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We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold.
Using the fractional integration and differentiation on R we build the fractional jet fibre bundle on a differentiable manifold and we emphasize some important geometrical objects. Euler-Lagrange fractional equations are described. Some…
The aim of the paper is to study some dynamic aspects coming from a tangent form, i.e. a time dependent differential form on a tangent bundle. The action on curves of a tangent form is natural associated with that of a second order…
This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or…
Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case…
The classical fields with fractional derivatives are investigated by using the fractional Lagrangian formulation.The fractional Euler-Lagrange equations were obtained and two examples were studied.
We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fr\'echet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.
Lagrangians linear in velocities were analyzed using the fractional calculus and the Euler-Lagrange equations were derived. Two examples were investigated in details, the explicit solutions of Euler-Lagrange equations were obtained and the…
Let $C\to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density $\Lambda $ on $C$ satisfying a weak condition of regularity, are shown to admit an…
Analysing an application in liquid film dynamics, a guide for obtaining the corresponding constrained functional derivatives for constraints coupling the functional variables is given. The use of constrained derivatives makes the proper…
We study variational problems for curves approximated by B-spline curves. We show that, one can obtain discrete Euler-Lagrange equations, for the data describing the approximated curves. Our main application is to the curve completion…
We study in this paper the continuous and discrete Euler-Lagrange equations arising from a quadratic lagrangian. Those equations may be thought as numerical schemes and may be solved through a matrix based framework. When the lagrangian is…
We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale…
Given a differential equation on a smooth fibre bundle Y, we consider its canonical vertical extension to that, called the deviation equation, on the vertical tangent bundle VY of Y. Its solutions are Jacobi fields treated in a very general…
We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in…
We give an account, in terms of fibered categories and their fibrewise duals, of aspects of the theory of bundle functors and star-bundle functors in differential geometry.
In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on…
Motivated by mirror symmetry, we consider a Lagrangian fibration $X\to B$ and Lagrangian maps $f:L\hookrightarrow X\to B$, when $L$ has dimension 2, exhibiting an unstable singularity, and study how their caustic changes, in a neighbourhood…
The application of a gauge covariant derivative to the Euler-Lagrange equation yields a shortcut to the equations of motion for a field subject to an external force. The gauge covariant derivative includes an external force as an intrinsic…
In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to…