The synchronous motor itself does not have the ability to self-start. However, the addition of an additional winding known as damper winding (similar to squirrel cage motor) enables it to self-start.
This damper winding is linked with the rotor, and works on the principle of induction motor. When voltage is applied on the stator, current flows through it causing the production of rotating flux. This rotating flux induces a current in damper winding, causing flux and then rotation of the rotor, which is linked with damper winding.
Just like an induction motor, the rotor cannot reach the speed of rotating stator flux and lags behind the synchronous speed. However, when the rotor reaches closer to synchronous speed, it is connected to the main DC source, which locks its speed with a synchronous speed of stator and the motor starts to run as a synchronous motor with stator and rotor rotating at the same speed.
So I think you learn how to make synchronous motor self-starting