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Generalized Coordinates Functions (Fields) Other smooth coordinatization qi,i=1,...D Generalized Coordinates Physics 504, Physics 504, Spring 2011 qi(~r) and ~r( qi ) are well defined (in some domain) Spring 2011 i Electricity { } Electricity Cartesian coordinates r ,i=1, 2,...D for Euclidean and mostly 1—1, so Jacobian det(∂qi/∂rj) =0. and space. Magnetism 6 Magnetism 2 i 2 Shapiro Distance between P qi and P qi + δqi is given Shapiro Distance by Pythagoras: (δs) = (δr ) . { } 0 { } i by X i Generalized Generalized Unit vectorse ˆi, displacement ∆~r = i ∆r eˆi Coordinates Coordinates i Cartesian k k Cartesian Fields are functions of position, or of ~r or of r . coords for ED 2 k 2 ∂r i ∂r j coords for ED P { } Generalized (δs) = (δr ) = i δq j δq Generalized Scalar fields Φ(~r), Vector fields V~ (~r) Coordinates ∂q ! ∂q Coordinates Vector Fields Xk Xk Xi Xj Vector Fields ∂Φ Derivatives i j Derivatives ~ Φ= eˆi, Gradient = gijδq δq , Gradient ∇ i Velocity of a Velocity of a i ∂r particle ij particle Derivatives of X Derivatives of X Vectors Vectors ∂Vi Differential where Differential ~ ~ Forms k k Forms V = i , ∇· ∂r 2-forms ∂r ∂r 2-forms i 3-forms gij = i j . 3-forms X Cylindrical ∂q ∂q Cylindrical ∂2Φ Polar and k Polar and 2Φ=~ ~ Φ= , Spherical X Spherical ∇ ∇·∇ i 2 gij is a real symmetric matrix called the metric tensor. i ∂r X In general a nontrivial function of the position, gij(q). ∂Vk ~ V~ = ijk eˆi, 3D only To repeat: ∇× ∂rj 2 i j ijk (δs) = g δq δq . X ij Xij k Physics 504, Another problem:e ˜k is not normal to the surface q = Physics 504, Functions (fields) Spring 2011 Spring 2011 Electricity constant. Electricity A scalar field f(P ) f(~r) can also be specified by a and and function of the q’s, f˜(q)=f(~r(q)). Magnetism Magnetism What to do? Two approaches: What about vector fields? V~ (~r) has a meaning Shapiro Shapiro independent of the coordinates used to describe it, but I Give up on orthonormal basis vectors. Define Generalized ∂Φ k Generalized components depend on the basis vectors. Should have Coordinates differential forms, such as dΦ= dx . The Coordinates Cartesian k Cartesian D ∂x D i coords for E k coords for E basis vectorse ˜i aligned with the direction of q .Howto Generalized Generalized Coordinates k X Coordinates define? Vector Fields coefficients of dx transform as covariant vectors. Vector Fields Consider Derivatives Contravariant vectors may be considered coefficients Derivatives Gradient k Gradient 1 1 2 3 1 2 3 k Velocity of a of directional derivatives ∂/∂q . This is the favored Velocity of a ~r(q + δq ,q ,q ) ~r(q ,q ,q ) ∂r eˆk particle particle e˜1 = lim − = Derivatives of approach for working in curved spaces, differential Derivatives of 1 1 Vectors Vectors δq 0 δs ∂q √g11 Differential Differential → k Forms geometry and general relativity. Forms X 2-forms 2-forms 3-forms I Restrict ourselves to orthogonal coordinate systems 3-forms and the similarly definede ˜2 ande ˜3. In general not good Cylindrical i j Cylindrical Polar and — surface q = constant intersects q = constant at Polar and orthonormal bases, because Spherical Spherical right angles. Then ~ qi ~ qj =0. k k ∇ · ∇ i j ∂r ∂r ij i j ∂q ∂q e˜1 e˜2 = /√g11g22 = g12/√g11g22, In general define g := ~ q ~ q = . · ∂q1 ∂q2 ∇ · ∇ ∂rk ∂rk k k X ij X Note that g is not the same as gij. which need not be zero. In fact Physics 504, The set eˆk and the set e˜i are each orthonormal, Physics 504, Spring 2011 { } { } Spring 2011 i ` m m i m ande ˆk = Akie˜i,soA is orthogonal, and B = A. i` ∂q ∂q ∂r ∂r ∂q ∂r Electricity i ∴ Electricity g g`j = k k ` j = k δkm j and Can also check as and ∂r ∂r ∂q ∂q ∂r ∂q Magnetism Magnetism ` ` k m km P X X X X X i k j k ij .. Shapiro AkiAkj = hihj (∂q /∂r )(∂q /∂r )=hihjg = δij. Shapiro = δij,sog is the inverse matrix to g... k k Generalized X X Generalized If ~ qi ~ qj = 0 for i = j, gij = 0 for i = j, g.. is diagonal, Coordinates Coordinates Cartesian Thus A can be written two ways, Cartesian ∇ · ∇ 6 2 6 D D so g.. is also diagonal. And as (δs) > 0 for any non-zero coords for E coords for E Generalized i k Generalized Coordinates ∂q 1 ∂r Coordinates δ~r, g.. is positive definite, so for an orthogonal coordinate Vector Fields − Vector Fields 2 Aki = hi k = hi i . system the diagonal elements are positive, gij = hi δij, Derivatives ∂r ∂q Derivatives ij 2 Gradient Gradient and g = h− δij. Then the unit vectors are Velocity of a Velocity of a i particle Note that Ajk is a function of position, not a constant. In particle Derivatives of Derivatives of k Vectors Vectors 1 ∂r Euclidean space we say thate ˆx is the same vector Differential 1 Differential e˜i = hi− eˆk i =: Bkieˆk (1) Forms regardless of which point ~r the vector is at . But then Forms ∂q 2-forms 2-forms k k 3-forms 3-forms X X Cylindrical e˜i = Aji(P )ˆej is not the same vector at different Cylindrical with the inverse relation Polar and Polar and Spherical Xj Spherical ∂qi points P . eˆk = hie˜i =: Akie˜i = Aki B`ieˆ`. (2) ∂rk 1 i i i ` That is, Euclidean space comes with a prescribed parallel X X X X transport, telling how to move a vector without changing it. As thee ˆk are independent, this implies i AkiB`i = δk` or ABT =1I. P Physics 504, Physics 504, Vector Fields Spring 2011 Gradient of a scalar field Spring 2011 Electricity Electricity So for a vector field and and Magnetism Magnetism ~ ˜ V (P )= Vj(q)˜ej = Vi(~r)ˆei = Vi(~r)Aike˜k, Shapiro ˜ ` Shapiro ~ ∂f ∂f ∂q j i ik f = k eˆk = ` k (Akme˜m) X X X Generalized ∇ ∂r ∂q ∂r ! Generalized Coordinates Xk Xk`m Coordinates ˜ Cartesian Cartesian so Vk(q)= Vi(~r(q))Aik(q). coords for ED ˜ ` m ˜ coords for ED Generalized ∂f ∂q ∂q ∂f m` Generalized i Coordinates = ` k hme˜m k = ` hme˜mg Coordinates X Vector Fields ∂q ∂r ∂r ∂q Vector Fields ˜ k`m ! `m Also Vk(~r)= Aki(~r)Vi(q(~r)). X X Derivatives ˜ ˜ Derivatives i Gradient ∂f 2 1 ∂f Gradient X Velocity of a − − Velocity of a particle = ` hme˜mhm δm` = hm m e˜m, particle Derivatives of ∂q m ∂q Derivatives of Let’s summarize some of our previous relations: Vectors `m Vectors Differential X X Differential Forms Forms 2-forms or 2-forms eˆk = Akie˜i, e˜i = Aji(P )ˆej 3-forms 3-forms Cylindrical Cylindrical Polar and ˜ Polar and i j 1 ∂f X X Spherical ~ f = h− e˜ . (3) Spherical ∇ m m m i k m ∂q ∂q 1 ∂r X Aki = hi k = hi− i . ∂r ∂q Both f(~r) and f˜(q) represent the same function f(P )on 2 ij 2 gij = hi δij,g= hi− δij space, so we often carelessly leave out the twiddle. Physics 504, Physics 504, Spring 2011 Spring 2011 Velocity of a particle Electricity Electricity and and Magnetism Magnetism k k i j dr ∂r dq ∂q Shapiro Shapiro ~v = eˆk = hj e˜j By looking at distances from varying one coordinate, dt ∂qi dt ∂rk 2 2 2 2 2 2 2 k k i ! j comparing to (ds) = hr(dr) + h (dθ) + h (dφ) , X X X X Generalized θ φ Generalized i j Coordinates we see that Coordinates dq dq Cartesian Cartesian = hjδije˜j = hj e˜j. coords for ED coords for ED dt dt Generalized Generalized ij j Coordinates hr =1,hθ = r, hφ = r sin θ. Coordinates X X Vector Fields Vector Fields j Derivatives Derivatives Note it is hjdq which has the right dimensions for an Gradient Thus Gradient j Velocity of a Velocity of a particle ~v =˙re˜r + rθ˙e˜θ + r sin θφ˙e˜φ particle infinitesimal length, while dq by itself might not. Derivatives of Derivatives of Vectors Vectors Differential Differential Forms and Forms Example, spherical coordinates. 2-forms 2 2 2 2 2 2 2 2-forms 3-forms v =˙r + r θ˙ + r sin θφ˙ . 3-forms r spherical shells Cylindrical Cylindrical Polar and Polar and Surfaces of constant θ cone, vertex at 0 Spherical Spherical φ plane containing z with the shells centered at the origin. These intersect at right angles, so they are orthogonal coords. Physics 504, Physics 504, Derivatives of Vectors Spring 2011 Differential Forms Spring 2011 Electricity Electricity and i and Magnetism A general 1-form over a space coordinatized by q is Magnetism Shapiro Shapiro i ∂eˆi ω = Ai(P )dq , Euclidean parallel transport: j =0 Generalized i Generalized ∂r Coordinates X Coordinates Cartesian Cartesian k ` coords for ED and is associated with a vector. But if we are using coords for ED 1 ∂ 1 ∂r ∂ 1 ∂r Generalized Generalized hj− e˜i = hj− hi− eˆ` Coordinates Coordinates ∂qj ∂qj ∂rk ∂qi Vector Fields orthogonal curvilinear coordinates, it is more natural to Vector Fields k` X Derivatives express the coefficients as multiplying the “normalized” Derivatives k Gradient i i ˜ Gradient 1 ∂r ∂A`i Velocity of a 1-forms ωi := hidq , with ω = i Aidq = Viωi. Velocity of a = hj− j k eˆ` particle particle ∂q ∂r Derivatives of ~ ˜ Derivatives of k` Vectors Then ω is associated with the vector V = i Vie˜i.
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