My question is, during the evolution of a solar system is it possible to end up with a massive planet around a star with two small planets around their $L_4$ and $L_5$.
Since we know it is not possible to have any bodies around $L_1, L_2, L_3$ due to their low stability, hence any planet trying to evolve their can be easily nudges out of its position.
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$\begingroup$ The objects orbiting in the L4 and L5 probably can't be classified as planets due to the impossibility of them clearing their orbits by definition. $\endgroup$– A. C. A. C.Commented Aug 4, 2017 at 19:19
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Yes, provided the star is big enough, the "Jupiter" is big enough and the planet is small enough.
The details are discussed herein: "https://astronomy.stackexchange.com/questions/3595/is-there-a-ceiling-for-stable-l4-or-l5-masses"
To summarise: The sun needs to be at least 25 times more massive than the giant planet, and the small planet needs to have negligible mass.
It is worth noting that the minor body could not be said to be dominating its orbit, and so would be considered a dwarf planet by the IAU.
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$\begingroup$ These limits will be applicable for an already existing system where rearrangement of bodies would be required to achieve the result, is there any model that can predict the odds of system evolving in that manner from a protoplanetary disk? $\endgroup$ Commented Mar 9, 2016 at 15:29
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$\begingroup$ The planet would probably have to start in a regular orbit, and be captured to the L4/5 orbit, in the same way that the existing trojans are captured asteroids. I don't think it's likely, and I don't know of any model that predicts the formation of earth sized trojans $\endgroup$– James KCommented Mar 9, 2016 at 18:57