Short Intro to Coorinate Transformation 1 A Vector A vector can basically be seen as an arrow in space pointing in a specific irection with a specific length. The following problem arises: How o we represent this quantity mathematically? Iea: choose an arbitrary reference point an introuce three linear inepenent vectors with length 1, starting at the chosen reference point. Now the vector ~v can be represente in terms of our three known, so-calle basis vectors. The whole arrangement is calle a reference frame or coorinate system. ~v = a ~e x + b ~e y + c ~e z (1) or ~v = 0 B @ a b c 1 C A (2) 2 Transformation of Coorinates 2.1 Which Frame to use? Depening on the reference frame which is chosen the representation of the vector will iffer, of course. However, as can be seen in Figure 5.2 the vector itself is an inepenent quantity in space. This implies that there must exist some relation between ifferent reference frames. Otherwise it is not possible to have ifferent representations for the same quantity. 1
2.2 Transformation Mathematically spoken, this can be expresse as follows. ~e 1 = ~e 1 (~e x,~e y,~e z ) (3) ~e 2 = ~e 2 (~e x,~e y,~e z ) (4) ~e 3 = ~e 3 (~e x,~e y,~e z ) (5) For our cases the inverse transform must always exist as well! 3 Time Depenency 3.1 Time Depenent Vector So far, only constant vectors in space have been examine. However, especially in Mechanics III, we are intereste what happens when the vector, like a velocity vector evolves in time. Representation or ~v (t) =a(t) ~e x + b(t) ~e y + c(t) ~e z (6) ~v (t) = 0 B @ a(t) b(t) c(t) 1 C A (7) Note that there is time-epenency of each coefficient now. Time Derivative As we are taking erivatives the representation where the basis vector are inclue is favorable. (easier to take all necessary terms into account) t ~v (t) = t (a(t) ~e x + b(t) ~e y + c(t) ~e z )=ȧ(t) ~e x + ḃ(t) ~e y +ċ(t) ~e z (8) 3.2 Time Depenent Basis Vectors Now consier the case that the basis itself moves in space. However now we nee at least two reference frames. This of course is possible. Why: Since our frame moves in space, we nee again a fixe reference to which we can relate the motion of the coorinate system. This fixe reference frame is a so-calle inertial frame. In every problem in Mechanics III at least one inertial frame is neee! ~e 4 = ~e 4 (~e x,~e y,~e z,t) (9) 2
~e 5 = ~e 5 (~e x,~e y,~e z,t) (10) ~e 6 = ~e 6 (~e x,~e y,~e z,t) (11) Note that: t ~e 4 6= 0 t ~e 5 6= 0 t ~e 6 6= 0 (12) 3.3 Time Depenent Vectors in a Time Depenent Basis Of course we can also express a changing vector in a changing basis. Do not forget that we are still looking at the same vector in space. So in case a velocity vector is expresse in such way, we are still looking at the same vector as when expresse in the inertial frame! Representation ~v (t) =a(t) ~e 4 (t)+b(t) ~e 5 (t)+c(t) ~e 6 (t) (13) Note that only this notation is use to explicitly express that the basis is time-epenent. Time Derivative We have to apply prouct rule. In case, the unit vectors are implicitly epenent on time, o not forget to apply the general chain rule as well! t ~v (t) =ȧ(t) ~e 4(t)+a(t) ~e 4 (t)+ḃ(t) ~e 5(t)+b(t) ~e 5 (t)+ċ(t) ~e 6 (t)+c(t) ~e 6 (t) (14) 4 Motivation for Mechanics 3 The reason to even consier such moving frame is the following: As the time erivatives might see very complicate however working in such reference frames can rastically reuce the computational effort. Transformation of coorinates often yiels very simple results (see e.g. Section 5.3 for a circular motion). However since LMP, AMP are efine for an inertial frame only we cannot neglect the time epenency of our unit vectors. As long as we o not o this it is perfectly fine an highly recommene to use moving frame. So always pay attention in which reference frame you are using AMP, LMP. This is critical since you have to use the corresponing formulas for velocity, acceleration, etc...! 5 Important Transformations seen so far 5.1 Spherical coorinates ~e r = cos( ) cos( ) ~i + sin( ) cos( ) ~j + sin( ) ~k ~e = sin( ) ~i + cos( ) ~j ~e = cos( ) sin( ) ~i sin( ) sin( ) ~j + cos( ) ~k (15) 3
As can be seen the basis vectors are implicitly epenent on time. (, an r epen on time). Attention: Definition of spherical coorinates iffers between textbooks. Thumb rule: If is efine from above (between z-axis an r-vector), then switch all cos( ) to sin( ) an vice versa. Now ue to the efinition of spherical coorinates the position vector of our point mass or of the center of mass will always be collinear to ~e r. Therefore the position vector can be expresse as follows, with r as the absolute istance to the origin. ~r = r ~e r (16) The absolute velocity an acceleration follow from the time erivatives: ~v =ṙ ~e r + r cos( ) ~e + r ~e (17) ~a = ( r r 2 cos 2 ( ) r 2 ) ~e r (18) +(2ṙ cos( )+r cos( ) 2r sin( )) ~e +(2ṙ + r 2 sin( )cos( )+r ) ~e 5.2 Cylinrical Coorinates ~e r = cos( ) ~i + sin( ) ~j ~e = sin( ) ~i + cos( ) ~j ~e z = ~ k (19) Due to the efinition(i.e. position vector can be expresse as follows: projection of ~r on the x,y-plane is collinear with ~e r )the ~r = r ~e r + z ~e z (20) 4
Velocity an acceleration follow from the time erivatives: ~v =ṙ ~e r + r ~e +ż ~e z (21) ~a =( r r 2 ) ~e r +(r +2ṙ ) ~e + z ~e z (22) 5.3 Circular Motion As has been seen before, a circular motion with solely tangential velocity arises in a lot of ifferent application. E.g., when a point mass or the center of mass of a rigi boy moves in a circular motion. The formulas for the position vector, acceleration an velocity are a special case of cylinrical motion with the following conitions: z=0 the cylinrical reference frame rotates with the same (!) angular velocity as the center of mass. i.e.: ~e r is always collinear with the position vector ~r (has been state above but just to be clear) ṙ =0! r=const we now efine v = r Uner these circumstances our equations yiel: ~r = r ~e r (23) ~v = r ~e = v ~e (24) 5
~a = r 2 ~e {z } r + r ~e = v2 {z } r ~e r + v ~e (25) centripeal tangential 5.4 Circular Motion in inertial frame Just to see a practical example, that even such a simple example as a circular motion is much more ifficult to eal with in the inertial frame. (here we use sketch) instea ue to the ~r = R (cos( )~i + sin( )~j) (26) ~v = R ( sin( )~i + cos( )~j) (27) ~a = R( sin( ) 2 cos( ))~i + R( cos( ) 2 sin( ))~j (28) 6 TAKE HOME MESSAGES Always be aware of which reference frame you are using an apply the appropriate formulas accoringly! Use ifferent reference frames when appropriate. If it helps, always inicate the reference system you are currently using. (also goo for the exams since it is then clear what you i). KEEP IN MIND WHAT CONDITIONS WE IMPOSED WHEN WE DERIVED THE FORMULAS (especially for circular motion) In the en, it might be easier to express all your quantities in a ifferent coorinate system an compute everything in this coorinate system rather than the inertial frame. 6