Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle over the time axis coordinated by .
This bundle is trivial, but its different trivializations correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection
on which takes a form with respect to this trivialization. The corresponding covariant differential
determines the relative velocity with respect to a reference frame .
As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonian field theory) on . Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold of provided with the coordinates . Its momentum phase space is the vertical cotangent bundle of coordinated by and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form .
One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle of coordinated by and provided with the canonical symplectic form; its Hamiltonian is .
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Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle over the time axis coordinated by . This bundle is trivial, but its different trivializations correspond to the choice of different non-relativistic reference frames.
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.
This course provides an overview of the phenomenological concepts and mathematical tools that have been developed to study the thermodynamics, kinetics and mechanics of solid-state phase transformatio