What is "Phase Space" in differential equations and classical mechanics?

Question

I started reading a book on ordinary differential equations by Vladimir Arnold. He started his book of with the idea of phase space and phase points. I seem to be confused what the general idea of what phase space, phase points and phase velocity vectors are. I have a general idea about them, but they don't make full sense to me. Could someone explain to me in the simplest way possible what those things are, and the general idea about them ?

Also, how do they relate to classical mechanics and physics ?

Thank You!

Answer 1

In physics you have a couple of different but related formalisms to describe dynamical systems. It starts with Newton and forces $F = m \ddot{x}$, then refines into Lagrangians in general coordinates $q_i$ and general velocities $\dot{q_i}$, via Legendre transformation to Hamiltonians in general coordinates $q_i$ and general momenta $p_i$.

Those pairs $q=(q_i), \dot{q} = (\dot{q}_i)$ or $q=(q_i), p = (p_i)$ are considered as independent variables (in classical physics) and describe a state of the dynamical system, they form the coordinates of the phase space. The time evolution of those quantities leads to trajectories in phase space.

Below some phase diagrams of a pendulum. $x$ axis shows the generalized coordinate $\phi$ (angle), $y$ axis velocity/momentum:

(Graphics taken from here)

I have not read the ODE book by V.I. Arnol'd, so I do not know if he deviates much from this.

Answer 2

There are several equivalent formulations of classical mechanics. The most basic one is by Newton. His second law states that the rate of change of momentum of a particle is equal to the net force acting on the particle, that is $\dot{\vec{p}} = \vec{F}$. Since the momentum $\vec{p} = m\vec{v}$, $m$ being the mass and $\vec{v}$ being the velocity, we can write Newton's equation as $m{\ddot{\vec{r}}} = \vec{F}$. This is a second order ordinary differential equation in the position vector $\vec{r}$. Equivalently, it is a set of three scalar equations $m\ddot{x_i} = F_i$, $i = 1, 2, 3$. In order to solve them, we need six initial conditions. The trajectory of the particle is then a curve in 3 dimensional Euclidean space.

An equivalent formulation was developed by Hamilton and his predecessors. Instead of the finding out the force acting on a particle, which is sometimes impossible, we write down the particle's Hamiltonian function $H$. The equations of motion are $\dot{p}_i = -\partial H/\partial q$ and $\dot{q}_i = \partial H/\partial p_i$, where $q_i$ are the (generalized) coordinated and $p_i$ the (generalized) momenta. We call them 'generalized' because they need not be the usual Euclidean position coordinates. Since our differential equations are now in both $p_i$ and $q_i$, it helps if we consider both of them on an equal footing. Therefore, instead of following a trajectory in a 3 dimensional space of Euclidean coordinates $x_i$, we consider the trajectory in a 6 dimensional 'phase space'. The six dimensions are the three generalized coordinates and the three generalized momenta. Further, the phase space is usually not Euclidean.

The state of the particle in phase space is given by the six "coordinates" $\{q_i, p_i\}$ of a 'phase point'. Phase flow is the motion of the phase point in the phase space.

As you proceed further in Arnold's book you will read how Hamilton's formulation is mathematically elegant, insightful and surprisingly similar to quantum mechanics, which was developed 70 years after Hamilton wrote down his equations. You may find reading Cornelius Lanczos's 'Variational Principles of Mechanics' helpful to understand the key ideas behind these equivalent formulations.