Works out where the spacecraft is, and where it will be.

Flight dynamics is orbit determination and manoeuvre planning: taking tracking data, fitting an orbit, and computing the burn that puts the spacecraft where the mission needs it, with the least propellant.
The maths matters more than the department name.
Aerospace engineers · US Bureau of Labor Statistics, Occupational Outlook Handbook, May 2024 wages and 2024–34 projections.
Counted inside "aerospace engineers", along with every other engineering discipline that works on air and space vehicles. There is no separate published line for this specialism.
The maths is genuinely hard and stays hard. A manoeuvre executed on a wrong number cannot be recalled, and the confirmation comes back minutes or hours later.