scholarly journals The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 294
Author(s):  
Michael M. Tung
Keyword(s):  
Gravitational Field ◽  
Harmonic Oscillator ◽  
Order Term ◽  
Classical Model ◽  
Relativistic Equation ◽  
Correct Treatment ◽  
Relativistic Limit ◽  
Starting Point ◽  
Relativistic Equation Of Motion ◽  
Lagrange Formalism

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

scholarly journals WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

2007 ◽  
Vol 22 (35) ◽  
pp. 2675-2687 ◽  
Author(s):  
LUIS F. BARRAGÁN-GIL ◽  
ABEL CAMACHO
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Field ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Approximation Method ◽  
Bound States ◽  
The Other ◽  
Homogeneous Gravitational Field ◽  
Definition Of ◽  
Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

Lyapunov-stable control of mechatronic systems

2005 ◽  
Vol 219 (2) ◽  
pp. 173-185 ◽  
Author(s):  
O Enge ◽  
P Maißer
Keyword(s):  
Inverse Dynamics ◽  
Dynamic Control ◽  
Linear Feedback ◽  
Mechatronic Systems ◽  
Electromechanical Systems ◽  
Starting Point ◽  
Reaction Forces ◽  
The Stability ◽  
Lagrange Formalism ◽  
Stable Control

In this paper, a method for controlling mechatronic systems using inverse dynamics is proposed. The starting point is a unified mathematical approach to modelling electromechanical systems based on Lagrange formalism. This mathematical theory is used to represent such systems taking into account all interactions between their substructures. The concept of Lagrange formalism for electromechanical systems is given and the complete governing equations are presented. The Voronetz equations of a partially kinematically controlled electromechanical system (EMS) are derived. The corresponding reaction forces and voltages following from the Voronetz equations are determined. Using these reactions with small modifications, a so-called ‘augmented proportional-derivative (PD) dynamic control law’ is generated. This controller consists of a non-linear feedforward - based on inverse dynamics - and a linear feedback. The stability of the controller is proved using a Lyapunov function. The controller can also be applied to pure multibody systems or a sheer electrical system, both of which are borderline cases of mechatronic systems.

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