Question: Am I allowed to jog or jump on a trampoline on Shabbos?
Answer: The Mishna (Shabbos 22:6) writes that there is a prohibition against being misamel on Shabbos. While Rashi translates this as massage, Rambam (Shabbos 21:28) understands it to mean exercise. Exerting oneself enough to bring out a sweat is a form of refuah, which is generally forbidden on Shabbos (See Shulchan Aruch OC 328:42). Thus, R’ Eliezer Waldenberg (Tzitz Eliezer 6:4) writes that one can’t work out with gym equipment.
The Gemara (Shabbos 113a) writes that one’s actions on Shabbos should be different to those during the week. According to Ramban (Vayikra 23:24), there is a mitzva deoraisa to rest on Shabbos. Thus, the Shulchan Aruch (OC 301:1) writes that one must not run on Shabbos unless it is to do a mitzva, such as running to shul (See Aruch Hashulchan OC 301:44). While the Mishna Berura (301:7) forbids jogging on Shabbos, R’ Shmuel Wosner (Shevet Halevi 1:58) writes that one may run to escape the rain.
The Shulchan Aruch (OC 301:2) does allow one to jump and run for pleasure. One would therefore be allowed to jump on a trampoline for fun, though not for health reasons (See Shemiras Shabbos Kehilchasa 16:39).R’ Dr. Avraham Avraham (Nishmas Avraham OC 301:1) points out that if one had a medical condition and had to jog for medical reasons they can certainly do so.