Mediterr. J. Math.
Springer International Publishing AG 2017
Second-Order Time-Dependent Tangent
Bundles and Geometric Mechanics
Abstract. The aim of this paper is to geometrize time-dependent La-
grangian mechanics in a way that the framework of second-order tan-
gent 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 × T
M. Then we turn
our attention to regular time-dependent Lagrangians and their interac-
tion with R × T
M in diﬀerent situations such as mechanical systems
with potential ﬁelds, external forces and holonomic constraints. Finally,
we propose some examples to support our theory.
Mathematics Subject Classiﬁcation. Primary 58B20; Secondary 58A05.
Keywords. semisprays, connections, time-dependent Lagrangian,
second-order tangent bundle.
2.1. Second-Order Time-Dependent Tangent Bundles
3. Time-Dependent Lagrangians
3.1. Motion in a Time-Dependent Potential Field
3.2. External Forces
3.3. Holonomic Constraints
3.4. Invariant Lagrangians and Second-Order Vector Bundle
4. Applications and Examples
4.1. R × T
and the Motion of an Incompressible Fluid