Physics Notes - Herong's Tutorial Notes - v3.24, by Herong Yang
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:
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