Hamiltonian on Free Fall Motion

This section provides an example of calculating the Hamiltonian on a simple mechanical system of an single object in free fall motion and applying the Law of Conservation of Energy.

What Is Free Fall Motion? Free Fall Motion is simple mechanical system of a single object with mass, m, falling freely to the ground under the force of gravity.

To calculate the Hamiltonian of the free fall motion system, we can express the object's kinetic energy, T, and potential energy, V, as:

```T = m*v*v/2
# m is the mass of the object
# v is the velocity of the object

V = m*g*x
# g is the standard gravity (9.80665)
# x is the height of the object
```

So the Hamiltonian, H, of the free fall motion system can be expressed as:

```H = T + V

or:
H = m*v*v/2 + m*g*x
```

Since this free fall motion can be considered as an isolated conservative system, we can apply the Law of Conservation of Energy:

```H = constant

or:
m*v*v/2 + m*g*x = constant

or:
d(m*v*v/2)/dt + d(m*g*x)/dt = 0
# Since d(constant)/dt = 0
```

The last equation can be simplified as:

```m*v*dv/dt + m*g*dx/dt = 0
# The chain rule for derivatives applied

m*v*a - m*g*v = 0
# a = dv/dt, is the acceleration of the object
# v = -dx/dt, is the velocity of the object

a - g = 0
a = g                              (H.4)
# Cancel out m*v from the equation
```

Cool. Equation H.4 matches perfectly with Newton's second law of motion:

```F = m*a                            (H.5)
# Newton's second law of motion

m*g = m*a
# Gravity force, F = m*g, applied.

a = g                              (H.4)
# Cancel out m from the equation
```

With equation H.4, we can easily figure out the position h, the velocity v, and the acceleration a as functions of time t:

```a(t) = g

v(t) = g*t + v0
# v0 is the initial velocity

h(t) = -g*t*t/2 - v0*t + h0
# h0 is the initial position
```

The following picture illustrates an object in free fall motion (source: owlcation.com):