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L. Mangiarotti

Publications and source records attributed to L. Mangiarotti.

48 records · Page 3Linked to original sources
arXiv1999

The Koszul-Tate Cohomology in Covariant Hamiltonian Formalism

We show that, in the framework of covariant Hamiltonian field theory, a degenerate almost regular quadratic Lagrangian $L$ admits a complete set of non-degenerate Hamiltonian forms such that solutions of the corresponding Hamilton equations, which live in the Lagrangian constraint space, exhaust solutions of the Euler--Lagrange equations for $L$. We obtain the characteristic splittings of the configuration and momentum phase bundles. Due to the corresponding projection operators, the Koszul-Tate resolution of the Lagrangian constraints for a generic almost regular quadratic Lagrangian is constructed in an explicit form.

hep-th↗
arXiv1999

On the Geodesic Form of Non-Relativistic Dynamic Equations

It is shown that any second order dynamic equation on a configuration bundle $Q\to R$ of non-relativistic mechanics is equivalent to a geodesic equation with respect to a (non-linear) connection on the tangent bundle $TQ\to Q$. The case of quadratic dynamic equations is analyzed in details. The equation for Jacobi vector fields is constructed and investigated by the geometric methods.

math-ph↗
arXiv1999

Nonrelativistic Geodesic Motion

We show that any second order dynamic equation on a configuration space $X\to R$ of nonrelativistic mechanics can be seen as a geodesic equation with respect to some (nonlinear) connection on the tangent bundle $TX\to X$ of relativistic velocities. We compare relativistic and nonrelativistic geodesic equations, and study the Jacobi vector fields along nonrelativistic geodesics.

gr-qc↗
arXiv1999

Constraints in Hamiltonian time-dependent mechanics

In Hamiltonian time-dependent mechanics, the Poisson bracket does not define dynamic equations, that implies the corresponding peculiarities of describing time-dependent holonomic constraints. As in conservative mechanics, one can consider the Poisson bracket of constraints, separate them in first and second class constraints, construct the Koszul-Tate resolution and a BRST complex. However, the Poisson bracket of constraints and a Hamiltonian makes no sense. Hamiltonian vector fields for first class constraints are not generators of gauge transformations. In the case of Lagrangian constraints, we state the comprehensive relations between solutions of the Lagrange equations for an almost regular Lagrangian and solutions of the Hamilton equations for associated Hamiltonian forms, which live in the Lagrangian constraint space. Degenerate quadratic Lagrangian systems are studied in details. We construct the Koszul-Tate resolution for Lagrangian constraints of these systems in an explicit form.

math-ph↗
arXiv1999

Covariant Hamiltonian field theory

We study the relationship between the equations of first order Lagrangian field theory on fiber bundles and the covariant Hamilton equations on the finite-dimensional polysymplectic phase space of covariant Hamiltonian field theory. The main peculiarity of these Hamilton equations lies in the fact that, for degenerate systems, they contain additional gauge fixing conditions. We develop the BRST extension of the covariant Hamiltonian formalism, characterized by a Lie superalgebra of BRST and anti-BRST symmetries.

hep-th↗
arXiv1998

Energy-Momentum and Gauge Conservation Laws

We treat energy-momentum conservation laws as particular gauge conservation laws when generators of gauge transformations are horizontal vector fields on fibre bundles. In particular, the generators of general covariant transformations are the canonical horizontal prolongations of vector fields on a world manifold. This is the case of the energy-momentum conservation laws in gravitation theories. We find that, in main gravitational models, the corresponding energy-momentum flows reduce to the generalized Komar superpotential. We show that the superpotential form of a conserved flow is the common property of gauge conservation laws if generators of gauge transformations depend on derivatives of gauge parameters. At the same time, dependence of conserved flows on gauge parameters make gauge conservation laws form-invariant under gauge transformations.

gr-qc↗
arXiv1998

Nonholonomic Constraints in Time-Dependent Mechanics

The constraint reaction force of ideal nonholonomic constraints in time-dependent mechanics on a configuration bundle $Q\to R$ is obtained. Using the vertical extension of Hamiltonian formalism to the vertical tangent bundle $VQ$ of $Q\to R$, the Hamiltonian of a nonholonomic constrained system is constructed.

math-ph↗
arXiv1998

Dynamic Connections in Analytical Mechanics

It is shown that any dynamic equation on a configuration bundle $Q\to R$ of non-relativistic time-dependent mechanics is associated with connections on the affine jet bundle $J^1Q\to Q$ and on the tangent bundle $TQ\to Q$. As a consequence, any non-relativistic dynamic equation can be seen as a geodesic equation with respect to a (non-linear) connection on the tangent bundle $TQ\to Q$. Using this fact, the relationship between relativistic and non-relativistic equations of motion is studied. The geometric notions of reference frames and relative accelerations in non-relativistic mechanics are introduced in the terms of connections. The covariant form of non-relativistic dynamic equations is written.

math-ph↗
arXiv1998

Relativistic and Non-relativistic Equations of Motion

It is shown that any second order dynamic equation on a configuration space $X$ of non-relativistic time-dependent mechanics can be seen as a geodesic equation with respect to some (non-linear) connection on the tangent bundle $TX\to X$ of relativistic velocities. Using this fact, the relationship between relativistic and non-relativistic equations of motion is studied.

gr-qc↗
arXiv1997

Universal Spin Structure in Gauge Gravitation Theory

Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.

gr-qc↗
arXiv1997

Differential Geometry of Time-Dependent Mechanics

The usual formulations of time-dependent mechanics start from a given splitting $Y=R\times M$ of the coordinate bundle $Y\to R$. From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical transformations. Our goal is to formulate time-dependent mechanics in gauge-invariant form, i.e., independently of any reference frame. The main ingredient in this formulation is a connection on the bundle $Y\to R$ which describes an arbitrary reference frame. We emphasize the following peculiarities of this approach to time-dependent mechanics. A phase space does not admit any canonical contact or presymplectic structure which would be preserved under reference frame transformations, whereas the canonical Poisson structure is degenerate. A Hamiltonian fails to be a function on a phase space. In particular, it can not participate in a Poisson bracket so that the evolution equation is not reduced to the Poisson bracket. This fact becomes relevant to the quantization procedure. Hamiltonian and Lagrangian formulations of time-dependent mechanics are not equivalent. A degenerate Lagrangian admits a set of associated Hamiltonians, none of which describes the whole mechanical system given by this Lagrangian.

dg-ga↗