Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory

We study the S-matrix of planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory when external momenta are restricted to a two-dimensional subspace of Minkowski space. We find significant simplifications and new, interesting structures for tree and loop amplitudes in two-dimensional kinematics; in particular, the higher-point amplitudes we consider can be obtained from those with lowest-points by a collinear uplifting. Based on a compact formula for one-loop N${}^2$MHV amplitudes, we use an equation proposed previously to compute, for the first time, the complete two-loop NMHV and three-loop MHV octagons, which we conjecture to uplift to give the full $n$-point amplitudes up to simpler logarithmic terms or dilogarithmic terms.Comment: v2: important typos fixed. 38 pages, 4 figures. An ancillary file with two-loop NMHV "remainders" for n=10,12 can be found at http://www.nbi.dk/~schuot/nmhvremainders.zi

Multibond Graph Method For Flexible Multibody System Dynamics

In this thesis, a general method for the modelling and simulation of the dynamics of flexible multibody systems undergoing special motion is developed using the multiband graph approach. Until now, bond graph was mostly used to model rigid multibody system dynamics and two dimensional flexible bodies. The multibond graph approach developed in this thesis for the computer-aided modelling and simulation of flexible multibody system dynamics provides an alternative to current computer-aided modelling and simulation of 3-D flexible multibody system dynamics with the added advantage of including a mixed energy domain. The flexible multibody system considered consists of a number of arbitrarily connected flexible and rigid bodies connected through translational and rotational joints. The non-linear vibration behaviour of flexible members subjected to axial forces, which is known as geometric stiffness/softness effect, is investigated and modelled by non-linear strain-stress relations. The governing equations of motion of a flexible body including this geometric nonlinearity is derived in the form that can be represented by multibond graphs. The application of different kinds of floating frames to the local reference frame is discussed, which can simplify the mathematical and bond graph models.;Multibonds, which represent the three translational and the three rotational motions of a body, and multiport elements, which represent physical parameters of the body, are employed instead of the conventional single bond graph. This makes the bond graph representation of the flexible multibody system much more compact and clear. The inertial coupling between rotation and vibration of the flexible body is modelled in the multibond graph by a summation element. This makes it possible for the multibond graph of a flexible body to keep a similar bond graph structure to that of a rigid body, except for the vibration velocity. The thesis also discuses the multibond graph representation of common mechanical joints and the procedure of establishing the multibond graph of the entire system.;The application of this multibond graph approach is illustrated by a set of examples of contemporary interest in flexible multibody system dynamics. They include a rotating beam which shows the stiffening effect on the vibration of the beam in the rotation plane, a four bar mechanism which indicates the softening effect, and a four body manipulator system with two flexible links which investigates the influence of the flexibility on 2-D and 3-D maneuvers. The significant advantages of the method developed in this thesis are the ease of modelling, particularly evident in the case of mixed energy domains. It has been revealed from the solutions of these example problems, that the geometric stiffening and softening of flexible beams under large dynamic axial forces have significant effects on the prediction of dynamic behavior and digital simulation of the flexible systems. It has also been shown that the inverse dynamic control of a flexible multibody system without consideration of the flexibility of the system would cause unaccepted errors in position control. These examples demonstrate the validity of this multibond graph approach