May one go walking on Shabbos for weight loss and exercise purposes?
Going for an exercise walk on Shabbos touches upon two possible prohibitions; 1) Perhaps it transgresses the Rabbinical command that requires for one to change his walking habits on Shabbos. 2) Perhaps it transgresses the Rabbinical prohibition of causing oneself to sweat which was enacted due to the general medicine prohibition on Shabbos. The following is the background of these two prohibitions.
The Shabbos walking restriction: The verse states “Im Tashiv Mishabbos Raglecha…Vechibadeto Meiasos Derachecha etc/If from walking you will rest and honor Shabbos by not doing weekday activity.” From here the Sages learned that one’s walking on Shabbos should not be like one’s walking during the week. Thus, it is forbidden to run on Shabbos. Likewise, one is not to walk more than necessary in order to get to a certain location. Hence, if there are two ways to walk, the shorter route should always be chosen. Nonetheless, it is permitted to go for a Shabbos walk [and take a longer route for that purpose].
The Shabbos exercise restriction: Just as it is forbidden to massage one’s body in order to exercise and sweat, similarly it is forbidden to exercise in other forms in order so one sweats for healing purposes. The reason for this is due to a decree that one may come to grind and drink herbs which cause one to sweat. [Accordingly, it is certainly forbidden to run for the sake of exercise and health purposes on Shabbos. This applies even if one runs in small strides.]
The law by walking:
Plain and regular walking: Some Poskim rule that it is permitted for one to go for a walk in a normal pace on Shabbos without limitation even if he intends to do so for the sake of exercise purposes and to sweat. Accordingly, one who takes a daily walk for weight loss and exercise purposes, may do so even on Shabbos. However, other Poskim rule that it is forbidden to even walk on Shabbos if his intent is to sweat or exercise, and hence one may not go for his daily walk on Shabbos unless it is being done for a different purpose, and not simply for the sake of exercise. Practically, from the letter of the law one may choose to be lenient, and so should be done when medically required. However, it is proper to be stringent and push off the daily walk for Motzei Shabbos and the like, especially in light of the fact that one should use his time on Shabbos for Torah learning.
Power walking/Running/Jugging for exercise: All the above discussion is only regarding walking a normal pace in which it is not recognizable that one is doing so for exercise purposes. However, according to all it is forbidden for one to run or jog or even power walk for the sake of exercise on Shabbos, as in such a case it is apparent to all that he is doing so for exercise purposes, and is likewise forbidden due to the Rabbinical restrictions against running on Shabbos. Likewise, even when walking a normal pace, if ones hand movements or dress make it evident that he is doing an exercise walk, then it is forbidden to do so.
From the letter of the law, it is permitted for one to go on a regular walk on Shabbos even if he intends to do so as part of his daily exercise, so long as his form of walking is not recognizable that it is being done for an exercise purpose. Nonetheless, it is proper to be stringent to push off such activities until after Shabbos. According to all, only plain walking is allowed, while jogging and power walking remains forbidden.
 See Piskeiy Teshuvos 301:4
 Admur 301:1-2 [printed in the new Edition of the Shulchan Aruch, based on manuscript of Maharil, first published in Yagdil Torah N.Y. 10]; Michaber 301:1-3; Shabbos 113; Ketzos Hashulchan 117:2-3
 Yeshaya 58:13
 Shabbos 113a; Tur 301; Levush 301
 Admur ibid; Michaber 301:1; Tosefta Shabbos 17:16
 Rama 301:2; Beis Yosef in name of Tosefta; Igros Moshe 5:18; Piskeiy Teshuvos 301:5
 Admur 328:47
 M”A 301:5; Beis Yosef in name of Tosefta; Admur 328:47 regarding sweating for exercise
The reason: It is forbidden to exercise in order so one sweats for healing purposes due to a decree that one may come to grind and drink herbs which cause one to sweat. [Admur ibid] This is in addition to the prohibition against running on Shabbos due to the above quoted verse.
 P”M 301 A”A 1
 M”A 301:5; P”M 301 A”A 5 in negation of Elya Raba; Tosefes Shabbos 301:5; Pesach Hadvir 301:3; 1st and Stam opinion in M”B 301:7; Kaf Hachaim 301:13; Shaar Hatziyon 301:8 that so is the implication of Tosefta ibid; Orchos Rabbeinu 3:246 that the Chazon Ish took his daily walk for exercise purposes also on Shabbos
 See Beis Yosef 301, brought in Taz 301:1, that he may walk even for the entire day
 The reason: As the Sages did not restrict pleasure walking on Shabbos [Rama ibid] and the exercise and sweat prohibition only applies if it is apparent to others that the activity that one is doing is for this purpose. Thus, since going for a regular walk does not appear to onlookers as exercise, therefore it is permitted to be done even if in truth he is doing so for exercise. [P”M 301 A”A 5; Machatzis Hashekel ibid; M”B 301:7; See Admur 327:1; 328:28 and 43]
 Elya Raba 301:7 in implication of Levush 301:2 and that so is implied from the Tosefta and Rama ibid, brought, and negated, in P”M 301 A”A 5; 2nd opinion in M”B ibid
 See M”A ibid; Piskeiy Teshuvos ibid
Other opinions: See Shut Milameid Lehoil 53 and SSH”K 16 footnote 106 in name of Rav SZ”A and Or Letziyon 3:36-12 that if he enjoys the actual exercise and is not intending do so in order to sweat, then it is permitted
 M”A ibid
 Taz 301:1 that whenever the purpose of the run is not intrinsically pleasurable, then it is forbidden due to the Rabbinical running restriction; However, from the M”A ibid it is evident that whenever there is a purpose of pleasure involved in doing so, then it is permitted to be done, if not for the secondary prohibition against exercise which is applicable in such a case; See Machatzis Hashekel on M”A ibid