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Bifurcation Behaviour of Non-Linear Systems
Published in T. Thyagarajan, D. Kalpana, Linear and Non-Linear System Theory, 2020
Trajectory: The solution to a differential equation is called a trajectory. Flow: The collection of all such solutions of differential equation is called flow.Map: The collection of solutions of difference equation is called the map.Dissipative system: It is a system where energy loss takes place. (May be due to friction or damping, etc.)Hamiltonian system: It is a system where the total mechanical energy is preserved.
Longitudinal Dynamics and Acceleration
Published in Volker Ziemann, ®, 2019
The longitudinal dynamics of the particles is governed by Equation 5.36, which also describes an equivalent Hamiltonian system, the mathematical pendulum. It can be shown [8] that the action variables in such Hamiltonian systems remain constant if a system parameter, such as the length of the pendulum, or V^ of the RF-system, changes slowly. In other words, the action variable is an adiabatic invariant. Here “slowly” means that dΩs/dt≪Ωs2. Since the action variable of each particle remains constant, their average, the longitudinal emittance, also stays constant in such an operation. In the following, we assume that the bunch extensions are sufficiently small to stay in the linear regime of synchrotron oscillations.
The Emergence of Chaos in Time
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
A concrete example of a chaotic system that can be easily found in the physics departments, but it can also be built at home, is the double pendulum (Shinbrot et al. 1992). A double pendulum consists of two simple double pendula, which are bound to one another as shown in Figure 10.2. The first point mass m1 is suspended from a fixed point (that is also the origin of the reference system) by a rigid weightless rod of length L1 and the second point mass m2 is suspended from m1 by another weightless rod of length L2 (see Figure 10.2). The top and center pivots are assumed frictionless, and the pendula rotate under the action of gravity in the absence of air. The double pendulum’s total energy, which is the sum of its kinetic and potential energies, is conserved. The double pendulum is a Hamiltonian system that exhibits chaotic behavior.
Practical perspectives on symplectic accelerated optimization
Published in Optimization Methods and Software, 2023
Valentin Duruisseaux, Melvin Leok
A hyperregular Lagrangian on induces a Hamiltonian system on via where is the conjugate momentum of . There is a Hamiltonian variational principle on the Hamiltonian side in momentum phase space which is equivalent to Hamilton's equations, and these equations are equivalent to the Euler–Lagrange equations (1), provided the Lagrangian is hyperregular. Hamiltonian systems possess a long list of structural invariants and constants of motion, the most important of which are the conservation of the Hamiltonian energy and the conservation of the symplectic 2-form.