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Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
which is just the transformation rule for the components of a dual vector, cf. Eq. (9.27). The phase space of a system is therefore equivalent to the cotangent bundleT*M of the configuration space, i.e., the union of all cotangent spaces Tp*M. An element of the cotangent bundle is any cotangent vector in configuration space and just as for the tangent bundle we can define a projection π such that π(ξ)=p if ξ is a dual vector in Tp*M. As any element in the cotangent bundle can be specified by N coordinates qa and N components of the canonical momentum pa, it is in itself also a manifold of dimension 2N with the coordinates yr that we have already introduced.
Abnormal Sub-Riemannian Minimizers
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
Wensheng Liu, Hector J. Sussmann
If M is a C∞ manifold, and p ε M, we use TpM to denote the tangent space of M at p, and TM to denote the tangent bundle of M. If Ε is any C∞ vector bundle over M, then Γ(E) denotes the set of all C∞ sections of E. A subbundle Ε of TM is sometimes called a distribution on M. A nonholonomic subbundle (also known as a bracket-generating distribution) is a subbundle Ε of ΤM such that the Lie algebra L(Γ(E)) of vector fields generated by the sections of Ε has the full rank property, i.e. satisfies {X(p) : X ε L(Γ(E))} = TpM for all p ε M. An Ε-admissible arc is an absolutely continuous curve γ on M, defined on some compact interval [a, b], such that γ˙(t)∈E(γ(t)) for almost all t ε [α, b]. If Ε is nonholonomic and M is connected, then any two points in M can be joined by an E-admissible arc.
Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
Lemma 10.14.2 states that compatible errors of a fuzzy-state automaton (A,τ) are inessential and remain inessential under the action of any input word. The maximal compatible state relation of (A,τ) is TQ=(τ,ττ), and ρτ,ττ is its maximal inessential state relation if pr1ρτ⊇Qp. Since the tangent bundle of the tolerance space (Q,τ) is the composite map t : TQ⊆Q×Q→pr1Q,[179], and the tangent space TqQ to Q at state q is the tolerance space TqQ=(tq,ττ), the compatible changes of state q determine the tangent space at q in a sense. In [37], such “geometric” properties of errors and t-modifications are studied. There modification tolerance of automata is considered, i.e., with masked and correctable t modification of an automaton. However, in the following, relational (settheoretic) properties of temporary errors are considered.
New subgradient extragradient algorithm for solving variational inequalities in Hadamard manifold
Published in Optimization, 2023
Let M be a m-dimnesional manifold and let be the tangent space of M at We denote by the tangent bundle of M. An inner product is called the Riemannian metric on The corresponding norm to the inner product on is denoted by We will drop the subscript x. A differentiable manifold M endowed with a Riemannian metric is called a Riemannian manifold. In what follows, we denote the Riemannian metric by when no confusion arise. Given a piecewise smooth curve joining x to y (i.e and ), we define the length of c by The Riemannian distance is the minimal length over the set of all such curves joining x to y including the original topology on M. Let ∇ be the Levi–Civita connection associated with the Riemannian metric.
Exergetic port-Hamiltonian systems: modelling basics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Markus Lohmayer, Paul Kotyczka, Sigrid Leyendecker
For tensorial quantities, we use (abstract) index notation with Einstein’s convention: Indices of contravariant slots are written as superscript and indices of covariant slots are written as subscript. Repeated indices (up-down pairs) imply contraction. With a smooth manifold, denotes the tangent bundle and the cotangent bundle over . We write for a general vector bundle with total space and base space . When the latter is clear from the context, we just write . For vector bundles and , is the vector bundle over where . A section of a bundle is a function satisfying where is the bundle projection. The set of all sections of is denoted by . Given a contravariant -tensor field , the sharp map is the (curried) function defined by . Dually, the flat map corresponding to a covariant -tensor field is a bundle map from the tangent to the cotangent bundle. Its name derives from the fact that in index notation, it lowers the up-index of a tangent vector into the down-index of the covector .
Motion analysis of a multi-joint system with holonomic constraints using Riemannian distance
Published in Advanced Robotics, 2022
Masahiro Sekimoto, Suguru Arimoto
The book [21] mentioned that the configuration space of a double planar pendulum is a two-dimensional differentiable manifold. Hence, the configuration space M of the walking model is an n-dimensional differentiable manifold. For a given chart , where U is an open set in Euclidean space and ϕ represent the homeomorphism (a one-to-one mapping) of U onto the open subset of M, two points are associated with and in local coordinates, as illustrated in Figure 2. Now, consider a curve (a motion) from to on the manifold as for , , and . The velocity of , i.e. the tangent vector at a point on is expressed as a linear combination of the basis tangent vectors defined by the coordinate chart : Note that we adopted this form because the points and tangent vectors on a manifold are conventionally represented in scalar forms associated with the points on the manifold in Riemannian geometry [22, 23]. At the point , the tangent vectors for any motion passing through the point constitute a tangent space . The tangent space is naturally constructed at every point on the manifold, and the union is called a tangent bundle. The section of the tangent bundle of M along is called a tangent vector field. To measure the curve length on the differentiable manifold, we introduce the following metric associated with the kinetic energy: where is given by the ij-component of the inertia matrix . Because the metric takes a positive-definite symmetric quadratic form over every tangent space [34], the differentiable manifold with the metric is considered a Riemannian manifold.