What Is Phase Portrait

This section provides an introduction to Phase Portrait, which is the trajectory curve of a system in the the Phase Space for a given period of time.

What Is Phase Portrait? Phase Portrait is the trajectory curve of a system in the the Phase Space for a given period of time.

For a single-object system with 1 degree of freedom, the Phase Space is a 2 dimensional space of (q1, p1). In this case, the Phase Portrait becomes a 2 dimensional curve.

Phase Portrait of Free Fall Motion

Let's take a look at Phase Portrait of the Free Fall Motion of a single object with mass m. In this case, the generalized position has only 1 component x, representing the height of the object. So we can express Canonical Coordinates (q,p) of the system as below:

```q = (x)
p = (m*x')
# x is the height of the object
# m is the mass of the object
# x' is the velocity of the object
```

The Hamilton Function can be expressed as:

```H = T + V

or:
H = p*p/(2m) + m*g*q                  (P.1)
# g is the standard gravity (9.80665)
```

If we apply Hamilton Equations, we have:

```∂H/∂q = -p'                                (P.2)
∂H/∂p = q'                                 (P.3)

or:
∂(p*p/(2m) + m*g*q)/∂q = -p'
∂(p*p/(2m) + m*g*q)/∂p = q'

or:
m*g = -p'                             (P.4)
p/m = q'                              (P.5)
```

Equations P.4 and P.5 are the equations of the free fall motion, which has the following solution:

```q(t) = -g*t*t/2 + p0*t + q0
p(t) = -m*g*t + p0
# q0 is the initial position
# p0 is the initial momentum
```

The Phase Portrait of this system is a parabolic curve in the Phase Plane of (q,p). For example, if the system has a initial condition of (q,p) = (0,1), its Phase Portrait will look like this: