Complex Spaces in Finsler, Lagrange and Hamilton Geometries (Fundamental Theories of Physics)
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Contents:
Geometry of the Total Space of a Tangent Bundle. Finsler Spaces. Lagrange Spaces. Generalized Lagrange Space. Relativistic Geometrical Optics. Geometry of Time Dependent Lagrangians.
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Les mer. Om boka The Geometry of Lagrange Spaces: Theory and Applications Differential-geometric methods are gaining increasing importance in the understanding of a wide range of fundamental natural phenomena. Very often, the starting point for such studies is a variational problem formulated for a convenient Lagrangian.
From a formal point of view, a Lagrangian is a smooth real function defined on the total space of the tangent bundle to a manifold satisfying some regularity conditions. The main purpose of this book is to present: a an extensive discussion of the geometry of the total space of a vector bundle; b a detailed exposition of Lagrange geometry; and c a description of the most important applications. F is positive. It is not difficult to see that this definition has a geometrical meaning, gij from 3.
F is a Finsler space []. We can see, without difficulties that the following theorem holds: Theorem 3. The fundamental tensor gfjj is O-homogeneous. Example 3.
In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [ ]:. In the talk, Ulam discussed a number of important unsolved mathematical problems. In particular, we get 4. These models are: the so called simplest scalar used in the expression for the Hilbert-Einstein action, i. Of course, by quantum field theory the dichotomy between matter and fields in the sense of a dualism is minimised as every field carries its particle-like quanta.
Then, theorem 1. It is not hard to see that these coefficients are homogeneous of degree l,2, This property implies that the d-Liouville vector field 3. Consider the function 3. It follows that F from 3. Concluding, we have: Theorem 3. The variational problem leads to the Euler - Lagrange equations. Applying the theory from the section 2, ch. In particular, Theorem 2. P o z- Then we have Theorem 3.
On some vertical cohomologies of complex Finsler manifolds
The Cartan differential 1-forms are the followings 3. Canonical mean here that all these object fields depend only on the fundamental function F. It will be called the Craig-Synge equation. Using 2. Applying the Theorem 2. It is a fc-spray, since it is a 2-homogeneous vector field.
The Geometry of Lagrange Spaces: Theory and Applications
By means of the Theorems 1. Some properties of N. The dual coefficients 3. The local adapted basis to the direct decomposition 3. Now we can determine the differential operators dk, dk-i, do defined in 1. One obtains: 3. So we have the following main 1-form fields Finsler Spaces of Order k 53 3.
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The 1-form fields 9o, The exterior differentials of 6o, The Lagrangians Fi, The lift of the fundamental tensor field gij is given by 2. The terms of G are 0,2, Notice that G is not homogeneous.
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We can construct an homogeneous one using the Lagrangians 3. Namely 3. In the following we consider the Riemannian structure G from 3. Such that we have 3. Finally, consider the tensor field F determined by the Cartan nonlinear con- nection A''. Now, we consider this problem for the general case. Sometimes cf.
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The Jacobian matrix of the transformation 4. The form 4. Also, we can prove without difBculties the following theorem: Theorem 4. Consider the category Man of differentiable manifolds. It is convenient to adopt the notation: 4. Its dimension is n and it is an integrable distribution, too.
Fundamental Theories of Physics. Free Preview Complex Spaces in Finsler, Lagrange and Hamilton Geometries Complex and holomorphic vector bundles. Compre Complex Spaces in Finsler, Lagrange and Hamilton Geometries ( Fundamental Theories of Physics Book ) (English Edition) de Gheorghe Munteanu.
We conclude by Proposition 4. Using again 4. Proposition 4. Rem,ark The differential dH being invariant under the coordinate transforma- tions 4. The exterior differential doj of the Liouville 1-form oj is expressed by 4. Now, based on the previous results we obtain: Theorem 4. By means of 4. According to the above theorem we may call J the k — 1 -tangent structure. The endomorphism J applied to the Liouville vector fields gives us: 4.
If X is given by 4. Using 4.
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In particular, we get 4. As we already have seen doH, dk-2H are expressed in 4. In this case do, dk-2 and d are the antiderivations of degree 1. The equalities 4. As we shall see in ch.