## On the mechanics with noncontinuous Hamiltonians.(English)Zbl 0606.58021

The authors attempt a formulation of mechanics with nondifferentiable or noncontinuous Hamiltonians. For this purpose they introduce the notion of an overflow, which is more general than the notion of a flow and more adequate to describe a dynamics of mechanical systems. For example, they consider the following Hamiltonians, given by (1) $${\mathbb{R}}^ 2\ni (x,p)\to H(x,p)=p^ 2/2m+V(x)\in {\mathbb{R}}^ 1$$, where for all x: ${\mathbb{R}}^ 1\ni x\to V(x)= \begin{cases} 0 &\text{ for $$x\in {\mathbb{R}}^ 1\{0\}$$} \\ \infty &\text{ for $$x=0,$$} \end{cases}$ or ${\mathbb{R}}^ 1\ni x\to V(x)= \begin{cases} -\alpha x &\text{ for $$x\in (-\infty,0), \alpha >0$$} \\ \beta x &\text{ for $$x\in [0,\infty)$$, $$\beta >0$$;} \end{cases}$ (2) $${\mathbb{R}}^ 2\ni (x,p)\to H(x,p)=p^ 2/2m+V(x)$$, where ${\mathbb{R}}^ 1\ni x\to V(x)= \begin{cases} 0 &\text{ for $$x\in (-\infty,0)$$} \\ E &\text{ for $$x\in [0,\infty)$$} \end{cases}$ or $V(x)= \begin{cases} 0 &\text{ for $$x\in {\mathbb{R}}^ 1-\{0\}$$} \\ E &\text{ for $$x=0,$$} \end{cases}$ and interesting results are obtained.

### MSC:

 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H05 Hamilton’s equations