The constant of motion of a free fall body

[jomoonrain]

1.what are the the constants of motion of a free fall body(treat this as a one-dimension motion,so there are two [independent] constants of motion )?

2.are they initial velocity and initial displacement？

3.and i know the total energy(T+V) is one,but they are not independent with the initial velocity and initial displacement, isn't it?

4.so i means can we just treat the initial velocity and initial displacement as constants of motion ?

5.if we can,do these constant of motin (initial velocity and initial displacement) correspond to some symmetry of the system (free fall body)?

thanks

 

[atyy]

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

 

[astrokreng]

I would like to clarify that the concept of "constants of motion" refers to quantities that remain constant throughout the motion of a body, regardless of any external forces acting on it. In the case of a free fall body, the two constants of motion are the initial velocity and the initial displacement. These quantities do not change as the body falls, as long as there are no external forces acting on it.

However, it is important to note that the total energy (T+V) of a free fall body is also a constant of motion. This is because the total energy of a system remains constant in the absence of external forces. In the case of a free fall body, the total energy is equal to the sum of its kinetic energy (T) and potential energy (V).

Therefore, while the initial velocity and initial displacement can be considered as constants of motion, they are not completely independent from the total energy of the system. This is because the total energy is determined by the initial velocity and initial displacement, as well as the mass and the acceleration due to gravity.

In terms of symmetry, the constants of motion (initial velocity and initial displacement) do correspond to certain symmetries of the system. For example, the initial velocity is a reflection of the symmetry of time translation, as it remains constant throughout the motion. Similarly, the initial displacement reflects the symmetry of space translation.

In conclusion, the constants of motion of a free fall body are the initial velocity and initial displacement, which correspond to symmetries of the system. However, it is important to also consider the total energy as a constant of motion, as it is determined by the initial conditions as well.