Systems with One Degree of Freedom

Definition

A system with one degree of freedom is a system described by one differential equation. $$\begin{gathered} \ddot{x}=f(x) \\ x\in \mathbb{R} \end{gathered}$$ The kinetic energy is $T=\frac{1}{2}{\dot{x}}^2$ The potential energy is $U(x)=-{\int }_{x_0}^{x}f(\xi )d\xi$ Total energy $E=T+U$

The law of conservation of energy

The total energy of points moving according to $$\begin{gathered} \ddot{x}=f(x) \\ x\in \mathbb{R} \end{gathered}$$ is conserved, $E(x(t),\dot{x}(t))$ is independent of t.

Phase Plane

$\ddot{x}=f(x)$ is equivalent to the system of two equations $\dot{x}=y,\dot{y}=f(x)$. The plane with coordinate x and y is called phase plane. The solution of the equations is a motion $\phi :\mathbb{R}\to {\mathbb{R}}^2$ of a phase point in a phase plane, the image of $\phi$ is called phase curve.

Proposition

Through every phase point, there is one and only one phase curve.

Phase Flow

Let M be a point in the phase plane. Consider solution to $\dot{x}=y,\dot{y}=f(x)$ whose initial conditions at $t=0$ are represented by the point $M$, we get $M(t)=g^{t}M$, with $g^t:{\mathbb{R}}^2\to {\mathbb{R}}^2$ a diffeomorphism. $g^t$ form a one-parameter group and we call this group the phase flow.