Second-Order Time-Dependent Tangent Bundles and Geometric Mechanics

The aim of this paper is to geometrize time-dependent Lagrangian mechanics in a way that the framework of second-order tangent bundles plays an essential role. To this end, we first introduce the concepts of time-dependent connections and time-dependent semisprays on a manifold M and their induced vector bundle structures on the second-order time-dependent tangent bundle R×T2M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}\times T^2M$$\end{document}. Then we turn our attention to regular time-dependent Lagrangians and their interaction with R×T2M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}\times T^2M$$\end{document} in different situations such as mechanical systems with potential fields, external forces and holonomic constraints. Finally, we propose some examples to support our theory.