# Momentum is a covector?

## Main Question or Discussion Point

So this question has been asked elsewhere, yet I haven't found a clear explanation.

Definition. momentum := mass * velocity, or
p=m*v

Now if we know p, then we know the direction v points in, and we get a function from R to R, which maps the magnitude of v to the mass, inverse proportionally. This seems like the only way of interpreting momentum, given the definition above.

In order for momentum to be a covector, two independent vectors would need to evaluate to the same real value, provided they're both in the same level set (codimension 1 plane). What does this say about momentum? Can a bullet headed North have the same momentum as a bullet heading East?

Rather, i suppose the answer is, given a bullet headed north, we can ask "what is the momentum in the direction NE?"

Ah, now it makes sense to use a dot product, then to clean things up by considering momentum a covector. The above definition is misleading, then, we should say momentum of q _in_the_direction_v_ is defined as <mq',v>, whereby <mq'|.> is the momentum covector.

Great!

Andrew Marshall

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