Increasing Endurance
Before we get started on this topic, I would highly recommend that you read the articles I wrote on range, pressure, and breathing first. This article makes the assumption that you are familiar with the ideas presented earlier. Before we can work on increasing our endurance, it is important that our trumpet playing "machine" is functioning properly. If not, it would be like trying to tune up a car that is in need of new spark plugs, cables, distributor cap, etc. without replacing these parts. You can adjust the timing, but it really isn't going to help much.
Also, after writing this article, I'm not really happy with the way I've worded it. So, I will probably re-write it in the near future. If you have any questions or suggestions, email me. I certainly don't want there to be any confusion.
Assuming that we are breathing correctly and our embouchure is set up correctly, their are two main things that we can do to increase our endurance. They are:
Lighten up on the mouthpiece pressure
Support your airflow with the lower abs and back
In order to lighten up on the mouthpiece pressure, we must rely on the strength of the muscles in our corners. I believe this topic was covered adequately here. Supporting our sound with the lower abs and back is an easy thing to do that really will help your endurance. It is imperative that you be careful not to constrict your throat while doing this. Let me dissect this support function into two sections: The abs, and the back.
The Abs
I'm of the opinion that when most people say "support with the diaphragm", they really mean the abs. Extensive studies have shown that the diaphragm cannot be willingly controlled anymore then the heart can be made to "beat". Generally, when you here people say "support with the diaphragm" they mean to suck in your stomach so as to support the release of the air stream (or at least, that is what it means to me). IMO, this is the correct way to exhale. When taking a full breath, your lungs will fill and cause your chest and "gut" area to expand. It is only natural that when exhaling, these areas should "collapse". By collapse I don't mean let them sag. I mean you should suck in your stomach like you are trying to impress members of the opposite sex :) This will compress the lungs from underneath, allowing a smooth, powerful release of air.
The Back
As for the support of the back, this one is pretty simple. When really blowing, I "flex" the muscles in the back that cause you to sit up as tall as you can. These muscles help support the air by compressing the lungs from the back side.
Jim Manley once said to me that after a long gig, his face wasn't tired, but his abs and back muscles would be tired like in after a long, grueling work out. After adopting these ideas, I understand where he is coming from. Your trumpet playing endurance will increase, and you may even feel some soreness after using these muscles.