Siacci’s resolution of the acceleration vector for a space curve (original) (raw)
A Note on the Acceleration and Jerk in Motion Along a Space Curve
Analele Universitatii "Ovidius" Constanta - Seria Matematica
The resolution of the acceleration vector of a particle moving along a space curve is well known thanks to Siacci [1]. This resolution comprises two special oblique components which lie in the osculating plane of the curve. The jerk is the time derivative of acceleration vector. For the jerk vector of the aforementioned particle, a similar resolution is presented as a new contribution to field [2]. It comprises three special oblique components which lie in the osculating and rectifying planes. In this paper, we have studied the Siacci’s resolution of the acceleration vector and aforementioned resolution of the jerk vector for the space curves which are equipped with the modified orthogonal frame. Moreover, we have given some illustrative examples to show how the our theorems work.
On the instantaneous acceleration of points in a rigid body
Mechanism and Machine Theory, 2011
This work reexamines some classical results in the kinematics of points in space using modern vector-matrix methods. In particular, some very simple Lie theory allows the velocities and accelerations of points to be found in terms of the instantaneous twist of the motion and its derivative. From these results many of the classical results follow rather simply. Although most of the results are well known, some new material is presented. In particular, the discriminant curve that separates cases with one or three real acceleration axes is found and plotted. Another new result concerns the chords to the cubic of inflection points. It is shown that for points on such a chord the osculating planes of the point's trajectories are parallel. Also a new result is found which distinguishes between cases where the Bresse hyperboloid of points whose velocities and accelerations are perpendicular, has one or two sheets.
Multivariable Calculus with MATLAB®, 2017
The study of curves in space is of interest not only as a topic in geometry but also for its application to the motion of physical objects. In this chapter, we develop a few topics in mechanics from the point of view of the theory of curves. Additional applications to physics will be considered in Chapter 10.
Kinematics of point motion along curves of the second order
2024
In the scientific literature, the differential equation of an ellipse is derived through dynamic quantities and laws. Kepler's laws are then derived. However, Kepler's laws are kinematic. This article examines the kinematic equation of an ellipse. The equation is derived through the oscillations of a parametric pendulum. It is shown that Kepler's laws are properties of kinematic equations of motion of a point along second-order curves. The motion of a material point along second-order curves is represented by kinematic equation (1.10). The kinematics of second-order curves is studied on an ellipse. Formulas for the dependence of acceleration and radius, speed and radius are derived. The direction of the velocity and acceleration vectors is determined. The conditions for the conservation of Kepler's laws when a material point moves along an ellipse are shown. The article "Kinematics of point motion along curves of the second order" https://www.academia.edu/116906587/ describes the kinematics of uniform, uniformly accelerated, elliptical (Keplerian) motion along an ellipse.
Differential Geometry: An Introduction to the Theory of Curves
Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.
On the Use of a Theorem by V. Vâlcovici in Planar Motion Dynamics
2005
A theorem by V. Vâlcovici that generalizes Koenig's well known angular momentum theorem finds its use in planar motion dynamics. The reference point in this type of motion is the instant center. The advantage in choosing the instant center as the reference point in angular momentum theorems is that unknown constraint forces do not appear in the equations of motions.
As an aid to teachers and students who are learning to apply Geometric Algebra (GA) to high-school-level physics, we provide this third installment in our guide guide to Hestenes's treatment of constant-acceleration motion. A key element of this installment is our detailed discussion of how Hestenes uses "extended hodographs" as sources of geometric insights that can be expressed and transformed via GA to deduce relationships among factors that affect constant-acceleration motion.
A study of the action from kinematical integral geometry point of view
J Geom Physics, 1990
We develop here an interpretation of the classical nonrelativistic and relativistic action fora pointparticle as related to geometric measures of sets of straight lines (inertial motions)associated in anatural way to closed timeike circuits in spacetime. This allows a point of view for the action common to classical and relativistic mechanics. Fw-thennore the results are not restricted to the free case and also holds forpartides in some potentials (homogeneous field and the harmonic oscillator).