FROM the beginning of civilization one need must make itself felt, namely, for some form of calendar. In turn, this gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon. When the planets are added in the next stage, the astronomer needs an ephemeris or almanac ; the modern Nautical Almanac, with its perfection within narrow limits, is the final outcome. There have been many stages on the road, and some apparent interruptions. But the urge has always been the same, essentially a practical one. The Analytical Foundations of Celestial Mechanics By Aurel Wintner. (Princeton Mathematical Series, No. 5.) Pp. xii + 448. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Analytical Foundations of Celestial Mechanics

On the interaction of the elementary particles

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Ordinary Differential Equations With Applications

Various problems concerning the geometry of the space of Hermitian operators on a Hilbert space are addressed. In particular, we study the canonical Poisson and Riemann?Jordan tensors and the corresponding foliations into K?hler submanifolds. It is also shown that the space of density states on an n-dimensional Hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space of rank-k states, k = 1, ..., n, is a smooth manifold of (real) dimension 2nk ? k2 ? 1 and this stratification is maximal in the sense that every smooth curve in , viewed as a subset of the dual to the Lie algebra of the unitary group , at every point must be tangent to the strata it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition , an abstract criterion of entanglement is proved.

https://iopscience.iop.org/article/10.1088/0305-4470/38/47/011/pdf

Geometry of quantum systems: Density states and entanglement

This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems

Periodic flows, rank-two Poisson structures, and nonholonomic mechanics

Noether theorem plays a central role in linking symmetries and first integrals in Lagrangian mechanics. The situation is different in the nonholonomic context, but in the last decades there have been several extensions of Noether theorem to the nonholonomic setting. We provide an overview of this subject which is as elementary as possible.

/pdf/an-elemental-overview-of-the-nonholonomic-noether-theorem-jdu8jwhie5.pdf

An elemental overview of the nonholonomic Noether theorem

Remarks on geometric quantum mechanics

We characterize the conditions for the conservation of the energy and of the components of the momentum maps of lifted actions, and of their ‘gauge-like’ generalizations, in time-independent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the so-called reaction-annihilator distribution.

Conservation of energy and momenta in nonholonomic systems with affine constraints

https://www.math.unipd.it/~fasso/research/papers/gauge.pdf

Nicola Sansonetto

Papers

Periodic flows, rank-two Poisson structures, and nonholonomic mechanics

An elemental overview of the nonholonomic Noether theorem

Remarks on geometric quantum mechanics

Conservation of energy and momenta in nonholonomic systems with affine constraints

Gauge conservation laws and the momentum equation in nonholonomic mechanics