China
Africa
Explore chapters and articles related to this topic
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.
Complexity Analysis of Pathogenesis of Coronavirus Epidemiological Spread in the China Region
Published in Jyoti Mishra, Ritu Agarwal, Abdon Atangana, Mathematical Modeling and Soft Computing in Epidemiology, 2020
Rashmi Bhardwaj, Aashima Bangia, Jyoti Mishra
Phase space is generally symbolized through cotangent bundle: T∗M:={(q,p):q∈M,p∈Tq∗M},
Stabilisation of a relative equilibrium of an underactuated AUV on SE(3)
Published in International Journal of Control, 2019
Local representations, such as Euler anglesand Rodrigues parameters, which have singular coordinates. Although, the unit quaternion can globally describe evolution of a rigid body, unwinding phenomena appear, which is called ambiguities (Magro, Ishihara, & Ferreira, 2017). Instead of these coordination representations above, SE(3) is employed to describe the evolution of the body's rotation and translation simultaneously without singularities or ambiguities. A Special Euclidian group SE(3) is defined as follows: where R consisting of body-fixed reference frame's axes is a 3 × 3 Special Orthogonal (SO) matrix that represents the body's attitude. Let r denote the position of the centre of an AUV. Thereafter, SE(3) represents the body's attitude and position, simultaneously. Let TSE(3) and T*SE(3) denote the tangent bundle and cotangent bundle of SE(3), respectively. Let TgSE(3) and T*gSE(3) denote the tangent space and cotangent space of SE(3) at an element g. The tangent space of SE(3) at the identity element e, TeSE(3), is called Lie algebra after defining a Lie bracket operation, denoted as , which is defined as follows: where the 3 × 3 skew-symmetric matrix represents the body's rotational velocity in a twister form, and v represents the translational velocity. They are both expressed in the body-fixed reference frame.
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 .