108 research outputs found
Symmetry Reduction of Optimal Control Systems and Principal Connections
This paper explores the role of symmetries and reduction in nonlinear control
and optimal control systems. The focus of the paper is to give a geometric
framework of symmetry reduction of optimal control systems as well as to show
how to obtain explicit expressions of the reduced system by exploiting the
geometry. In particular, we show how to obtain a principal connection to be
used in the reduction for various choices of symmetry groups, as opposed to
assuming such a principal connection is given or choosing a particular symmetry
group to simplify the setting. Our result synthesizes some previous works on
symmetry reduction of nonlinear control and optimal control systems. Affine and
kinematic optimal control systems are of particular interest: We explicitly
work out the details for such systems and also show a few examples of symmetry
reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure
Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products
There is a well developed and useful theory of Hamiltonian reduction for
semidirect products, which applies to examples such as the heavy top,
compressible fluids and MHD, which are governed by Lie-Poisson type equations.
In this paper we study the Lagrangian analogue of this process and link it with
the general theory of Lagrangian reduction; that is the reduction of
variational principles. These reduced variational principles are interesting in
their own right since they involve constraints on the allowed variations,
analogous to what one finds in the theory of nonholonomic systems with the
Lagrange d'Alembert principle. In addition, the abstract theorems about
circulation, what we call the Kelvin-Noether theorem, are given.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure
On the global version of Euler-Lagrange equations
The introduction of a covariant derivative on the velocity phase space is
needed for a global expression of Euler-Lagrange equations. The aim of this
paper is to show how its torsion tensor turns out to be involved in such a
version.Comment: 5 pages, 1 figur
Lagrangian reduction, the Euler--Poincaré Equations, and semidirect products
There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given
Analisa Karakteristik Karbon Aerosol (Oc dan Ec) dari Emisi Pm2.5 dan Rekomendasi Perlindungan Lingkungan dari Emisi Pm2.5 Kebakaran Lahan Gambut secara Pembaraan (Smouldering) (Studi Kasus : Kabupaten Siak dan Kabupaten Kampar Provinsi Riau)
Penelitian ini memiliki tujuan untuk mengetahui karakteristik karbon organik (OC) dan karbon elemental (EC) dalam PM2.5 yang diketahui sebagai salah satu polutan udara akibat kebakaran lahan gambut dan rekomendasi perlindungan lingkungan. Metode yang digunakan untuk mengetahui konsentrasi PM2.5 adalah gravimetri dengan bantuan alat Sartorius ME5-F dan metode analisa konsentrasi karbon aerosol adalah metode pemantulan cahaya dan thermal (IMPROVE A) dengan bantuan alat Carbon Analyzer Model DRI 2001. Konsentrasi rata-rata dan tertinggi PM2.5 emisi kebakaran lahan gambut terutama pada fase pembaraan adalah 996,71 ± 531,01 µm/g3 dan 2163.49 µg/m3. Nilai tersebut lebih tinggi dari konsentrasi PM2.5 ketika tidak terjadi kebakaran (background) sebesar 48 kali. Rata-rata komposisi OC (sebagai salah satu penyusun utama PM2.5) dan EC dalam karbon total (TC) adalah 98,58 ± 0,91% dan 1,42 ± 0,91%. Fraksi OC (Organic Carbon) yang dominan adalah OC1 dan OC2 dengan rata-rata komposisi dalam karbon total (TC) adalah 40,34 ± 5,43% dan 31,58 ± 5,58%. Rasio OC/EC pada penelitian ini lebih besar dari rasio OC/EC pada kebakaran reruntuhan kayu dan kebakaran pohon pinus pada fase yang sama. Rasio OC/EC menunjukkan pengaruh emisi kebakaran lahan gambut terhadap emisi sumber kebakaran lain. Perlindungan lingkungan dari dampak yang ditimbulkan dari kebakaran lahan gambut dapat dilakukan dengan pencegahan penyebaran kebakaran dan penurunan konsentrasi PM2.5. Pencegahan penyebaran kebakaran dilakukan dengan menciptakan sistem pelindung lahan terhadap kebakaran dengan bantuan parit buatan. Penurunan konsentrasi PM2.5 dilakukan dengan menyediakan zona penyangga/penyerapan (buffer zone) menggunakan vegetasi khusus pada luas dan jarak tertentu
Routh's procedure for non-Abelian symmetry groups
We extend Routh's reduction procedure to an arbitrary Lagrangian system (that
is, one whose Lagrangian is not necessarily the difference of kinetic and
potential energies) with a symmetry group which is not necessarily Abelian. To
do so we analyse the restriction of the Euler-Lagrange field to a level set of
momentum in velocity phase space. We present a new method of analysis based on
the use of quasi-velocities. We discuss the reconstruction of solutions of the
full Euler-Lagrange equations from those of the reduced equations.Comment: 30 pages, to appear in J Math Phy
Routh reduction for singular Lagrangians
This paper concerns the Routh reduction procedure for Lagrangians systems
with symmetry. It differs from the existing results on geometric Routh
reduction in the fact that no regularity conditions on either the Lagrangian
or the momentum map are required apart from the momentum being a
regular value of . The main results of this paper are: the description of
a general Routh reduction procedure that preserves the Euler-Lagrange nature of
the original system and the presentation of a presymplectic framework for Routh
reduced systems. In addition, we provide a detailed description and
interpretation of the Euler-Lagrange equations for the reduced system. The
proposed procedure includes Lagrangian systems with a non-positively definite
kinetic energy metric.Comment: 34 pages, 2 figures, accepted for publicaton in International Journal
of Geometric Methods in Modern Physics (IJGMMP
Routhian reduction for quasi-invariant Lagrangians
In this paper we describe Routhian reduction as a special case of standard
symplectic reduction, also called Marsden-Weinstein reduction. We use this
correspondence to present a generalization of Routhian reduction for
quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total
time derivative. We show how functional Routhian reduction can be seen as a
particular instance of reduction of a quasi-invariant Lagrangian, and we
exhibit a Routhian reduction procedure for the special case of Lagrangians with
quasi-cyclic coordinates. As an application we consider the dynamics of a
charged particle in a magnetic field.Comment: 24 pages, 3 figure
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