Systems with Two Degrees of Freedom

Systems with Two Degrees of Freedom

A system with two degrees of freedom is a system defined by $\ddot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t)),\mathbf{x}:\mathbb{R}\to {\mathbb{E}}^2$ where $\mathbf{f}$ is a vector field on ${\text{E}}^2$.

Conservative System

A system with two degrees of freedom is conservative if there exists a function $U:{\text{E}}^2\to \mathbb{R}$ such that $\mathbf{f}=-\nabla U$.

Remark

In a system of one degree of freedom, it is always possible to construct the potential energy by taking the integral, but in a system of two degrees of freedom, the this is not so.

Law of Conservation of Energy

The total energy of a conservative system is conserved, that is, $E=T+U=\frac{1}{2}⟨\dot{\mathbf{x}},\dot{\mathbf{x}}⟩+U$ is a constant along the trajectory of the system.

Proof $\frac{\text{dd}E}{\text{dd}t}=⟨\dot{\text{x}},\ddot{\text{x}}⟩+⟨\nabla U(\text{x}),\dot{\text{x}}⟩=⟨\ddot{\text{x}}+\nabla U(\text{x}),\dot{\text{x}}⟩=0$

Corollary

If at the initial moment the total energy is equal to $E$, then all trajectories lie in the region where $U(\mathbf{x})\le E$, i.e., a point remains inside the potential well $U(x_1,x_2)\le E$ for all time.

Phase Space

The equation of motion can be written as the system: ${\dot{x}}_1=y_1{\dot{x}}_2=y_2$ ${\dot{y}}_1=-\frac{\partial U}{\partial x_1}{\dot{y}}_2=-\frac{\partial U}{\partial x_2}$