The magazine aims to attract funding to launch fully as a monthly publication next January, with an increased circulation of between 300,000 and 500,000.
Norton, a former sports writer at the Telegraph, set up the company Rude Health Media to publish the magazine.
He hopes to capitalise on the success of Shortlist Media's free titles Stylist and Shortlist, and for Vigour to achieve "double the circulation" of Hearst's Men's Health magazine, the leading health monthly which recorded a circulation of 216,336 in the last six months of 2012.
Vigour positions itself as "the first freemium outdoor and health magazine" and is aimed primarily at men aged 25- to 40. It will offer health advice on training, nutrition, travel and sports gear, as well as stories of sporting adventure to inspire readers.
The pilot issue carries free ads from health brands including Vita Coco, the coconut water drink, and "barefoot" shoe brand, Vivo Barefoot.
The magazine is being given out at high-footfall train and Tube stations in central London by Contact Field Marketing.
Norton said: "We would plan to start monthly and then go weekly, with a number somewhere between 300,000 to 500,000 when we are confidently up and running and we have built partnerships with the appropriate advertisers."
The pilot issue's editorial includes a cover feature interview with ex-England rugby captain Lewis Moody, a Q&A with Bear Grylls, Bradley Wiggins' "fridge secrets", and recipes from chefs such as Tom Aikens.
Norton said: "It's been quite a tortuous challenge launching a pilot in a recession, a bit like a gruelling ultra-marathon in a snow storm, particularly as we are a start-up company with a small team.
"But I really feel there's a big market there for the first widespread free health magazine, which does not insult men that life is all about six-packs. Shortlist Media have been an inspiration in this marketplace."