On Periodic Solutions to Lagrangian System with Singularities and Constraints

Abstract

A Lagrangian system with singularities is considered. The configuration space is a non-compact manifold that depends on time. A set of periodic solutions has been found.

APPENDIX

Lemma 8. Fix a positive constant $\delta$ . Let $z_{1},z_{2}$ be functions from $\mathcal{V}_{p}$ such that

$$\inf\{|z_{i}(t)-\sigma^{\prime}||t\in[0,\omega],\quad\sigma^{\prime}\in\sigma,\quad i=1,2\}\geq\delta.$$

There exists a positive number $\varepsilon>0$ such that if

$$\max_{t\in[0,\omega]}|z_{1}(t)-z_{2}(t)|<\varepsilon$$

then these functions are homotopic.

Proof of Lemma 8. Our argument is quite standard. So we present a sketch of the proof.

It is convenient to consider $z_{1},z_{2}$ as functions with values in $\mathcal{C}$ (see Remark 1). In the same sense $F(t)$ is a submanifold in $\mathcal{C}$ and the functions $z_{1},z_{2}$ define a pair of closed curves in $\mathcal{C}$.

Choose a Riemann metric in $\mathcal{C}$, for example as follows

$$d\tau^{2}=\sum_{k=1}^{m+n}(dz^{k})^{2}.$$

This metric inducts a metric in $F(t)$.

Under the conditions of the Lemma any two points $z_{1}(t),z_{2}(t)\in F(t)$ are connected in $F(t)$ with a unique shortest piece of geodesic $\chi(\xi,t)$, $\xi\in[0,\tilde{\xi}(t)]$ such that

$$\chi(0,t)=z_{1}(t),\quad\chi(\tilde{\xi}(t),t)=z_{2}(t).$$

Here $\xi$ is the arc-length parameter.

Define the homotopy as follows $z(s,t)=\chi(s\tilde{\xi}(t),t)$, $s\in[0,1]$.

The Lemma is proved.