Theory of Singular Lagrangians

Cariñena, José F.

doi:10.1002/prop.2190380902

Cited by 54 publications

(69 citation statements)

References 24 publications

(8 reference statements)

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“…Such constraints are known to be hidden in some (consequences) of the Lagrange equations though currently the most efficient tool to analyze the situation is based on canonical Hamiltonian formalism (see [2][3][4][5][6][7]). However, let us here only mention that these constraints are somehow trivial as long as conservation laws are considered since their associated currents vanish on-shell (once initial conditions obey constraints).…”

Section: Introductionmentioning

confidence: 99%

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

2010

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Abstract:We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called "Natural Theories" and "Gauge-Natural Theories" that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincaré-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether symmetries", energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-called ADM laws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation "à la Palatini" and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).

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Section: Introductionmentioning

confidence: 99%

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

2010

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“…However, this is not the final form of the dynamics. To get the final dynamics we must perform a stabilization algorithm: [5][6][7]21,22 consistency requirements, that is, the tangency of X H to the surface of constraints, may lead to new constraints and also to the determination of some of the Lagrangian multipliers as functions in phase space.…”

Section: A Characterization In Phase Spacementioning

confidence: 99%

“…This is nothing but the framework first studied by Dirac to deal with gauge theories or, more generally, constrained systems. [4][5][6][7][8][9] The regular case is recovered when no Hamiltonian constraints occur.…”

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confidence: 99%

Canonical Noether symmetries and commutativity properties for gauge systems

Gràcia

Pons

2000

Journal of Mathematical Physics

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“…Thus only candidates to be optimal trajectories, called extremals, are found. Maximum Principle gives rise to different kinds of them and, particularly, the so-called abnormal extremals have been studied because they can be optimal, as Liu and Sussmann, and Montgomery proved in subRiemannian geometry [5,7].We build up a presymplectic algorithm, similar to those defined in [2,3,4,6], to determine where the different kinds of extremals of an optimal control problem can be. After describing such an algorithm, we apply it to the study of extremals, specially the abnormal ones, in optimal control problems for affine connection control systems [1].…”

mentioning

confidence: 99%

“…We build up a presymplectic algorithm, similar to those defined in [2,3,4,6], to determine where the different kinds of extremals of an optimal control problem can be. After describing such an algorithm, we apply it to the study of extremals, specially the abnormal ones, in optimal control problems for affine connection control systems [1].…”

mentioning

confidence: 99%

Optimal Control Problems for Affine Connection Control Systems: Characterization of Extremals

Barbero-Lin͂án

Muñoz-Lecanda

2008

AIP Conference Proceedings

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Abstract. Pontryagin's Maximum Principle [8] is considered as an outstanding achievement of optimal control theory. This Principle does not give sufficient conditions to compute an optimal trajectory; it only provides necessary conditions. Thus only candidates to be optimal trajectories, called extremals, are found. Maximum Principle gives rise to different kinds of them and, particularly, the so-called abnormal extremals have been studied because they can be optimal, as Liu and Sussmann, and Montgomery proved in subRiemannian geometry [5,7].We build up a presymplectic algorithm, similar to those defined in [2,3,4,6], to determine where the different kinds of extremals of an optimal control problem can be. After describing such an algorithm, we apply it to the study of extremals, specially the abnormal ones, in optimal control problems for affine connection control systems [1]. These systems model the motion of different types of mechanical systems such as rigid bodies, nonholonomic systems and robotic arms [1].Keywords: Pontryagin's Maximum Principle, abnormal extremals, optimal control problems, affine connection control systems. PACS: 02.30Yy, 45.80.+r, 46.15.Cc, 02.40Yy, 02.30Xx OPTIMAL CONTROL PROBLEM FOR AFFINE CONNECTION CONTROL SYSTEMSLet Q be a smooth n-dimensional manifold, ∇ be an affine connection on Q, U be an open set in R m . We consider the control system with dynamics given bywhere the controls u : I ⊂ R → U ⊂ R m are piecewise smooth, Y k are the input vector fields on Q. This control system is called an Affine Connection Control System, ACCS,In fact, the affine connection control systems are a particular case of the general simple mechanical systems without external forces nor nonholonomic constraints [1].The second-order equation (1) is rewritten on T Q as follows,where ϒ : I → T Q is a piecewise smooth curve such that ϒ =γ, Y V k denotes the vertical lift of the vector field Y k and the drift vector field Z of the control-affine system (2) is the geodesic spray associated to ∇. The vector field Z on T Q is second order. In natural coordinates (x, v) for T Q, locally Z = v i ∂ /∂ x i −Γ i jl (x)v j v l ∂ /∂ v i with Christoffel symbols Γ i jl for ∇.

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Theory of Singular Lagrangians

Cited by 54 publications

References 24 publications

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

Canonical Noether symmetries and commutativity properties for gauge systems

Optimal Control Problems for Affine Connection Control Systems: Characterization of Extremals

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