I'm completely new to martial arts. I would like to know if there are techniques that can be used in self-defense if you had one hand tied behind your back or in the case you lose your hand?
Absolutely. Actually, one of the schools in the Bujinkan (Gikan-ryu) was reportedly heavily influenced by the inclusion of a one-armed soke.
There is, of course, a strong natural disadvantage (all else being equal) to having only one arm available (for example, the opponent knows your high attacks will largely come from that side, you are not naturally balanced, etc.), but martial arts are about learning to compensate for your disadvantages. Most (un- or under-trained) people fight as if they have an arm, an arm, a leg, and a leg; not as if they have one body.
To give an example, I have a couple of techniques from Gikan-ryu that exemplify a one-handed approach. NB: These are from my notes, according to my understanding at the time of learning, and may not be typical of all students in the Bujinkan. These should not be attempted without proper supervision as injury will likely occur. I've used the left hand so I can copy from my notes directly for ease, but the techniques are meant for either side and do not require both hands.
Gyaku Mune Dori
Reversing Lapel Grab (Left hand only, exemplifies lock and throw.)
Straight Line (Left hand only, exemplifies lock and throw.)
Hanging Bell (Left hand only, exemplifies striking and throwing.)
Breaking it down
At a very, very basic level (and since you're new, this will be the case), you fight with your arms and your legs. You likely move deliberately, each technique performed step-by-step. This is a very low-level style of training, though.
As you improve, your body is one body. The exclusion of an arm is compensated for by the actions of the unified body.
This is all well exemplified by the story of Senju Kannon (千手観音), the 1000-Armed Boddhisatva in esoteric Buddhism (mikkyo). As Takuan Souhou wrote in The Unfettered Mind:
By this same virtue, we can see that men have the preconceived notion of not understanding the idea of having more than two arms any better than having less than two. Through training, we let go of distracting thoughts to learn to act and move naturally.
Whether a man has no arms or ten, if he trains his body to move as best and as efficiently as possible, he will always be more capable than the man distracted by his own thoughts.
There are plenty of techniques available to you if you only have one hand available, but it really depends on which martial arts you're referring to.
In our Hapkido classes, we'll randomly practice our grabs in different scenarios (eyes closed, kneeling, lying down, sitting on a chair, one hand behind our back, etc).
It teaches you to really understand the concepts behind our a specific technique works and how to adapt to whatever situation you might be put in. It also forces us to learn how our entire body is used to apply techniques and generate power. From a kneeling or seated position, you really have to adapt how your apply certain locks when you no longer have the full strength/power/mobility of your body -- it forces you to rely on technique rather than brute strength.
I often practice my kicks with one or both hands behind my backs, not for any practical reason, but more so just to gain a greater understanding of how my body and balance is affected in different stances or scenarios.
In real life situations, you never know what you might injuries/limitations you might have to contend with, so practicing dealing with them in class can be very beneficial. Even if you don't put that one-handed technique into practice, it still helps you expand your understanding of how the technique works.
In one school I trained at we would change the sparring rules up to
Sure, you can can do various things one handed---blocking and punching with the other hand for instance---but it is much, much harder. Use your legs for trips or kicks. Run away if time and circumstances permit. Yell for help.
Being tied will also affect your balance, and the range of motions available to you.
Yes there are tools to work with when you can not really use your hands. The idea of loosing your hand is a bit ... As you would go through Hades.;-)) I can see use hands free techniques in sudden attack situation where you are taken aback and there is no time to rise your hands, or in a crowd where your hands are obstructed, or when you are holding someone and you get under fire.. (Just expression of being battered) So. Use your shoulders, this is a massive bone with massive muscles, if you doge with your head and get behind works well. Shoulder shrugs to deflect, get covered or even hit. Way of practising. First partner must attack slowly and be realistic you can not block Mae geri chudan with it;-) You keep your hands in your pockets and have fun. Practise in diferent distances but preferably close combat as common sense you would see attack comming from the long distance and there would be plenty time to use hands. Have fun