Can anybody explain the order of names in the United States?

Question

For example,

father's surname : Smith

mother's surname : Jones

Name : Michael

How would be his name? Michael J. Smith?

Answer 1

Children take their biological father's name at birth.

Answer 2

The Mother of the child decides what will go on the birth certificate. The child can carry her name : Michael Jones (Father unkown) or chooses not to disclose the Father's name.

If the Mother has decided on the name Michael she may also want to choose a middle name of her choice. (Henry) (named after some relative of hers). Without a blood test which is sometimes court ordered, if she is seeking child support, the Father is not always listed...thank you