What exercise builds your internal fortitude the best? the ability to mentally prepare you for life, exercise, encounters, etc.? Is it weight lifting that gives you the physical strength and appearance? Extreme aerobic that gives you endurance and stamina? If you had to choose, which would it be?
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Unfortunately, just participating in any kind of sports can't help you with determination (internal fortitude) by itself. I've tried mountaineering, running, aikido and yoga, and each of them can help with this goal if you have the right state of mind, but each of them can make you even weaker (mentally), when you are practicing with wrong thoughts.
My own method to improve determination is to state goals for yourself, which are just a little above your skill/strength, as you are feeling it. And successfully accomplishing these goals. For example, if you are in the middle of your (say, 10 km) running workout and sprain your ankle, you can choose to run home (though slower) or even to take a slightly longer route, instead of calling for help. I mean no torturing yourself, but just fighting with difficulties which are hard, but have no real danger (I've been in this situation a couple of times, last time yesterday, and all was well afterwards, but "your mileage may vary").
Being on the mountain and having to climb on and on because storm/night is coming, you find that you can go further (and even make right decitions) even when yor last sleep was 30 hours ago.
All this can make you believe, that you can handle a lot of difficult situations, and shows you that your effort gives its result, even if it takes time. So in the next difficult situation you'll just fight longer - this is internal fortitude.
On the other hand, this "going over your head", if overused, can be unsafe and lead you into trouble (or plainly make you feel weaker, if you fail to accomplish your goal again and again). Here you should use your common sense and wisdom.
You can begin experimenting with this by exercising longer. In particular, when you feel that you can endure no more, give a promise to yourself to exercise +10% of what you have done already. Running is ideal for this type of experiments.
As steed says, goal setting is one of the keys. One of the other things that I've found to develop that inner confidence is competition. Doesn't matter what kind, and in fact, the more varied the better. Doesn't necessarily have to be physical competition, mental and business lend themselves to the same structure in different ways.
When you are just working out, even if you push yourself, you may not always face the same adversities as you would during a competition. Plus, there is a certain immediacy to competition that promotes the ability to think on your feet, adapt to changing situations, fluid thinking, etc.
Team sports will generate one dynamic and way of thinking, individual sports another. You can develop leadership skills while in a team situation, as well as being able to excel within a role for the good of the team. Individual sports give you a chance to test your self reliance and adaptability.
Even if you don't win, do horribly, do average, you should always be able to learn something about yourself every time you do a workout or enter a competition. It's the accumulation of all of this over many many repetitions that gives someone that confidence in themselves, and points out places that you may need to work on for greater confidence.