(willpower is a muscle, you exercise it by using it

Willpower is a battery - you drain it by using it.

Making choices led to reduced self-control (i.e., less physical stamina, reduced persistence in the face of failure, more procrastination, and less quality and quantity of arithmetic calculations). A field study then found that reduced self-control was predicted by shoppers' self-reported degree of previous active decision making. Further studies suggested that choosing is more depleting than merely deliberating and forming preferences about options and more depleting than implementing choices made by someone else and that anticipating the choice task as enjoyable can reduce the depleting effect for the first choices but not for many choices.

Source: "Making choices impairs subsequent self-control: A limited-resource account of decision making, self-regulation, and active initiative." from Journal of Personality and Social Psychology by Vohs, Kathleen D.; Baumeister, Roy F.; Schmeichel, Brandon J.; Twenge, Jean M.; Nelson, Noelle M.; Tice, Dianne M.

However, I'm not arguing against pushing ourselves to work when our innate motivation falls a little short.

I would expect exercising willpower to train my self-image as a stronger person. That might be helpful (although often it seems to have the opposite effect, locally, of giving me license to relax)

Also, in my personal experience, it's easier to start caring more about a goal after investing sufficient work in it (recently, or total). Specific work is habit-forming if it offers rewards and validations along the way.

I try to begin some useful work the moment I begin to feel unhappy (this is most practical when you're alone). This often makes me feel better.