The prophet warns (Isaiah 58:13): “If you refrain from trampling (lit. turn back your foot from) the Shabbat... and if you honor it and go not your ways...” From this, the Sages derived the principle that “your walking on Shabbat should not be like your walking on weekdays,” since on weekdays people typically hurry and run about their business. Therefore, on Shabbat it is forbidden to run, to hop (i.e., lifting both feet off the ground at once), to skip (i.e., lifting one foot before the other has touched the ground), or to take large strides (i.e. approximately half a meter). One should walk calmly.
Running for a purpose: It is permitted—and even a mitzvah—to run for the sake of a mitzvah, such as going to synagogue, to Torah classes, or the like. A person may also run to watch an event they enjoy, provided there is nothing forbidden in the event itself. If someone encounters a puddle and has the option to go around it or jump over it—jumping is preferable to walking around, since the latter involves more walking on Shabbat. If jumping is not possible, it is permitted to go around.
Fleeing: It is permitted to run to avoid rain, or similar situations, to prevent getting dirty in the mud, or to avoid hearing or seeing forbidden things, or things that cause distress.
Running for its own sake: One who enjoys a morning run, or children who enjoy running and jumping, or when running is part of a game like tag or hide-and-seek—it is permitted to do so on Shabbat. However, if it is apparent that the person is doing so for medical or health reasons, it is not permitted—unless they are at least considered a “non-dangerous sick person” (choleh she’ein bo sakanah) who must do this activity that day. (In most cases, even those who exercise daily will not be harmed by skipping one day.)
Rabbi Yosef Simcha Ginzberg, translated from Sichat HaShevua
