When a person has his hand out in rishut harabim with something in it, there is a view in the gemara that Chazal would not make a knas that he has to keep his hand there until after shabbat because he would tire and drop it. In that case he would be chayav a korban chatat. But it is hard to see why that is so.
If he was holding his hand out for a long time that means that he realizes that it is shabbat and it is forbidden to put it down. He drops it only because he cannot hold it any longer. This means that it is not shogeg at the start and at the end, which the gemarah later states as a condition for being chayav a korban. . Furthermore, his dropping it should be in the category of "oness," since he drops it only because unable to hold it any longer. He should be patur from a korban in that case.
Dr. Yehuda Gellman, Yerushalayim
Baruch she'Kivanta! Your question is asked by the Rashba.
The Rashba answers that the Gemara is referring to a case where he was Shogeg at the beginning and afterwards he remembered. The Rashba writes that when the Gemara states he would be liable to bring a Korban Chatas, it does not mean literally that in the times of the Beis ha'Mikdash there would be an obligation for a Chatas, because one requires Shogeg at the start and at the end, as you point out. Rather, the Gemara means that he might come to Chilul Shabbos which involves a Chiyuv.
The Rashba writes that a similar explanation can be given concerning the Din of Rav Bibi which, according to the Gemara on 4a, is referring to a case where he put the bread in the oven deliberately. The Rashba writes that when the Gemara there states that he would come to a Chiyuv Sekilah, it does not literally mean Sekilah, because if the reason he does not remove the bread from the oven is that he thinks it is forbidden to do so, then this means that at the end he was Shogeg and therefore not Chayev Sekilah. Rather, the intention of the Gemara (4a) is to ask whether he may remove the bread -- which generally carries with it a Chiyuv Sekilah -- before the baking is concluded and Shabbos is desecrated.
Similarly, when the Gemara (3b) mentions a Chiyuv Chatas it does not mean literally a Chiyuv Chatas, but rather that by dropping the item this represents Chilul Shabbos which generally is liable for a Chatas when the necessary conditions to be liable are fulfilled.