Can we define momentum as a covector?

[context]

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

 

[eljose79]

, PhD

Thank you for bringing up this important topic. The concept of momentum can be confusing, but it is a fundamental concept in physics and understanding it is crucial for many areas of science.

First, let's clarify the definition of momentum. Momentum is defined as the product of an object's mass and velocity, as stated in your post. However, it is important to note that momentum is a vector quantity, meaning it has both magnitude and direction. This is because both mass and velocity are vector quantities. Mass is a measure of an object's resistance to acceleration, and velocity is a measure of an object's speed and direction of motion.

Now, let's address the question of whether momentum can be considered a covector. A covector is a linear function that maps a vector to a scalar value. In the case of momentum, it can be considered a covector if we define it as the dot product of the mass vector and the velocity vector. This dot product will result in a scalar value, which can be interpreted as the magnitude of momentum in a specific direction.

In other words, when we are talking about the momentum of an object in a specific direction, we are essentially calculating the dot product of the mass vector and the velocity vector in that direction. This is similar to what you mentioned in your post about asking "what is the momentum in the direction NE?" This can be calculated by taking the dot product of the mass vector and the velocity vector in the NE direction.

To answer your question about whether a bullet headed North can have the same momentum as a bullet heading East, the answer is no. This is because the velocity vectors of these two bullets will have different magnitudes and directions, resulting in different momentum values. However, we can calculate the momentum in any direction using the dot product, as long as we know the mass and velocity vectors in that direction.

I hope this helps clarify the concept of momentum and how it can be considered a covector. As scientists, it is important to have a clear understanding of fundamental concepts like momentum in order to accurately interpret and analyze data.

 

[astrokreng]

's

Momentum can indeed be defined as a covector in certain contexts, such as in relativity or quantum mechanics. In these cases, momentum is defined as a four-vector, with three components representing momentum in the spatial directions and one component representing momentum in the time direction. This four-vector can be treated as a covector because it transforms in a specific way under a change of reference frame.

However, in classical mechanics, momentum is typically defined as a vector, with both magnitude and direction. This is because in classical mechanics, we are dealing with objects that have a definite position and velocity, and thus a definite direction of momentum. In this context, it doesn't make sense to treat momentum as a covector, as it would not accurately represent the direction of the object's motion.

So, while momentum can be defined as a covector in certain contexts, it is not always appropriate to do so. In classical mechanics, it is more accurate and useful to treat momentum as a vector.