As an obligation, yes they mean the same thing. Just remember that "have got (to)" is used in contractions. We say "I've got to..." or "she's got to", but not "I have got to" or "she has got to".

Have got to and have to: forms Affirmative form Have (got) to comes before the main verb: You have to try these cakes. They are so good. It is often contracted, especially in speaking. You’ve got to press very hard on the doorbell. Have (got) to cannot be followed by a modal verb: We’ve to take the car to the garage this morning. Not: We’ve to must take the car to the garage this morning. Have (got) to is only used alone when the main verb and any complement of the verb is obvious: A: Does he have to do exams this year? B: Yes, he has (got) to. See also: Have Negative form The negative of have to is formed using do not, does not, did not or don’t, doesn’t, didn’t: We don’t have to pay for the food. The negative of have got to is formed by adding not after have. We don’t use don’t, doesn’t, didn’t: We haven’t got to pay for the food. Not: We don’t have got to pay for the food. See also: Modality: forms Have Question form Do, does, did are used before the subject to form questions with have to: Do we have to take our shoes off? The subject and have change position to form questions with have got to: Have we got to take our shoes off? Source : http://dictionary.cambridge.org/grammar/british-grammar/have-got-to-and-have-to