Discrete Hamilton's Equations for Magnetothermoelastic-plastic Media

2002 ◽  
Author(s):  
Eric Fahrenthold ◽  
Seunghan Lee
Keyword(s):  
Hamilton's Equations ◽  
Hamilton’S Equations ◽  
Plastic Media

Global Dynamics of an Autoparametric System With Multiple Pendulums

2005 ◽  
Vol 1 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Ashwin Vyas ◽  
Anil K. Bajaj
Keyword(s):  
Initial Conditions ◽  
Hamiltonian Formulation ◽  
Forced Response ◽  
Global Dynamics ◽  
Equilibrium Solutions ◽  
External Excitation ◽  
Poincaré Sections ◽  
Poincare Sections ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

The Hamiltonian dynamics of a resonantly excited linear spring-mass-damper system coupled to an array of pendulums is investigated in this study under 1:1:1:…:2 internal resonance between the pendulums and the linear oscillator. To study the small-amplitude global dynamics, a Hamiltonian formulation is introduced using generalized coordinates and momenta, and action-angle coordinates. The Hamilton’s equations are averaged to obtain equations for the first-order approximations to free and forced response of the system. Equilibrium solutions of the averaged Hamilton’s equations in action-angle or comoving variables are determined and studied for their stability characteristics. The system with one pendulum is known to be integrable in the absence of damping and external excitation. Exciting the system with even a small harmonic forcing near a saddle point leads to stochastic response, as clearly demonstrated by the Poincaré sections of motion. Poincaré sections are also computed for motions started with initial conditions near center-center, center-saddle and saddle-saddle-type equilibria for systems with two, three and four pendulums. In case of the system with more than one pendulum, even the free undamped dynamics exhibits irregular exchange of energy between the pendulums and the block. The increase in complexity is also demonstrated as the number of pendulums is increased, and when external excitation is present.

On the concept of kinetic energy

2000 ◽  
Vol 78 (10) ◽  
pp. 883-899 ◽  
Author(s):  
A F Antippa
Keyword(s):  
Kinetic Energy ◽  
Scalar Product ◽  
Mass Function ◽  
Degree Of Freedom ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

We analyze, in their historic perspective, the three alternative traditional definitions of kinetic energy and then propose a new definition according to which the change in kinetic energy is equal to the scalar product of the velocity and the change in momentum. We justify this definition on conceptual grounds, and show that it follows from Hamilton's equations. From this definition, the explicit classical and relativistic expressions for kinetic energy follow simply upon using the appropriate mass function. The definition given here has the added advantage of being naturally generalizable to a canonical kinetic energy per degree of freedom. This latter takes into account both particle and field kinetic energies. PACS Nos.: 01.55+b, 03.30+p, 03.20+i, 46.10+z

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