As all natural laws, Newtonian dynamics should be governed by Einstein’s Covariance Principle;i.e., being covariant under all coordinate transformations, even time-dependent transformations. But Newton’s Second Law,...As all natural laws, Newtonian dynamics should be governed by Einstein’s Covariance Principle;i.e., being covariant under all coordinate transformations, even time-dependent transformations. But Newton’s Second Law, as it is generally understood, is unchanged only under Galilean transformations, which do not include time-dependent coordinate transformations. To achieve the covariant formulation of Newton’s Second Law, a distinction must be made between frames and coordinate systems, as advanced by the Principle of Material Frame-Indifference, and furthermore, the ordinary time derivative must be replaced by the rotational time derivative. Elevating Newton’s Second Law to covariancy has born many fruits in flight dynamics from the theoretical underpinning of unsteady flight maneuvers to the practical modeling of complex flight engagements in tensors, followed by efficient programming with matrices.展开更多
Tensor flight dynamics solves flight dynamics problems using Cartesian tensors, which are invariant under coordinate transformations, rather than Gibbs’ vectors, which change under time-varying transformations. Three...Tensor flight dynamics solves flight dynamics problems using Cartesian tensors, which are invariant under coordinate transformations, rather than Gibbs’ vectors, which change under time-varying transformations. Three tensors of rank two play a prominent role and are the subject of this paper: moment of inertia, rotation, and angular velocity tensor. A new theorem is proven governing the shift of reference frames, which is used to derive the angular velocity tensor from the rotation tensor. As applications, the general strap-down INS equations are derived, and the effect of the time-rate-of-change of the moment of inertia tensor on missile dynamics is investigated.展开更多
文摘As all natural laws, Newtonian dynamics should be governed by Einstein’s Covariance Principle;i.e., being covariant under all coordinate transformations, even time-dependent transformations. But Newton’s Second Law, as it is generally understood, is unchanged only under Galilean transformations, which do not include time-dependent coordinate transformations. To achieve the covariant formulation of Newton’s Second Law, a distinction must be made between frames and coordinate systems, as advanced by the Principle of Material Frame-Indifference, and furthermore, the ordinary time derivative must be replaced by the rotational time derivative. Elevating Newton’s Second Law to covariancy has born many fruits in flight dynamics from the theoretical underpinning of unsteady flight maneuvers to the practical modeling of complex flight engagements in tensors, followed by efficient programming with matrices.
文摘Tensor flight dynamics solves flight dynamics problems using Cartesian tensors, which are invariant under coordinate transformations, rather than Gibbs’ vectors, which change under time-varying transformations. Three tensors of rank two play a prominent role and are the subject of this paper: moment of inertia, rotation, and angular velocity tensor. A new theorem is proven governing the shift of reference frames, which is used to derive the angular velocity tensor from the rotation tensor. As applications, the general strap-down INS equations are derived, and the effect of the time-rate-of-change of the moment of inertia tensor on missile dynamics is investigated.
