Hamilton’s Principle as Variational Inequality for Mechanical Systems with Impact

The classical form of Hamilton’s principle holds for conservative systems with perfect bilateral constraints. Several attempts have been made in literature to generalise Hamilton’s principle for mechanical systems with perfect unilateral constraints involving impulsive motion. This has led to a number of different variants of Hamilton’s principle, some expressed as variational inequalities. Up to now, the connection between these different principles has been missing. The aim of this paper is to put these different principles of Hamilton in a unified framework by using the concept of weak and strong extrema. The difference between weak and strong variations of the motion is explained in detail. Each type of variation leads to a variant of the principle of Hamilton in the form of a variational inequality. The conclusion of the paper is that each type of variation leads to different necessary and sufficient conditions on the impact law. The principle of Hamilton with strong variations is valid for perfect unilateral constraints with a completely elastic impact law, whereas the weak form of Hamilton’s principle only requires perfect unilateral constraints and no condition on the energy.     

ESI‐MS/MS of expanded porphyrins: a look into their structure and aromaticity

Electrospray mass spectrometry/mass spectrometry was used to investigate the gas‐phase properties of protonated expanded porphyrins, in order to correlate those with their structure and conformation. We have selected five expanded meso‐pentafluorophenyl porphyrins, respectively, a pair of oxidized/reduced fused pentaphyrins (22 and 24 π electrons), a pair of oxidized/reduced regular hexaphyrins (26 and 28 π electrons) and a regular doubly N‐fused hexaphyrin (28 π electrons). The gas‐phase behavior of the protonated species of oxidized and reduced expanded porphyrins is different. The oxidized species (aromatic Hückel systems) fragment more extensively, mainly by the loss of two HF molecules. The reduced species (Möbius aromatic or Möbius‐like aromatic systems) fragment less than their oxidized counterparts because of their increased flexibility. The protonated regular doubly fused hexaphyrin (non‐aromatic Hückel system) shows the least fragmentation even at higher collision energies. In general, cyclization through losses of HF molecules decreases from the aromatic Hückel systems to Möbius aromatic or Möbius‐like aromatic systems to non‐aromatic Hückel systems and is related to an increase in conformational distortion. Copyright © 2016 John Wiley & Sons, Ltd.     

A parameterisation of impact laws in perfect multi-contact collisions

Impacts in rigid multibody systems cause instantaneous jumps in the generalised velocities. Naturally, the velocity after impact depends on the chosen impact law. An impact law should fulfil kinematic, kinetic as well as energy restrictions. In this paper, we study the domain of possible post-impact velocities for arbitrary impact laws. For single-contact collisions, this domain is at most one-dimensional but the domain becomes higher dimensional in the multi-contact case. The domain of possible post-impact velocities is a compact convex subset of the tangent space to the configuration manifold. Using a complete canonical parameterisation, the post-impact velocity of all impact laws can be addressed. For instance, the impact law corresponding to maximal dissipation as well as Newtons (extended) impact law are examples of incomplete canonical parameterisations. These impact laws are not complete as non-local impact effects are not addressed. Here, we try to find a complete canonical parameterisation which covers non-local impact effects. Moreover, the relationship between symmetries and conservation laws in this context will be elucidated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Euler-Maupertuis principle of least action as variational inequality

Starting from Hamilton's Principle, the current paper discusses how we can derive the Euler-Maupertuis Principle of Least Action in the context of non-smooth dynamics. This variational principle allows us to directly obtain the space curve y (x) of a point-mass in a potential field V (x, y) without referring to the temporal dynamics. This paper generalises the Euler-Maupertuis Principle of Least Action to systems with impact by formulating the principle as a variational inequality. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)