changing fields - Is it easy to change your major after starting your PhD or masters, if it is necessary?

I’m a mechanical engineering student and I am about to apply for a PhD. I am certain about my desire to pursue advanced studies. However, I have serious doubts in what I want to pursue my studies on.

I have done research on structural health monitoring and power electronics. At some point I thought that I had interest in electrical engineering and after some research I got interested in mathematical physics and then in applied mathematics.

I started doing undergraduate research when I was a sophomore in the area of structural health monitoring. However my job was to write an algorithm to implement the methodology. I realize now that what interested me was not structural health monitoring, but to study the origin and derivation of the theoretical equations. I saw the profiles of various faculty in applied mathematics and I found some of their research interesting. Doubts about applying:

• 
  

  Although I have a GPA above 3.60 some programs require certain number of courses in mathematics. I only have the mathematics included in the mechanical engineering program and one course of elementary statistics. I have taken 5 courses in mathematics. However I don’t show strength because out of the 5 courses, in 2 of them I got B when I was a freshman. I think that the taking the elementary statistic course last semester was not a good idea. I should have taken a more advanced course.

  
  


• 
  

  I have not taking any advanced course in mathematics.

  
  


• The application to the program requires the GRE, TOEFL and subject test in math. Some graduate programs say that the GRE math are not mandatory but not taking it puts you in disadvantage. I am not sure I will have time to prepare for it and as a result I don't know if I will do well.


• I have research experience, but it is not in this area. In addition, the reference letters will be made by engineering professors and not math professors.


• Since I am about to finish next semester, I don’t have much time to take more courses or do research in the area.

I looked at a few topics that I found interesting in applied mathematics but I haven’t done research in them. As a result I don’t know if I will get tired or if I will change my mind once I start on the program. If I were to apply for this program and I change my mind would it be easier to change even if it means going to other department?