Questioning Relativity 11: Galileo’s Law of Free Fall

· theory of relativity, Uncategorized
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Let us ask how to motivate Galileo’s Law of Free Fall stating that all bodies irrespective of size and composition fall freely under gravitation in the same way according to

• $\frac{du}{dt}= -\nabla\varphi$

where $u=\frac{dx}{dt}$ is the velocity of a body with position $x$ as a function of time $t$, $\frac{du}{dt}$ its acceleration and $\varphi$ is a gravitational potential with field strength $\nabla\varphi$.

Let us now show that Galileo’s Law is a consequence of

1. Galilean invariance
2. conservation of gravitational plus kinetic energy.

Assume then that a gravitational potential $\varphi$ is given and let us assign the gravitational energy $\varphi (x)$ to a body of unknown size and composition at position $x$. Let us further assign to the body kinetic energy of size $f(\frac{dx}{dt})$ where $f(u)$ is a given function of $u$ with $u=\frac{dx}{dt}$, so that the kinetic energy only depends on velocity and not position.

Let us then assume that the motion of the body is such that the total energy as the sum of gravitational and kinetic energy, is equal to a constant $C$:

• $\varphi (x) + f(\frac{dx}{dt}) = C$.

Differentiation with respect to time $t$ then gives

• $\nabla\varphi\, u + f^\prime (u)\frac{du}{dt} = 0$ with $f^\prime =\frac{df}{du}$ and $u=\frac{dx}{dt}$,

that is

• $\frac{du}{dt} = -\frac{u}{f^\prime (u)}\nabla\varphi$.

We now argue that $\frac{u}{f^\prime (u)}$ is constant as a consequence of Galilean invariance stating that the motion of the body is not influenced by a change of Galilean coordinate system representing shift of origin and translation with constant velocity (change between inertial (non-accelerated) frames). We can then normalize the constant to 1 and thus arrive at Galileo’s Law of Free Fall.

To sum up, we obtain Galileo’s Law of Free Fall as a consequence of Galilean invariance and conservation of total energy as the sum of gravitational and kinetic energy.

Recalling the previous post we see that Newtonian mechanics including Newton’s 2nd Law can be seen to be a consequence of Galilean invariance and balance of gravitational and kinetic energy, with the Equivalence Principle appearing as a definition.