Radial velocity and angular momentum, with geometric algebra

Transcript

now we're going to look at radial Vector representations this picks up where we left off with circular and spherical Vector representations we're going to write Vector x equals R times R hat where R is the length of the vector X and R hat is the vector x divided by the length the vector R hat implicitly encodes any angular dependence along in a path on the unit sphere the vector X Prime by Chain rule as R Prime R hat plus r r hat Prime we're going to calculate our hat Prime this is an illustration of our hat and R hat Prime where R hat Prime is scaled to Unity tracing out an arbitrary path on a unit sphere observe that R hat Prime is always tangential to the unisphere and perpendicular to R hat at all points we'll need a little Lemma namely R Prime is X Prime dotted with r hat to derive this we start with r squared equals x dot X take derivatives of both sides you have 2R Prime equals 2x Prime dotted with x divide through by 2R on both sides we're left with r Prime equals x Prime dotted with X over R which is R hat we're going to derive the expression for R hat Prime that's the derivative of the vector x divided by its length R we use chain rule to find X Prime divided by R minus r Prime over r squared the derivative of one over R times X Factor out R in the denominator to find one over R all times x Prime minus r Prime R hat now we factor out R hat we can do this by writing one equals R hat squared for the X Prime term come up with r hat divided by R all times R hat times the geometric product with X Prime minus r Prime we use our Lemma to write R Prime as R hat dotted with X Prime and we notice that we have the geometric product of R hat with X Prime minus the dot product of R hat with X Prime which is the wedge product of our hat with X Prime this gives us our final result our hat Prime is R hat divided by R all times bi Vector R hat wedge with X Prime here's a sneakier way to come up with the same result I can write X Prime equals 1 times x Prime where one is our hat times R hat factor out R hat and write the geometric product or hat with X Prime as a DOT product plus the which product you can use our Lemma to identify R hat dotted with X Prime as the derivative of the scalar length R Prime to strip X Prime as R Prime Times R hat plus a rejection of our hat from X Prime we can also expand the derivative of x Prime as R Prime R hat plus r r hat Prime and notice that we must have R times R hat Prime equal to the rejection R hat times the bi Vector R hat wedged with X Prime that gives us the same result we can expand our hat Prime as a conventional triple Vector cross product we start by taking the vector or had times the wedge product and closing that in a grade one selection as this is a vector in the first place it doesn't change the result we then expand the wedge product as a cross product as I times cross product commute the I and are left with the geometric product of R hat with a wedge product a vector expand that as dot product plus I times cross product our DOT product term is a scalar all multiplied by a pseudoscaler which is grade three so the great one selection of that term is zero we're left with just the triple cross product incorporate the sine of I squared equals minus one into the order of that cross product as an application of these results let's compute the kinetic energy of a particle with mass m found that our velocity is R Prime R hat plus r hat times R hat wedge with v since R hat and R hat Prime are perpendicular we can compute the velocity Square by squaring each of these terms independently we have energy equals MV squared over two which is M by 2 of R Prime squared plus M by 2 of our hat times R hat which V all square root let's insert a factor of r squared changing one of our R hats into an r that leaves us with for that term one over two m r squared times R hat times R wedge with MV MV is our momentum and we'll write a bi-vector angular momentum as L equals R wedged with MV or R wedged with P we can now write our energy in terms of L the angular momentum term is one over two m r squared times R hat times L times R hat times L but R hat and L and to commute we can change the order flipping the sign leaving us with r hat L squared r hat L squared is a scalar so we can write this as L squared times R hat squared and R hat squared is one we're left with the energy one-half MV squared equals m over two of R Prime squared minus one over two m r squared L squared the minus sign is because we've used a bi-vector angular momentum representation which scores the negative values to summarize given x equals R times R hat we found that X Prime is R Prime R hat plus r r hat Prime we found a geometric algebra expression for R hat Prime our hat Prime because R hat divided by R times r h wedged with X Prime that's just one of our times the rejection of R hat from X Prime all the non-radial components of X Prime we also found the conventional triple cross product representation of R hat Prime finally we found an expression for the kinetic energy of a point particle with mass m that kinetic energy is one over two M times Mr Prime squared minus 1 over 2 m r squared times the angular momentum squared Mr Prime is the radial component of the particles momentum and L is the angular momentum R which p this video was created with Mana for more content be sure to like subscribe and click the Bell check out my blog Peter yo.com for more geometric algebra material where you can also find a free PDF copy of my book geometric algebra for electrical engineers and detailed latex typeset notes from a number of physics and Engineering courses