By PPJM Schram
ISBN-10: 0792313925
ISBN-13: 9780792313922
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Extra resources for Kinematical theory of spinning particles: classical and quantum formalism
Sample text
N and s = 0, 1, . . , 2 k– 1) at the initial instant t 1. However the variational problem has been stated by the requirement that the solution goes through the two fixed endpoints, a condition that does not guarantee either the existence or the uniqueness of the solution. Nevertheless, let us assume that with the fixed endpoint conditions of the variational problem, q (i s ) (t1) and q ( s ) (t 2), i = 1, . . , n a n d i s = 0, 1, . . 6) perhaps non-unique. This implies that the 2kn boundary conditions at time t l required by the existence and uniqueness theorems, can be expressed perhaps in a non-uniform way, as functions of the kn conditions at each of the two endpoints.
K – 1 including the time t l , and of the corresponding kn + 1 variables at final time t 2 . We write it as We thus arrive at the following Definition: The kinematical variables of the system are the time t and the n degrees of freedom q i and their time derivatives up to order k – 1. The manifold X they span is the kinematical space of the system. The kinematical space for ordinary Lagrangians is just the configuration space spanned by variables q i enlarged with the time variable t. It is usually called the enlarged configuration space.
The kinematical variables with the time excluded. 30), will be 19 GENERAL FORMALISM 6. LIE GROUPS OF TRANSFORMATIONS Let us introduce the notation and general features of the action of Lie groups on continuous manifolds to analyze the transformation properties of the different magnitudes we can work with in either classical or quantum mechanics. We shall use these features all throughout this book. , there exists a group composition law c = φ ( a, b) ∈ G, ∀ a, b ∈ G, in terms of r continuous and differentiable functions φ σ .
Kinematical theory of spinning particles: classical and quantum formalism by PPJM Schram
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