Are Jews allowed to pump their muscles?
What if the intent is to make yourself look good?
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The Rambam (Hil. De'os 4:2,14) recommends exercising (and then resting briefly) before eating.
R' Moshe Feinstein rules (Igros Moshe C.M. 2:65) that dieting is permitted even if solely for the sake of looking good (even if it is not necessary from a health perspective). The halachic question is regarding whether it is a legitimate reason to put oneself through pain/hunger, being that the halachos of "chovel ba'chaveiro" apply just as much to oneself.
It seems reasonable that the same would apply to exercise, since the same halachic issues seem to apply. (R' Moshe might even mention it in his t'shuva; I just can't remember where it is.)
physical health and well being are an integral part of being a human being. it is important that we take care of our seichal, goof, and neshama. Physical exercise is important to taking care of one's physical health. Even in the case of exercising for the sake of how one looks this can impact a person' mental well being in a positive manner. Furthermore, exercise is known to release endorphins which help regulate positive moods as well as improve blood circulation which can improve mental concentration as well as all manners of physical and mental health. Vayikra parshas achrei mois in the beginning of chapter 18 it says "you shall live by them". not only should you live by them you should live for them for our purpose in being Jews is to serve Hashem and one cannot properly do so with an unhealthy body. (on a separate note one should of course be knowledgeable about your exact state of health and know or find out what course of exercise is best for you based on your current state of health)