4.2 Conservative Systems

First Integral

We have a dynamical system $\dot{x} = F (x)$ defined on the phase space $U \subset R^{n}$ and $ϕ^{t} (x_{0})$ is the flow.

If $G : R^{n} \to R$ is a differentiable function such that:

$\forall x_{0} \in U, \forall t \in R, G (ϕ^{t} (x_{0})) = G (x_{0})$
$H$ is non-constant on any open set

Mechanical System With Potential

A mechanical system can be described by the differential equation:

\ddot{x} = - \nabla V (x) .

Here, $x$ is the position vector, and $V : R^{n} \to R$ is the potential function.

Dynamical System Formulation

By introducing the velocity $y = \dot{x}$ , the system is converted into a first-order dynamical system in $2 n$ -dimensions:

\begin{aligned} \dot{x} & = y, \\ \dot{y} & = - \nabla V (x) . \end{aligned}

Total Energy (First Integral)

The function $H (x, y)$ is the total energy or Hamiltonian of the system:

H (x, y) = \frac{| y |^{2}}{2} + V (x) .

This function is a first integral of the system, which means it is conserved over time.

$K = \frac{| y |^{2}}{2} = \frac{1}{2} \sum_{i = 1}^{n} y_{i}^{2}$ is the kinetic energy.
$V (x)$ is the potential energy.

Conservation Principle: Since $H$ is a first integral, a mechanical system with a potential (a conservative system) preserves its total energy along its motion. That is, $H (x (t), y (t))$ is constant for all time $t$ .

Hamiltonian System

If we have the dynamical system:

there exists and $H$ such that:

{\dot{x}}_{i} = \frac{\partial H (x, y)}{\partial y_{i}}, {\dot{y}}_{i} = - \frac{\partial H (x, y)}{\partial x_{i}}

Then $H (x, y)$ is a first integral.

Conservative System

A system that has a first integral

Periodic Orbits in Conservative Systems

If $\dot{x} = F (x)$ is conservative with a first integral $H : R^{2} \to R$ . If $H^{- 1} {c}$ is a non-empty connected component of a regular compact set and there is no fixed point on $H^{- 1} ({c})$ , then $H^{- 1} ({c})$ contains a periodic orbit.

To find fixed points we should look for $F (x) = 0$ , however if we have a Hamiltonian then $F (x, y) = 0$ $i f f$ $\nabla H (x, y) = 0$

No Attracting Fixed Points

Suppose that $\dot{x} = F (x)$ is a conservative system with first integral $H : R^{2} \to R$ . Let $x^{*}$ be a fixed point. Then $x^{*}$ cannot be $ω - a t t r a c t i n g$ .

For contradiction assume that $x^{*}$ is indeed $ω$ -attracting. By definition, it means that

\exists δ > 0 s.t. \forall x_{0} \in B_{δ} (x^{*}), lim_{t \to \infty} ϕ^{t} (x_{0}) = x^{*}

Now if $H$ is the first integral, one sees that along the flow ( $ϕ^{t}$ ) for any point $x_{0}$ in the neighborhood $B_{δ} (x^{*})$ , it holds that

\forall t \in R^{+}, H (x_{0}) = H (ϕ^{t} (x_{0})) = lim_{t \to \infty} H (ϕ^{t} (x_{0})) = H (x^{*})

As a result, $H (x_{0})$ is a constant in $B_{δ} (x^{*})$ , which contradicts the definition of a first integral