In this study, we investigate trajectories involving multiple impulses within the framework of a generic spacecraft dynamics. 

Revisiting the age-old query of "How many impulses?", we present novel manipulations heavily leveraging on the properties of the state transition matrix.

Surprisingly, we are able to rediscover classical results leading to the introduction of a primer vector, albeit not making use of Pontryagin Maximum Principle as in the original developments by Lawden. 

Furthermore, our mathematical framework exhibits great flexibility and enables the introduction of what we term a "surrogate primer vector" extending a well known concept widely used in mission design. This enhancement allows to derive new simple optimality conditions that provide insights into the possibility to add and/or move multiple impulsive manoeuvres and improve the overall mass budget. This proves especially valuable in scenarios where a baseline trajectory arc is, for example, limited to a single impulse—an instance where traditional primer vector developments become singular and hinder conclusive outcomes.

In demonstrating the practical application of the surrogate primer vector, we examine a specific case involving the four-body dynamics of a spacecraft within an Earth-Moon-Sun system. The system is characterized by the high-precision and differentiable VSOP2013 and ELP2000 ephemerides models. The focal point of our investigation is a reference trajectory representing a return from Mars, utilizing the weak stability boundary (WSB) of the Sun-Earth-Moon system. The trajectory incorporates two consecutive lunar flybys to insert the spacecraft into a lunar distant retrograde orbit (DRO). Conventionally, this trajectory necessitates a single maneuver at the DRO injection point.

Prior to implementing the surrogate primer vector, a local optimization of the trajectory is performed. Upon application of the surrogate primer vector, we successfully identify potential maneuver injection points, strategically reducing the overall mission cost. The introduction of these additional maneuvers, followed by local optimization, validates that the revised trajectory indeed incurs a lower cost compared to the original configuration.