I am flying from LA to NYC and i'm looking at some flights. on Kayak, there is a cheap flight from LA, stopping in ORD prior LGA. Kayak is showing a warning that there is a 35 minute connection between landing in ORD and taking off on the connection for LGA. I know this is a tight window.

My question is, what happens if i miss the connecting flight. I assume if the first flight is late, aa in this scenario will take care of putting me on the next available flight at no cost since the connection is on aa as well. What happens if the first plane lands in chicago with 15 minutes before take off. Technically i could make the connecting flight. Does the airline care? or do they charge me a rebooking fee or do they just put me on the next flight from Chicago to NYC without issue? i know if the first flight is late, it's on them and they'll take care of me. But i'm just curious on if i really am sprinting to the other plane and can't make it. Will i be screwed?

i've never done this before, but hte price is so good, i just want to see what people's experiences are. Thanks for any help you can give me.