1045-6 |year=2001 |publisher=AMS Bookstore}}
+* {{cite book |title=Fundamentals of multibody dynamics: theory and applications |author=Farid M. L. Amirouche |chapter-url=https://books.google.com/books?id=_nlEcQYldeIC&pg=PA46 |page=46 |chapter=§2.4: Generalized coordinates |publisher=Springer |isbn=0-8176-4236-6 |year=2006}}
+* {{cite book |title=Mechanics: From Newton's Laws to Deterministic Chaos |author=Florian Scheck |chapter-url=https://books.google.com/books?id=yUDo7VptDgIC&pg=PA286 |page=286 |chapter=§5.1 Manifolds of generalized coordinates |isbn=978-3-642-05369-6 |publisher=Springer |edition =5th |year=2010}}
+* {{cite book | last = Greenwood | first = Donald T. | year = 1987 | title = Principles of Dynamics  | edition = 2nd | publisher = Prentice Hall | isbn = 0-13-709981-9 }}
+* {{cite book |title=Engineering dynamics |last1=Ginsberg |first1=Jerry H. |isbn=978-0-521-88303-0 |year=2008 |publisher=[[Cambridge University Press]] |location=Cambridge UK |edition=3rd }}
+* {{cite book |title=Classical Mechanics |last1=Goldstein |first1=Herbert |last2=Poole |first2=Charles |last3=Safko |first3 = John |isbn=0-201-65702-3 |year= 2002 |edition=3rd|publisher = Addison Wesley|location= San Francisco, CA}}
+* {{cite book |last1=Hand |first1=Louis N. |last2=Finch |first2=Janet D. |title=Analytical mechanics |date=1998 |publisher=Cambridge University Press |location=Cambridge |isbn=978-0521575720}}
+* {{cite book |last1=Kibble |first1=T.W.B |author1-link=Tom Kibble |last2=Berkshire |first2=F.H. |year=2004 |title=Classical Mechanics |edition=5th |publisher=[[Imperial College Press]] |location=River Edge NJ |isbn=1860944248 }}
+* {{cite book |last1=Landau |first1=L. D. |last2=Lifshitz |first2=E.M. |title=Mechanics |date=1976 |location=Oxford |isbn=978-0750628969 |edition=Third}}
+* {{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America  |isbn=0-03-063366-4 |chapter=Energy Methods}}
+* {{cite book |title=Analytical mechanics of space systems |author=Hanspeter Schaub, John L. Junkins |chapter=Rigid body kinematics |page=71 |chapter-url=https://books.google.com/books?id=qXvESNWrfpUC&q=attitude+%22rigid+body%22&pg=PA71 |isbn=1-56347-563-4 |year=2003 |publisher=American Institute of Aeronautics and Astronautics}}
+* {{cite web|url=http://liftoff.msfc.nasa.gov/academy/rocket_sci/shuttle/attitude/lvlh.html |title=LVLH Attitude |accessdate=2010-10-08 |url-status=dead |archiveurl=https://web.archive.org/web/20070714231221/http://liftoff.msfc.nasa.gov/Academy/ROCKET_SCI/SHUTTLE/ATTITUDE/LVLH.HTML |archivedate=2007-07-14 }}
+* {{cite web|url=http://liftoff.msfc.nasa.gov/academy/rocket_sci/shuttle/attitude/inertial.html |title=Inertial Attitude |accessdate=2010-10-08 |url-status=dead |archiveurl=https://web.archive.org/web/20070714231308/http://liftoff.msfc.nasa.gov/Academy/ROCKET_SCI/SHUTTLE/ATTITUDE/INERTIAL.HTML |archivedate=2007-07-14 }}
+* {{cite book | last = Hurt | first = H. H., Jr. | orig-year = 1960 | title = Aerodynamics for Naval Aviators |date=January 1965 | publisher = U.S. Navy, Aviation Training Division | location = U.S. Government Printing Office, Washington D.C. | id = NAVWEPS 00-80T-80 | page = 284 }}
 {{refend}}