B757-200 FLT CREW RECEIVES MULTIPLE STAR AND RWY CHANGES DURING LATE STAGES OF ARRIVAL TO LAX.

Narrative

WE ORIGINALLY HAD BEEN GIVEN THE CIVET 5 RWY 25L ARR INTO LAX. WE WERE NOW GIVEN DIRECT TNP AND THE SEAVU 1 RWY 25L ARR. WE WERE TOLD TO DSND VIA THE SEAVU 1; RWY 25L. JUST PASSING DIXXN ABOUT TO INTERCEPT THE RWY 25L COURSE WE WERE GIVEN DSND SEAVU 1 RWY 24R. THIS IS A COMPLICATED FMC CHANGE; SINCE WE HAVE TO REINSTALL THE SEAVU 1 WITH OUR ORIGINAL IFP (TNP); PUT NEW RWY IN; AND THEN FIND THE RIGHT FIX TO FLY TO. I WAS EXPECTING THIS SINCE SOCAL IS DOING THIS SO OFTEN NOW AND HAD FINISHED WITHIN 15 SECONDS OF HAVING NEW COURSE IN; BUT REALIZED THAT WE WERE ALMOST PAST THE MINZE POINT AND NEEDED TO TURN ACFT FOR RWY 24R. THE ACFT WOULDN'T ENGAGE IN LNAV EVEN THOUGH THERE WAS NO DISCONTINUITY AND THAT THE NEXT FIX WAS MAGENTA. I HAD THE FO JUST HDG SELECT THE ACFT ON THE COURSE. WE WERE NEVER OFF COURSE BUT LNAV WOULDN'T ENGAGE. WE WERE THEN CLRED FOR THE APCH RWY 24R BUT ALMOST IMMEDIATELY THAT WAS CANCELED AND TOLD MAINTAIN 4000 FT. WE WERE THEN CLRED FOR APCH TO RWY 24L; WITH A B747 FOR RWY 24R ONLY 3 AND 1/2 MI AHEAD FOR THE R. ALSO TOLD TO MAINTAIN 180 KTS; WHICH BECAME APPARENT WAS TOO FAST WITH THE HVY AHEAD OF US. WE HAD TO SLOW DOWN TO MAINTAIN SEPARATION. SOCAL WHEN GIVING US RWY 24L NEVER GAVE US A HDG; TOLD US TO 'JUST SLIDE IT OVER TO THE L.' THIS IS THE LATEST I HAVE EVER BEEN GIVEN A CHANGE IN THE SEAVU 1 ARR. FOR THIS APPLICATION WE WERE AT ONE TIME OR ANOTHER GIVEN AN APCH CLRNC TO 3 DIFFERENT RWYS! WE WERE IMC DURING MOST OF THIS MANEUVER. A VERY HIGH WORKLOAD ENVIRONMENT FOR A LARGE ACFT LIKE A B757 TO BE PERFORMING IN IMC.