\h*aSSKJrJrJrJrJrJrJrJr J r SSK J r SSK Jr SSKJr SSKJr SSKJr S/r"SS\\5r"S S \5rS rg ) ) SsympifyexpandsqrtAddzerosacosImmutableMatrixsimplify)trigsimp) Printable) filldedent) EvalfMixin) prec_to_dpsVectorcz\rSrSrSrSrSrSr\S5r Sr Sr Sr S r S rS rS rS rSrSrSrS)SjrSrSr\ r\rSrSr\ R\l\rSr\R\lSr\R\lS)SjrS*Sjr Sr!Sr"Sr#Sr$Sr%Sr&S r'S!r(S"r)S#r*S$r+S%r,S&r-S'r.g()+raThe class used to define vectors. It along with ReferenceFrame are the building blocks of describing a classical mechanics system in PyDy and sympy.physics.vector. Attributes ========== simp : Boolean Let certain methods use trigsimp on their outputs FcF/UlUS:Xa/n[U[5(aUnO20nUH*nUSU;aX#S==US- ss'M USX#S'M, UR5H4upEU[ /SQ5:wdMURR XT45 M6 g)a_This is the constructor for the Vector class. You should not be calling this, it should only be used by other functions. You should be treating Vectors like you would with if you were doing the math by hand, and getting the first 3 from the standard basis vectors from a ReferenceFrame. The only exception is to create a zero vector: zv = Vector(0) r)rrrN)args isinstancedictitemsMatrixappend)selfinlistdinpkvs S/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/vector/vector.py__init__Vector.__init__s Q;F fd # #AAq6Q;!fIQ'I #AA!fI  GGIDAF9%%   !(c[$)zReturns the class Vector. )rrs r"func Vector.func<s  r%c>[[UR55$N)hashtuplerr's r"__hash__Vector.__hash__AsE$))$%%r%clUS:XaU$[U5n[URUR-5$)zThe add operator for Vector. r) _check_vectorrrrothers r"__add__Vector.__add__Ds0 A:Ke$dii%**,--r%cSSKJnJn [X5(aJU"U5n[ S5nUR H%nXESUS-USR U5-- nM' U$[U5n[RnUR HHnUR H5nXhSRUSRUS5-US-S- nM7 MJ [R(a [USS9$U$)aDot product of two vectors. Returns a scalar, the dot product of the two Vectors Parameters ========== other : Vector The Vector which we are dotting with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dot >>> from sympy import symbols >>> q1 = symbols('q1') >>> N = ReferenceFrame('N') >>> dot(N.x, N.x) 1 >>> dot(N.x, N.y) 0 >>> A = N.orientnew('A', 'Axis', [q1, N.x]) >>> dot(N.y, A.y) cos(q1) rDyadic _check_dyadicrT) recursive)sympy.physics.vector.dyadicr8r9rrrdotr1rZeroTdcmsimpr ) rr3r8r9olr!outv1v2s r"r= Vector.dotKs8 F e $ $!%(EBZZdQqTkQqTXXd^44 Ie$ff))BjjARUYYr!u%56"Q%@!DD! ;;C40 0Jr%cFUR[RU- 5$)z6This uses mul and inputs self and 1 divided by other. )__mul__rOner2s r" __truediv__Vector.__truediv__xs||AEEEM**r%c\US:Xa [S5n[U5nUR/:XaUR/:XagUR/:XdUR/:XagURSSnUH%n[ X- R U55S:wdM% g g![a gf=f)a<Tests for equality. It is very import to note that this is only as good as the SymPy equality test; False does not always mean they are not equivalent Vectors. If other is 0, and self is empty, returns True. If other is 0 and self is not empty, returns False. If none of the above, only accepts other as a Vector. rFTr)rr1 TypeErrorrrr=)rr3framer!s r"__eq__ Vector.__eq__|s A:1IE !%(E IIO%**"2ii2o5::#3 ! QAt|((+,1  s B B+*B+c[UR5n[U5n[[ U55HnXUS-X#S4X#'M [ U5$)aLMultiplies the Vector by a sympifyable expression. Parameters ========== other : Sympifyable The scalar to multiply this Vector with Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> b = Symbol('b') >>> V = 10 * b * N.x >>> print(V) 10*b*N.x rr)listrrrangelenr)rr3newlistis r"rHVector.__mul__sT,tyy/s7|$A!*Q-/A?GJ%gr%c US-$Nr's r"__neg__Vector.__neg__s byr%c jSSKJn [U5nU"S5nURGHnURGHnX2"USSUSS-USRUSR4/5- nX2"USSUSS-USRUSR 4/5- nX2"USSUSS-USRUSR 4/5- nX2"USSUSS-USR USR4/5- nX2"USSUSS-USR USR 4/5- nX2"USSUSS-USR USR 4/5- nX2"USSUSS-USR USR4/5- nX2"USSUSS-USR USR 4/5- nX2"USSUSS-USR USR 4/5- nGM GM U$)aHOuter product between two Vectors. A rank increasing operation, which returns a Dyadic from two Vectors Parameters ========== other : Vector The Vector to take the outer product with Examples ======== >>> from sympy.physics.vector import ReferenceFrame, outer >>> N = ReferenceFrame('N') >>> outer(N.x, N.x) (N.x|N.x) r)r8rr:)r<r8r1rxyz)rr3r8rBr!rEs r"outer Vector.outersm* 7e$ AYAjjfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEEfqtAwAq11Q4662a577CDEE! r%cURn[U5S:Xa [S5$/nUHnSHnUSUS:Xa&URSUSRU-5 M5USUS:Xa&URSUSRU-5 MgUSUS:wdMuUR USU5n[ USU[5(aSU-nUSS:XaUSS nSnOSnURXv-USRU-5 M M S RU5nURS5(aUS S nU$URS 5(aUSS nU$) zLatex Printing method. rrrr:r + rZ - (%s)-N ) rrTstrr latex_vecs_printrrjoin startswith) rprinterarrBr!jarg_str str_startoutstrs r"_latex Vector._latexsqYY r7a<q6M AQ47a<IIeadooa&889qT!W]IIeadooa&889qT!W\&nnQqT!W5G!!A$q'3//"(7"2qzS(")!"+$) $) IIi1AaDOOA4FFG%(   U # #ABZF   s # #ABZF r%c ^^SSKJm /nUU4SjnURHupE[S5HnXFS:XaM XFS:Xa&UR T"UR U55 M;XFS:Xa/UR T"S5T"UR U5-5 MrUR U"XFUR U55 M M U(aTR "U6nU$T"S5nU$) zPretty Printing method. r) prettyFormc>TRU5nTRU5nUR(aT"UR56nTRX#/SS9$)Nrl) delimiter)rois_Addparens _print_seq)abpapbr{rrs r" juxtapose!Vector._pretty..juxtaposesN"B"Bxx-%%rh#%> >r%rkrrZz-10) sympy.printing.pretty.stringpictr{rrSr pretty_vecsr4) rrrtermsrMNrV pretty_resultr{s ` @r"_prettyVector._prettys? ?IIDA1X419TQYLLAMM!,TRZLLD!1Jq}}Q?O4P!PQLL14q1A!BC &..6M'sOMr%cSU-U-$rYr[r2s r"__rsub__Vector.__rsub__!sT U""r%c U(a[UR5S:Xa[UR5nO[UR5S:XaURS5$URVs0sH oDSUS_M nn[ UR 5SS9n/nUHnUR XWU45 M /nUHnSHn USU S:Xa&UR SUSRU -5 M5USU S:Xa&UR SUSRU -5 MgUSU S:wdMuURUSU 5n [USU [5(aS U -n U SS :XaU SS n Sn OSn UR X-S -USRU -5 M M S RU5n U RS5(aU SS n U $U RS5(aU SS n U $s snf)zPrinting method. rrcUR$r+)index)r_s r""Vector._sympystr..,s!''r%)keyrerfrZrgrhriN*rjrkrl) rTrrRrosortedkeysrstr_vecsrrrprq) rrrorderrsr!rrrrBrtrurvrws r" _sympystrVector._sympystr$sDII!+diiB ^q >>!$ $%)YY/Y1qtYA/!&&((9:DB 163-( AQ47a<IIeadmmA&667qT!W]IIeadmmA&667qT!W\&nnQqT!W5G!!A$q'3//"(7"2qzS(")!"+$) $) IIi1C7!A$--:JJK%(   U # #ABZF   s # #ABZF ?0s/Hc*URUS-5$)zThe subtraction operator. rZ)r4r2s r"__sub__Vector.__sub__Ls||EBJ''r%cSSKJnJn [X5(a^U"U5nU"S5n[ UR 5H3upVXFSUR US5RUS5-- nM5 U$[U5nUR /:Xa [S5$Sn/nUR n U HnUSRn USRn USRn XU /URU 5URU 5URU 5/[U/5RU 5[U/5RU 5[U/5RU 5//n X"U 5R - nM [U5$)aThe cross product operator for two Vectors. Returns a Vector, expressed in the same ReferenceFrames as self. Parameters ========== other : Vector The Vector which we are crossing with Examples ======== >>> from sympy import symbols >>> from sympy.physics.vector import ReferenceFrame, cross >>> q1 = symbols('q1') >>> N = ReferenceFrame('N') >>> cross(N.x, N.y) N.z >>> A = ReferenceFrame('A') >>> A.orient_axis(N, q1, N.x) >>> cross(A.x, N.y) N.z >>> cross(N.y, A.x) - sin(q1)*A.y - cos(q1)*A.z rr7rr:cUSSUSSUSS-USSUSS-- -USSUSSUSS-USSUSS-- --USSUSSUSS-USSUSS-- --$)zThis is needed as a little method for to find the determinant of a list in python; needs to work for a 3x3 list. SymPy's Matrix will not take in Vector, so need a custom function. You should not be calling this. rrr:r[)mats r"_detVector.cross.._detxsF1IQSVAY!6QSVAY9N!NO!fQi3q6!9s1vay#83q6!9F1I<$!$Qc!fQi#a&).CF1IAq )/*"++ ,r%)r<r8r9r enumeratercrossrbr1rr_r`rar=)rr3r8r9rBrVr!routlistrstempxtempytempztempms r"r Vector.crossPs`: F e $ $!%(EB!%**-dtzz!A$/66qt<==.Ie$ :: !9  , ZZAaDFFEaDFFEaDFFEU+xx%Iqc{u-vqc{u/Eqc{u-/0E tE{'' 'Ggr%cR0nURHn[U/5XS'M U$)as The constituents of this vector in different reference frames, as per its definition. Returns a dict mapping each ReferenceFrame to the corresponding constituent Vector. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> R1 = ReferenceFrame('R1') >>> R2 = ReferenceFrame('R2') >>> v = R1.x + R2.x >>> v.separate() == {R1: R1.x, R2: R2.x} True r)rr)r componentsr_s r"separateVector.separates/( A%qc{Jt r%c$URU5$r+)r=r2s r"__and__Vector.__and__sxxr%c$URU5$r+)rr2s r"__xor__Vector.__xor__zz%  r%c$URU5$r+)rbr2s r"__or__ Vector.__or__rr%cSSKJn U"U5 [U5n/nURHnUSnUSnX:XaXWR U5U4/- nM*U(a.UR U5R U5[ SS5:XaXWR U5U4/- nMw[U/5RU5n U RSSR U5n U[X4/5R- nM [U5$)aReturns the partial derivative of the vector with respect to a variable in the provided reference frame. Parameters ========== var : Symbol What the partial derivative is taken with respect to. frame : ReferenceFrame The reference frame that the partial derivative is taken in. var_in_dcm : boolean If true, the differentiation algorithm assumes that the variable may be present in any of the direction cosine matrices that relate the frame to the frames of any component of the vector. But if it is known that the variable is not present in the direction cosine matrices, false can be set to skip full reexpression in the desired frame. Examples ======== >>> from sympy import Symbol >>> from sympy.physics.vector import dynamicsymbols, ReferenceFrame >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> t = Symbol('t') >>> q1 = dynamicsymbols('q1') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'Axis', [q1, N.y]) >>> A.x.diff(t, N) - sin(q1)*q1'*N.x - cos(q1)*q1'*N.z >>> A.x.diff(t, N).express(A).simplify() - q1'*A.z >>> B = ReferenceFrame('B') >>> u1, u2 = dynamicsymbols('u1, u2') >>> v = u1 * A.x + u2 * B.y >>> v.diff(u2, N, var_in_dcm=False) B.y r) _check_framerrk) sympy.physics.vector.framerrrdiffr@rrexpress) rvarrN var_in_dcmrrvector_componentmeasure_numbercomponent_framereexp_vec_compderivs r"r Vector.diffsR <Ucl $ -a0N.q1O'//4e<=="eii&@&E&Ec&J&+Aqk'2 3 3C 8/JKKF%+-=,>%?%G%G%NN*//215::3?Efun%56;;;F!* f~r%cSSKJn U"XUS9$)a Returns a Vector equivalent to this one, expressed in otherframe. Uses the global express method. Parameters ========== otherframe : ReferenceFrame The frame for this Vector to be described in variables : boolean If True, the coordinate symbols(if present) in this Vector are re-expressed in terms otherframe Examples ======== >>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols >>> from sympy.physics.vector import init_vprinting >>> init_vprinting(pretty_print=False) >>> q1 = dynamicsymbols('q1') >>> N = ReferenceFrame('N') >>> A = N.orientnew('A', 'Axis', [q1, N.y]) >>> A.x.express(N) cos(q1)*N.x - sin(q1)*N.z r)r) variables)sympy.physics.vectorr)r otherframerrs r"rVector.expresss8 1t9==r%c[UVs/sHo RU5PM sn5RSS5$s snf)aReturns the matrix form of the vector with respect to the given frame. Parameters ---------- reference_frame : ReferenceFrame The reference frame that the rows of the matrix correspond to. Returns ------- matrix : ImmutableMatrix, shape(3,1) The matrix that gives the 1D vector. Examples ======== >>> from sympy import symbols >>> from sympy.physics.vector import ReferenceFrame >>> a, b, c = symbols('a, b, c') >>> N = ReferenceFrame('N') >>> vector = a * N.x + b * N.y + c * N.z >>> vector.to_matrix(N) Matrix([ [a], [b], [c]]) >>> beta = symbols('beta') >>> A = N.orientnew('A', 'Axis', (beta, N.x)) >>> vector.to_matrix(A) Matrix([ [ a], [ b*cos(beta) + c*sin(beta)], [-b*sin(beta) + c*cos(beta)]]) rkr)rr=reshape)rreference_frameunit_vecs r" to_matrixVector.to_matrixsDJ&(&/7xx)&())0A 7(s;c ~^0nURH nUSRU4Sj5X#S'M" [U5$)z(Calls .doit() on each term in the Vectorrc(>UR"S0TD6$)Nr[)doit)r_hintss r"rVector.doit..Hsqvvr%r)r applyfuncr)rrrr!s ` r"r Vector.doitDs9 Adnn%>?AdGayr%cSSKJn U"X5$)z Returns a Vector which is the time derivative of the self Vector, taken in frame otherframe. Calls the global time_derivative method Parameters ========== otherframe : ReferenceFrame The frame to calculate the time derivative in r)time_derivative)rr)rrrs r"dt Vector.dtKs 9t00r%ch0nURHn[US5XS'M [U5$)zReturns a simplified Vector.rr)rr r)rrr!s r"r Vector.simplify\s2 AqtnAdGayr%cv0nURHnUSR"U0UD6X4S'M [U5$)zSubstitution on the Vector. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> from sympy import Symbol >>> N = ReferenceFrame('N') >>> s = Symbol('s') >>> a = N.x * s >>> a.subs({s: 2}) 2*N.x rr)rsubsr)rrkwargsrr!s r"r Vector.subscs> Adii00AdGayr%c6[URU55$)zReturns the magnitude (Euclidean norm) of self. Warnings ======== Python ignores the leading negative sign so that might give wrong results. ``-A.x.magnitude()`` would be treated as ``-(A.x.magnitude())``, instead of ``(-A.x).magnitude()``. )rr=r's r" magnitudeVector.magnitudexsDHHTN##r%cT[UR/-5UR5- $)z9Returns a Vector of magnitude 1, codirectional with self.)rrrr's r" normalizeVector.normalizes!dii"n%(888r%c[U5(d [S5e0nURHnUSRU5X#S'M [ U5$)z/Apply a function to each component of a vector.z`f` must be callable.rr)callablerMrrr)rfrr!s r"rVector.applyfuncsL{{34 4 AdnnQ'AdGayr%czUR5nUR5n[URU55nU$)a; Returns the smallest angle between Vector 'vec' and self. Parameter ========= vec : Vector The Vector between which angle is needed. Examples ======== >>> from sympy.physics.vector import ReferenceFrame >>> A = ReferenceFrame("A") >>> v1 = A.x >>> v2 = A.y >>> v1.angle_between(v2) pi/2 >>> v3 = A.x + A.y + A.z >>> v1.angle_between(v3) acos(sqrt(3)/3) Warnings ======== Python ignores the leading negative sign so that might give wrong results. ``-A.x.angle_between()`` would be treated as ``-(A.x.angle_between())``, instead of ``(-A.x).angle_between()``. )rr r=)rvecvec1vec2angles r" angle_betweenVector.angle_betweens3B~~}}TXXd^$ r%c8URU5R$)anReturns the free symbols in the measure numbers of the vector expressed in the given reference frame. Parameters ========== reference_frame : ReferenceFrame The frame with respect to which the free symbols of the given vector is to be determined. Returns ======= set of Symbol set of symbols present in the measure numbers of ``reference_frame``. )r free_symbols)rrs r"rVector.free_symbolss$~~o.;;;r%cSSKJn U"XS9$)aReturns the free dynamic symbols (functions of time ``t``) in the measure numbers of the vector expressed in the given reference frame. Parameters ========== reference_frame : ReferenceFrame The frame with respect to which the free dynamic symbols of the given vector is to be determined. Returns ======= set Set of functions of time ``t``, e.g. ``Function('f')(me.dynamicsymbols._t)``. r)find_dynamicsymbols)r)!sympy.physics.mechanics.functionsr)rrrs r"free_dynamicsymbolsVector.free_dynamicsymbolss& J"4IIr%cUR(dU$/n[U5nURH%upEURURUS9U/5 M' [ U5$)N)n)rrrevalfr)rprecnew_argsdpsrrNs r" _eval_evalfVector._eval_evalfsTyyK$))JC OOSYYY-u5 6$hr%c/nURH(up4URU5nURX4/5 M* [U5$)aReplace occurrences of objects within the measure numbers of the vector. Parameters ========== rule : dict-like Expresses a replacement rule. Returns ======= Vector Result of the replacement. Examples ======== >>> from sympy import symbols, pi >>> from sympy.physics.vector import ReferenceFrame >>> A = ReferenceFrame('A') >>> x, y, z = symbols('x y z') >>> ((1 + x*y) * A.x).xreplace({x: pi}) (pi*y + 1)*A.x >>> ((1 + x*y) * A.x).xreplace({x: pi, y: 2}) (1 + 2*pi)*A.x Replacements occur only if an entire node in the expression tree is matched: >>> ((x*y + z) * A.x).xreplace({x*y: pi}) (z + pi)*A.x >>> ((x*y*z) * A.x).xreplace({x*y: pi}) x*y*z*A.x )rxreplacerr)rrulerrrNs r"r Vector.xreplacesCL))JC,,t$C OOSL )$hr%)rN)T)F)/__name__ __module__ __qualname____firstlineno____doc__rA is_numberr#propertyr(r.r4r=rJrOrHr\rbrxrrrrr__radd____rmul__rr__rand__rrrrrrrr rrrrrrrrr __static_attributes__r[r%r"rrs  DI):&.+Z+:8&P D>#&P(@DHH2kkGOH!mmGO!]]FN@D>>&7P1"* $9$L<(J. * r%c(^\rSrSrU4SjrSrU=r$)VectorTypeErroric x>[S[U5<SU<S[U5<S35n[TU] U5 g)NzExpected an instance of z, but received object 'z' of .)rtypesuperr#)rr3wantmsg __class__s r"r#VectorTypeError.__init__s0*.t*eT%[JK r%r[)r r rrr#r __classcell__)rs@r"rrs r%rcF[U[5(d [S5eU$)NzA Vector must be supplied)rrrM)r3s r"r1r1#s eV $ $344 Lr%N)sympyrrrrrrr r rr sympy.simplify.trigsimpr sympy.printing.defaultsr sympy.utilities.miscrsympy.core.evalfrmpmath.libmp.libmpfr__all__rrMrr1r[r%r"r*sN888,-+'+ *J  Y J  Zir%