Question

Problem 7. A) Show that the potential energy of two particles is invariant under a Galilean transformation. Recall that U = U (t)-rı (t)) that is the separation is evaluated at a given time t. B) Show that the kinetic energy of an object is not, generally, invariant under a Galilean transformation. C) Argue that the total energy of a system of objects is not, generally, invariant under a Galilean transformation

Answer 1

(A)

$U'=U'(\vec r'_2(t)-\vec r_1'(t))=U'(\vec r_2(t)-\vec v t-\vec r_1(t)+\vec v t)$

$=U(\vec r_2(t)-\vec r_1(t))$

So it remains invariant.

(B)

$r'(t)=r(t)-vt$

$K'=\frac {1}{2}mv'^2=\frac {1}{2}m(v-V)^2\neq K=\frac {1}{2}mv^2$

(C)

Since the kinetic energy is not, generally, invariant under a Galilean transformation so the total energy is not, generally, invariant under a Galilean transformation.