Olga Krupková

Hamiltonian field theory

Abstract In this paper, a general Hamiltonian theory for Lagrangian systems on fibred manifolds is proposed. The concept of a Lepagean ( n +1)- form is defined (where n is the dimension of the base manifold), generalizing Krupka’s concept of a Lepagean n-form. Lepagean ( n +1)-forms are used to study Lagrangian and Hamiltonian systems. Innovations and new results concern the following: a Lagrangian system is considered as an equivalence class of local Lagrangians (of all orders starting from a minimal one); a Hamiltonian system is associated with an Euler–Lagrange form (not with a particular Lagrangian); Hamilton equations are based upon a Lepagean ( n +1)-form, and cover Hamilton–De Donder equations (which are based upon the exterior derivative of the Poincare–Cartan form) as a special case. First-order Hamiltonian systems, namely those carying higher-degree contact components of the corresponding Lepagean forms, are studied in detail. The presented geometric setting leads to a new (more general than the standard one) understanding of the concepts of regularity and Legendre transformation in the calculus of variations, relating them directly to the properties of the arising exterior differential systems . In this way, new regularity conditions and Legendre transformation formulas are obtained, depending on a Lepagean ( n +1)-form, i.e., related with the corresponding Euler–Lagrange form .

Mechanical systems with nonholonomic constraints

A geometric setting for the theory of first-order mechanical systems subject to general nonholonomic constraints is presented. Mechanical systems under consideration are not supposed to be Lagrangian systems, and the constraints are not supposed to be of a special form in the velocities (as, e.g., affine or linear). A mechanical system is characterized by a certain equivalence class of 2-forms on the first jet prolongation of a fibered manifold. The nonholonomic constraints are defined to be a submanifold of the first jet prolongation. It is shown that this submanifold is canonically endowed with a distribution—this distribution (resp., its vertical subdistribution) has the meaning of generalized possible (resp., virtual) displacements. The concept of a constraint force is defined, and a geometric version of the principle of virtual work is proposed. From the principle of virtual work a formula for a workless constraint force is obtained. A mechanical system subject nonholonomic constraints is modeled as ...

Higher-order mechanical systems with constraints

A general mathematical theory covering higher-order mechanical systems subject to constraints of arbitrary order (i.e., depending on time, positions, velocities, accellerations, and higher derivatives) is presented, including higher-order holonomic systems as a particular case. Within differential geometric setting on higher-order jet bundles, the concept of a mechanical system (not necessarily regular, or Lagrangian) is introduced to be a class of 2-forms equivalent with a dynamical form. Dynamics are then represented by means of corresponding exterior differential systems. Higher-order constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution (canonical distribution, higher-order Chetaev bundle). With help of a constraint structure a constraint force is naturally introduced. Higher-order mechanical systems subject to different kinds of higher-order constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view....

Legendre Transformation for Regularizable Lagrangians in Field Theory

Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

A geometric setting for higher‐order Dirac–Bergmann theory of constraints

Hamilton equations for degenerate time‐dependent, and generally higher‐order mechanical systems are considered and their geometric meaning is analyzed. The local dynamics of constrained systems is clarified and an algorithm for finding it is presented.

Recent results in the geometry of constrained systems

Abstract Nonholonomic mechanical systems on fibered manifolds are investigated from a geometrical point of view. Regularity, variationality and existence of Legendre transformation is studied. A solution to the variational inverse problem for constrained systems is presented.

The relativistic particle as a mechanical system with non-holonomic constraints

The geometric theory of non-holonomic systems on fibred manifolds is applied to describe the motion of a particle within the theory of special relativity. General motion equations for material particles subjected to potential forces are found. They cover, as particular cases, standard motion equations as well as a generalization of the special relativity theory proposed by Dicke. Moreover, they offer new possibilities for studying the dynamics of relativistic particles interacting with an electromagnetic and/or a scalar field.

The Cartan form and its generalizations in the calculus of variations

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Euler-Lagrange and Hamilton equations for non-holonomic systems in field theory

A generalization of the concept of a system of non-holonomic constraints to fibred manifolds with n-dimensional bases is considered. Motion equations in both Lagrangian and Hamiltonian settings for systems subjected to such constraints are investigated. Regularity conditions for the existence of a non-holonomic Legendre transformation, and the corresponding formulae for Hamiltonian and momenta are found. In particular, Lagrangian constraints and semi-holonomic constraints, and simplifications arising in this case are discussed.

Variational first-order partial differential equations☆

Abstract Geometrical and variational properties of systems of first-order partial differential equations (PDE) on fibered manifolds are studied. Existence of Lagrangians is shown to be equivalent with the existence of a closed form which is global and unique; an explicit construction of this form is given. A bijective map between a set of dynamical forms on J 1 Y , representing first-order PDE, and forms on the total space Y is found, providing a geometric description of the equations by means of a (not necessarily closed) ideal generated by a system of n -forms on Y ( n = dimension of the base manifold). Conditions for this ideal to be closed are studied. Relations with Hamiltonian structures and with multisymplectic forms are discussed.