Be Very Carefull
Unless the person is going to harm somone or endanger themselves, be very cautious.
My family recently had to do this to my brother. In florida it's called the baker act and is usually court ordered for police to enforce with a phsychologist on hand. I was against it.
My brother was picked up by 10 to 15 police officers with weapons drawn in my mothers front yard and insitutionalised for three days. The trauma and psychological effects of that alone, along with the resentment and anger that it can cause can do more harm than good, and state care is not always the most trustworthy.
most states at least require that the person be given an explanation and an opportunity to go volunteerily without having to be picked up under court order.
I understand your concern for your mother, but I'd recommend trying all available alternatives before resorting to a court order, it can be very ugly and emotionally damaging to your family and relationship with your mother, even if it is what is best for her. Lawyers will not prepare you for what you may have to deal with when she is released.
Since my brothers release two months ago he does not speak to us and no one knows where he is. My mother loses more sleep after the baker act than she did before it, when she at least knew where he was. There is also lot's of room for abuse while in these institutions and as such these mental health laws are constantly under scrutiny for reform.
Also once a person is taken in, they are evaluated and at some point released, after being taken in at the request of family, family is typically the last place they will go for help afterwards. If you do this, you should at the very least attempt to contact and notify your mother that this is happenning and let her know that if she does not go willingly for her own good that you may have to have it court ordered. There is a huge amount of trust that can be violated in your personal relationship and the state and mental health officials don't care about that fact, you will be the one who has to deal with it later, and they won't be around to help.
It's a very tough situation, I hope you are able to make the right choice.
But you should research the matter on your own and make every effort to not just broadside her with this.