Rheonomic tangent and cotangent bundles and applications

V. A. Vujičić

Theoretical and applied mechanics, Tome 25 (1999) no. 1, p. 145 .

Résumé

Let us consider an extended configuration manifold $\mathcal M=M^n\times x^0$, where $M$ is Lagrange's configurations $n$-dimensional manifold and $x^0=\tau(\aleph t)$ is taken from the given relations generating the manifold $\mathcal M$. In some special cases $x^0=t$ can be taken, but $x^0\equiv t$ can not. The extended tangent bundle has dimension $2n+2$. Metric on $\mathcal M$ is given by $ds^2=g_{ij}(x)dx^idx^j+2g_{0i}dx^0dx^i+g_{00}dx^0dx^0$. The set of all covectors $p_i=g_{ij}\dot x^j+g_{i0}\dot x^0$ and $p_0=g_{0j}\dot x^j+g_{00}\dot x^0$ at $x=(x^0,x^1,\dots,x^n)$ forms an extended cotangent bundle $T^*\mathcal M$ having a natural symplectic structure. The adding coordinates $x^0$ to $M$, $\dot x^0$ to $TM$ or $p^0$ to $T^*M$ is not simply algebraic extension respectively of $M$, $TM$ and $T^*M$, but have geometric and physical meaning. The geometry of such spaces is not considered so far and we modified the whole standard Lagrange's and Hamilton's mechanics systems on such a geometric base. The D’Alembert-Lagrange's principle as well as Hamiltonian principle or the least action principle are appropriately restated. There are more independent Lagrange's and Hamilton's equations and more variations of rheonomic constraints $(x_0\neq0,\delta t=0)$. Poincare's integral invariant for nonautonomous systems is generalized.