What does this two momentum transform look like?

[rtransformation]

q_i is the cartesian coordinate, and Q_i is the Generalized coordinate, why the momentum under the two coordinates have this transformation way:
p_i=∑P_j(∂Q_j/∂q_j)
p_i and P_i are corresponding momentum under the two coordinate respectively.

 

[DEvens]

Hi rtransformation. Welcome to the forum.

This is how vectors transform under a coordinate transformation. You need to study vector transformations. For example, rotations, changes to spherical or cylindrical coordinates, etc.

So the reason that momentum transforms this way is that it is a vector.

 

[rtransformation]

Hi rtransformation. Welcome to the forum.

This is how vectors transform under a coordinate transformation. You need to study vector transformations. For example, rotations, changes to spherical or cylindrical coordinates, etc.

So the reason that momentum transforms this way is that it is a vector.

Thank you! I really need to study some basic knowledge now...Thank you again.

 

[rtransformation]

Hi rtransformation. Welcome to the forum.

This is how vectors transform under a coordinate transformation. You need to study vector transformations. For example, rotations, changes to spherical or cylindrical coordinates, etc.

So the reason that momentum transforms this way is that it is a vector.

actually, I didn't find the relation between the vector transformation and my question, could you please be more specific and help me solve this problem?Thank you very much.
Is this transformation a contact transformation?

 

[DEvens]

actually, I didn't find the relation between the vector transformation and my question, could you please be more specific and help me solve this problem?Thank you very much.
Is this transformation a contact transformation?

I have no idea if it is a "contact transformation." I looked up "contact transformation."

https://en.wikipedia.org/wiki/Contact_geometry

It was fun, but seemed to be somewhat far afield from your question.

What sort of answer would satisfy you? This is how a vector transforms under a coordinate change. It is part of the definition of a vector.

 

[rtransformation]

I have no idea if it is a "contact transformation." I looked up "contact transformation."

https://en.wikipedia.org/wiki/Contact_geometry

It was fun, but seemed to be somewhat far afield from your question.

What sort of answer would satisfy you? This is how a vector transforms under a coordinate change. It is part of the definition of a vector.

I just want to know how I can get this result through derivation.

 

[MisterX]

I would suggest looking at a graduate level classical mechanics book in a chapter on canonical transformations.

 

[rtransformation]

I would suggest looking at a graduate level classical mechanics book in a chapter on canonical transformations.

Thank you, I now get it.

 

[MisterX]

Thank you, I now get it.

I'm not sure I do though. Would you mind posting your explanation for the curious reader?

 

[rtransformation]

I'm not sure I do though. Would you mind posting your explanation for the curious reader?

I found it in the Walter Greiner‘s famous classical mechanics book “Classical Mechanics——Systems of Particles and Hamiltonian Dynamics” ，Chapter 19——Canonical Transformation，and what I asked is the point transformation which is discussed in detail in this book.