Search results for: time-dependent lagrangians

Authors: Khaled I. Nawafleh

Abstract:

In this paper, it provide the canonical approach for studying dissipated oscillators based on the doubling of degrees of freedom. Clearly, expressions for Lagrangians of the elementary modes of the system are given, which ends with the familiar classical equations of motion for the dissipative oscillator. The equation for one variable is the time reversed of the motion of the second variable. it discuss in detail the extended Bateman Lagrangian specifically for a dual extended damped oscillator time-dependent. A Hamilton-Jacobi analysis showing the equivalence with the Lagrangian approach is also obtained. For that purpose, the techniques of separation of variables were applied, and the quantization process was achieved.

Keywords: doubling of degrees of freedom, dissipated harmonic oscillator, Hamilton-Jacobi, time-dependent lagrangians, quantization

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Authors: Khaled I. Nawafleh

Abstract:

In this work, we apply the method of Hamilton-Jacobi to obtain solutions of Hamiltonian systems in classical mechanics with two certain structures: the first structure plays a central role in the theory of time-dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians, including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism. Then, another expression of geometry of the Hamilton Jacobi equation is retrieved for Hamiltonians with time-dependent and frictional terms. Both approaches shall be applied to many physical examples.

Keywords: Hamilton-Jacobi, time dependent lagrangians, dissipative systems, variational principle

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Authors: Ce Liu, Wei Huo

Abstract:

A nonlinear trajectory tracking controller for quadrotor helicopter based on controlled Lagrangian (CL) method is proposed in this paper. A Lagrangian system with virtual angles as generated coordinates rather than Euler angles is developed. Based on the model, the matching conditions presented by nonlinear partial differential equations are simplified and explicitly solved. Smooth tracking control laws and the range of control parameters are deduced based on the controlled energy of closed-loop system. Besides, a constraint condition for reference accelerations is deduced to identify the trackable reference trajectories by the proposed controller and to ensure the stability of the closed-loop system. The proposed method in this paper does not rely on the division of the quadrotor system, and the design of the control torques does not depend on the thrust as in backstepping or hierarchical control method. Simulations for a quadrotor model demonstrate the feasibility and efficiency of the theoretical results.

Keywords: quadrotor, trajectory tracking control, controlled lagrangians, underactuated system

Procedia PDF Downloads 93