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Design and Kinematic Analysis of Disk Cams
Published in Kevin Russell, Qiong Shen, Raj S. Sodhi, Kinematics and Dynamics of Mechanical Systems Implementation in MATLAB® and Simmechanics®, 2018
Kevin Russell, Qiong Shen, Raj S. Sodhi
The parabolic follower displacement profile (Figure 9.6a), when differentiated, produces a triangular velocity profile (Figure 9.6b). This velocity profile, when differentiated, produces a stepped acceleration profile (Figure 9.6c). In comparison to the acceleration profile for constant velocity motion (Figure 9.5c), the acceleration profile for constant acceleration motion is an improvement because it has a finite height. In addition, for a given angle of rotation and rise, constant acceleration motion produces the smallest acceleration among the motion types presented in this chapter [2]. However, because the acceleration profile changes abruptly, shock loads will be produced in the cam system. This discontinuous acceleration profile also violates the continuous acceleration condition for cam design.
Symmetries and conserved quantities
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
In classical mechanics, Noether’s theorem occupies a prominent position because according to this theorem if a symmetry is found to exist in a dynamical problem then there is a corresponding constant of motion. It provides a connection between global continuous symmetry and the resulting conservation law. The theorem for such an assertation was put forward by Emma Noether in the paper Invariante Variationsprobleme which came out in 1918.
Many-Particle Systems
Published in Mircea S. Rogalski, Stuart B. Palmer, Quantum Physics, 2020
Mircea S. Rogalski, Stuart B. Palmer
Similarly, a rotation through a constant angle φ0, about the z-axis, will be described by the associated operator of the form (8.6), which reads: R^(φ0)=eiJ^ziφ0/ℏ∏i=1NeiL^ziφ0/ℏ Since a rotation through φ0 linearly transforms the coordinates and momentum components of all particles, the operator (10.12), and hence Jz, is a constant of motion, if the Hamiltonian (10.3) can be written in terms of scalar products of coordinate and momentum vectors only. This is valid for any component of , since the direction of the z-axis is arbitrary. In the presence of external fields with an axis of symmetry, only the component of along this axis will be a constant of motion. If the total angular momentum includes the intrinsic spin of the particle, Eq.(8.67), the rotation operator (10.12) takes she form: R^(φ0)=eiJ^zφ0/ℏ=ei(L^z+S^z)φ0/ℏ∏i=1Nei(L^z+S^z)φ0/ℏ where Ŝz denotes the total spin component along the z-axis.
Finite Maxwell field and electric displacement Hamiltonians derived from a current dependent Lagrangian
Published in Molecular Physics, 2018
With the present contribution, we hope to have given a clarification of the mechanical foundations of the finite field Hamiltonians HE of Equation (1) and HD of Equation (2). As a byproduct, we obtained canonical expressions for these Hamiltonians, given in Equations (34) and (41). The canonical Hamiltonians contain the conjugate momenta of the time integrals of the uniform fields which are the fundamental electric dynamical degrees of freedom in the extended Lagrangian. In practice, explicit dependence on these conjugate momenta is, however, of no consequence for constant field MD. Both quantities are constants of motion. As a result, the fundamental relation of dielectrics (Equation (6)) is conserved by HD in its canonical form Equation (34) as can be seen from Equation (29). HE in canonical form Equation (41) conserves the Maxwell field E. According to Equation (39), the conjugate momentum of X is simply proportional to E.