Invariance of Newton's second law

[Emanuel84]

Is someone able to proove the invariance under Galilean transformations of F=dp/dt within a system of variable mass? In particular is the momentum invariant? i.e. p=p', as Goldstein states? Please answer me!

 

[pervect]

Do you have Goldstein? I think he proves this in the text. Online, try looking up Noether's theorem (and in Goldstein, too for that matter)

http://math.ucr.edu/home/baez/noether.html

Note that you need to know that your forces are derived from a Lagrangian to prove this is true, this is equivalent to assuming Hamilton's principle of least action.

 

[astrokreng]

Hello,

Thank you for your question regarding the invariance of Newton's second law. This is a fundamental principle in classical mechanics, and it states that the force acting on an object is equal to the rate of change of its momentum. In other words, F=dp/dt.

To answer your question, yes, it is possible to prove the invariance under Galilean transformations of F=dp/dt within a system of variable mass. Galilean transformations refer to the mathematical transformations that describe the relationship between two frames of reference in classical mechanics. These transformations include translations, rotations, and uniform motion.

Invariance means that a physical quantity remains unchanged under these transformations. In the case of Newton's second law, the momentum is the physical quantity of interest. The momentum of an object is defined as its mass multiplied by its velocity, p=mv. Therefore, in order to prove the invariance of Newton's second law, we need to show that the momentum remains unchanged under Galilean transformations.

In particular, we need to show that p=p' (prime), where p is the momentum in one frame of reference and p' is the momentum in another frame of reference. This can be done by using the laws of conservation of momentum, which state that the total momentum of a system remains constant in the absence of external forces.

In a system of variable mass, the mass of an object may change over time due to various factors such as mass being added or removed from the system. However, this does not affect the invariance of Newton's second law. The change in mass is accounted for in the rate of change of momentum, dp/dt, which remains constant under Galilean transformations.

In conclusion, the momentum is indeed invariant under Galilean transformations, and this is reflected in the invariance of Newton's second law. I hope this clears up any confusion and answers your question. If you have any further inquiries, please do not hesitate to ask.

Best regards,