330 Fluids 4.7 Equations of Motion in Moving Coordi- nate Systems Moving coordinate systems are discussed in this section in the context of general mechanical systems. The theory is applied to the equations of fluid dynamics and particle mechanics. 4.7.1 Moving Coordinate Systems An inertial coordinate system is a coordinate system in which Newton’s laws of motion are valid; in particular, a free particle moves along a (straight) line. Imagine an inertial coordinate system in Euclidean space, with rectangu- lar coordinates (⇠,⌘,⇣) and the motion of a particle whose position vector in this coordinate system is R. Suppose there is a second (moving) rectangular coordinate system (with coordinates (x, y, z)) such that, after the transla- tion in space that moves its origin to the origin of the inertial coordinate system, its translated frame (ordered basis) of coordinate unit direction vec- tors [ex,ey,ez] can be rigidly rotated in space to coincide with the inertial frame [e⇠,e⌘,e⇣ ]. In other words, assume that the moving frame is given at each instant of time by a translation and rotation of the inertial frame. Translation in space, which more properly should be called parallel trans- lation or parallel transport, is given simply by vector addition. For example, let denote the vector from the origin of the moving frame to the origin of theV inertial frame. The vector Q := R ,wheretheadditioniswith respect to the inertial vector space, represents−V the parallel translation of R along the straight line joining the origins to the position of the origin of the moving frame. Likewise, the inertial frame [e⇠,e⌘,e⇣ ], is parallel transported to frame [e1,e2,e3]. The vector connecting the origin of the moving frame to the point Q in space is expressed in the coordinates of the moving frame by a Q = b . 0 c 1 @ A In other words, Q = aex + bey + cez. Fluids 331 The basis vectors for the moving frame have coordinate representations in the parallel transported inertial coordinates: a11 a12 a13 e = a ,e= a ,e= a . x 0 21 1 y 0 22 1 x 0 23 1 a31 a32 a33 @ A @ A @ A Thus, the coordinate representation of Q in the parallel transported inertial coordinates is a11 a12 a13 a a + b a + c a . 0 21 1 0 22 1 0 23 1 a31 a32 a33 Or, equivalently, @ A @ A @ A ↵ a11 a12 a13 a β = a a a b 0 1 0 21 22 23 1 0 1 γ a31 a32 a33 c are the coordinates of R@inA the parallel@ transportedA @ A inertial frame. In a more compact form, R = + AQ, (4.78) V where A is the matrix with components aij.Thismatrixchangestheparallel transported inertial frame [e1,e2,e3]tothemovingframe;thatis, Ae1 = ex, Ae2 = ey, Ae3 = ez. The addition in equation (4.78) is with respect to the moving frame. 1 T The matrix A is an orthogonal transformation; that is, A− = A with respect to the usual inner product, which will be denoted by angled brackets (see Exercise 4.22). Because A is orthogonal, hi AAT = AT A = I. Using this fact, R = A(AT + Q). (4.79) V In other words, the coordinates of R in the parallel transported inertial frame are the components of the vector AT + Q. By Newton’s second law, the equationV of motion of the particle with position R moving in the inertial frame is mR¨ = F, (4.80) 332 Fluids where m is the mass of the particle and F is the (vector) sum of the forces acting on the particle. Our goal is to express the equation of motion of the particle in the moving coordinate system. To do this, the quantities r := AQ, v := AQ,˙ a := AQ¨ will be employed to simplify the calculations used to obtain the transformed equation of motion. They are the position, velocity, and acceleration of the particle in the parallel transported inertial frame. In view of equation (4.78), the velocity of the particle can be expressed in the form R˙ = ˙ + AQ˙ + AQ,˙ V = ˙ + AA˙ T (AQ)+v V = ˙ + AA˙ T r + v. (4.81) V To make further progress, note that the transformation ⌦:= AA˙ T has a special property: it is skew-symmetric; that is, ⌦T = ⌦. This fact is easily proved by di↵erentiation with respect to time in the− identity AAT = I.In fact, AA˙ T + AA˙ T =0; (4.82) therefore, ⌦T =(AA˙ T )T = AA˙ T = AA˙ T = ⌦. − − By using the definition of skew-symmetry, the matrix representation of ⌦ must have the form 0 ! ! − 3 2 ! 0 ! . (4.83) 0 3 1 1 ! ! −0 − 2 1 The action of ⌦on a vector@W given by ⌦WAcan also be represented using the vector cross product (with respect to the usual orientation of space) and the vector ! := (!1,!2,!3)by ⌦W = ! W, (4.84) ⇥ Thus, the velocity of the particle may be expressed in two ways: R˙ = ˙ +⌦r + v, V R˙ = ˙ + ! r + v. (4.85) V ⇥ Fluids 333 Using the first equation in display (4.85) for the velocity of the particle, its acceleration is R¨ = ¨ +˙v + ⌦˙ r +⌦r.˙ (4.86) V Since r = AQ and v = AQ˙ ,theirtimederivativesare r˙ = AA˙ T (AQ)+AQ˙ =⌦r + v, (4.87) and v˙ = AA˙ T (AQ˙ )+AQ¨ =⌦v + a. (4.88) These formulas forr ˙ andv ˙ are substituted into equation (4.86) to obtain the particle’s acceleration in the form R¨ = ¨ + a +2⌦v + ⌦˙ r +⌦2r;(4.89) V or, equivalently, R¨ = ¨ + a +2! v +˙! r + ! (! r). (4.90) V ⇥ ⇥ ⇥ ⇥ The equation of motion (4.80) may be recast in the forms ma = m( ¨ +2⌦v + ⌦˙ r +⌦2r)+F (4.91) − V −V or ma = m( ¨ +2! v +˙! r + ! (! r)) + F , (4.92) − V ⇥ ⇥ ⇥ ⇥ −V where the vectors a, v,andr give the acceleration, velocity and position of the particle, moving under the influence of the force F ,withrespecttoan observer in the moving coordinate system but referred to the frame [e1,e2,e3]. These equations in the frame [ex,ey,ez]areobtainedbymultiplicationonthe left by the orthogonal matrix AT .Forexample,usingequation(4.91),the equation of motion is given by mAQ¨ = m( ¨ +2⌦AQ˙ + ⌦˙ AQ +⌦2AQ)+F ;(4.93) − V −V and, in the moving coordinate frame [ex,ey,ez], it is mQ¨ = m(AT ¨ +2AT ⌦AQ˙ + AT ⌦˙ AQ + AT ⌦2AQ)+AT (F ). (4.94) − V −V 334 Fluids The last equation can also be written in the form mQ¨ = m(AT ¨ +2ΓQ˙ + Γ˙ Q +Γ2Q)+AT (F ), (4.95) − V −V where Γ:= AT ⌦A;thatis,Γand⌦arenamesforthesamelineartransfor- mation expressed in di↵erent bases. Since the equation of motion (4.91) (or (4.95)) is a valid expression of Newton’s equation of motion for the particle on the influence of the force F from the point of view of a non-inertial observer sitting at the moving origin and measuring with respect to the translated inertial coordinates, this observer treats the terms on the right-hand side of the equation of motion as additional forces. The vector ¨ is the acceleration of the origin of the non- inertial frame, ⌦˙ r =˙! r is theV acceleration due to the rotation of the moving frame, 2m⌦v =2m!⇥ v is the Coriolis force, and m⌦2r = m! (! r) is the centrifugal force.⇥ Of course, from Newtonian mechanics, F is⇥ the⇥ only force acting on the particle. The remaining fictitious forces are artifacts of reference to a non-inertial coordinate system. Exercise 4.22. Prove that the coordinate transformation A in display (4.78) is orthogonal. 4.7.2 Pure Rotation To consider pure rotation, imagine a rotating disk whose motion is measured with respect to a rectangular rotating coordinate system (with coordinates (x, y, z)), whose third coordinate-axis coincides with the axis of rotation of the disk, whose origin is at the intersection of the disk with this axis, and whose first two coordinates are fixed in the disk. In addition, suppose without loss of generality, that an inertial frame with coordinates (⇠,⌘,⇣)hasthesame origin. Of course, we could choose the inertial coordinates so that the ⇣-axis corresponds with the axis of rotation; but, we will not make this assumption to illustrate an important feature of three-dimensional space: Each rotation is determined by a vector (specifying the axis of rotation) and the rotation angle about this direction. Thus, under our assumption that there is a fixed axis of rotation, there is an orthogonal transformation B,whichdoesnot depend on time, taking the unit vector in the direction of the ⇣-axis of the inertial frame to one of the two unit vectors in the direction of the axis of rotation. The matrix A,whichdeterminesthecoordinatetransformation Fluids 335 R = AQ from the moving frame to the inertial frame, takes the form BT B, where R cos ✓ sin ✓ 0 − := sin ✓ cos ✓ 0 (4.96) 0 1 R 001 @ A is the rotation matrix (in inertial coordinates) for the motion about the z-axis in the moving frame. By a direct computation, it is easy to check that 0 ✓˙ 0 − ⌦=AA˙ T = ✓˙ 00 , (4.97) 0 1 000 @ A which is exactly the angular velocity in matrix form; that is, the action of this matrix on a vector W is β W, ⇥ where β is the vector with components (0, 0, ✓˙). For rotations in three-dimensional space, we have the useful identity AA˙ T = AT A.˙ (4.98) Thus, ⌦= AT A˙. For pure rotation where the origins of the inertial and rotating coordinates coincide, the equation of motion (4.93) reduces to mAQ¨ = m(2⌦AQ˙ + ⌦˙ AQ +⌦2AQ)+F ;(4.99) − and, in case the angular frequency is constant, to mAQ¨ = m(2⌦AQ˙ +⌦2AQ)+F.