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2002 | Buch

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Analytical Mechanics

verfasst von: A. I. Lurie

Verlag: Springer Berlin Heidelberg

Buchreihe : Foundations of Engineering Mechanics

Enthalten in: Professional Book Archive

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Über dieses Buch

According to established tradition, courses on analytical mechanics include general equations of motion of holonomic and non-holonomic systems, vari ational principles, theory of canonical transformations, canonical equations and theory of their integration (the Hamilton-Jacobi theorem), integral in variants, theory of last multiplier and others. The fundamental laws of mechanics are taken for granted and are not subject to discussion. The present book is concerned with those issues of the above listed sub jects which, in the author's opinion, are most closely related to engineering problems. Application of the methods of analytical mechanics to non-trivial prob lems at the very stage of constructing the equations requires detailed knowl edge of the issues that are normally only briefly touched upon. With this perspective considerable attention is paid to ways of introducing the gener alised coordinates, the theory of finite rotation, methods of calculating the kinetic energy, the energy of accelerations, the potential energy of forces of various nature, and the resisting forces. These introductory chapters, which have to some extent independent significance, are followed by those on methods of constructing differential equations of motion for holonomic and non-holonomic systems in various forms. In these chapters the issues of their interrelations, determination of the constraint forces and some prob lems of analytical statics are discussed as well. It is thought useful to include geometric considerations of the motion of a material system as motion of the representative point in Riemannian space.

Inhaltsverzeichnis

Frontmatter

1. Basic definitions

Abstract

From a dynamical point of view any material system can be regarded as a collection of material particles. The relationships between the quantities determining the position and the velocity of the system of particles are referred to as constraints. These relationships must hold regardless of the initial conditions and the forces acting on the system.

A. I. Lurie

2. Rigid body kinematics — basic knowledge

Abstract

As mentioned in Chapter 1 rigid bodies and systems of rigid bodies are the most important objects of analytical mechanics. For this reason it is a worthwhile exercise to briefly review the basic formulae for rigid body kinematics and discuss some special problems in greater detail.

A. I. Lurie

3. Theory of finite rotations of rigid bodies

Abstract

A rigid body having a fixed point O is subject to rotation through an angle x about an axis whose direction is given by unit vector e. The direction of e is chosen in such a way that watching from the end of vector e one observes the rotation through a positive angle ≤180°, that is counterclockwise for a right-handed coordinate system.

A. I. Lurie

4. Basic dynamic quantities

Abstract

The kinetic energy of a system of particles is equal to half the sum of the masses multiplied by the velocities squared

$$ T = \frac{1} {2}\sum\limits_{i = 1}^N {m_i v_i ^2 } = \frac{1} {2}\sum\limits_{i = 1}^N {m_i v_i \cdot v_i .}$$

(4.1.1)

We obtain an expression for the kinetic energy by replacing the generalised velocities v_i with the expression given in eq. (1.3.3), i.e.

A. I. Lurie

5. Work and potential energy

Abstract

The sum of the elementary work due to the application of forces F _i at points M _i of a system undergoing virtual displacements δr_i of these points from their positions at a fixed time instant is given by

$$ \delta 'W = \sum\limits_{i = 1}^N {F_i } \cdot\delta r_i .$$

(5.1.1)

The notation δ’ is used to indicate that we are dealing with an infinitesimal quantity which is not a variation of the quantity W.

A. I. Lurie

6. The fundamental equation of dynamics. Analytical statics

Abstract

Constraint forces are the forces exerted at the points in the system when the constraints are mentally removed. Introducing into consideration the constraint forces we distinguish between two categories of forces acting at the points within the system, namely the constraint forces and the active (or prescribed) forces. The resultant of the constraint forces exerted at point M _i is denoted by R _i whereas that of the active forces is denoted by F _i.

A. I. Lurie

7. Lagrange’s differential equations

Abstract

Differential equations of motion for the generalised coordinates can be obtained easily with the help of Lagrange’s central equation. The equations will be derived twice here. The first derivation will assume that the operations d and δ are not interchangeable, while the second one will assume that the operations are interchangeable. In the first case, i.e. if dδ ≠ δd, it is necessary to use Lagrange’s central equation.

A. I. Lurie

8. Other forms of differential equations of motion

Abstract

These equations differing from Lagrange’s equations by introducing quasi-velocities instead of the generalised velocities, were derived by Boltzmann [13] and Hamel [35] at approximately the same time. It was Hamel who suggested the above name for these equations. The equations of motion used by Voronets [91] also deal with the quasi-velocities, however their form differs slightly from the Euler-Lagrange equations.

A. I. Lurie

9. Dynamics of relative motion

Abstract

Let us consider a material system consisting of “carrying” rigid body and N “carried bodies” which are particles whose position with respect to axes Oxyz, fixed in the “carrying body” can be described by a finite or even a countable set (the case of a solid) of the generalised coordinates. While investigating the motion of such a system we can state two problems. The first problem is as follows. The motion of the “carrying body” is prescribed and the motion of the “carried bodies” is required under the assumption that the motion of the “carried bodies” does not affect the prescribed motion of the carrying body. The position of the system can be defined by n independent generalised coordinates q _s, the case of a solid included. Particularly, such a motion can occur when the mass of the carrying body is much greater than the masses of the carried bodies and their influence on the motion of the carrying body can be neglected. For example, the influence of a gyroscope on the motion of the earth can be unconditionally neglected. However the motion of the earth influences the motion of the gyroscope considerably. It is clear that this case can occur when the prescribed motion of the carrying body is caused by external forces. These forces can be determined from the equations of motion.

A. I. Lurie

10. Canonical equations and Jacobi’s theorem

Abstract

Let us consider a continuous function Φ (x _i,..., x _n) of variables x _l,..., x _n which has continuous derivatives of the first and second order with respect to all variables. The transformation from the “old” variables x _l,..., x _n to the “new” ones

$$ y_s = \frac{{\partial \Phi }} {{\partial x_s }}(s = 1,...,n).$$

(10.1.1)

is carried out by means of function Φ referred to as the generating function.

A. I. Lurie

11. Perturbation theory

Abstract

Along with the given system of equations of motion

$$ \dot q_s = \frac{{\partial H}} {{\partial p_s }},{\mkern 1mu} \dot p_s = - \frac{{\partial H}} {{\partial q_s }} + Q_s {\mkern 1mu} \left( {s = {\mkern 1mu} 1, \ldots ,{\mkern 1mu} n} \right)$$

(11.1.1)

we consider an auxiliary (simplified) canonical system

$$ \dot q_s = \frac{{\partial H_0 }} {{\partial p_s }},{\mkern 1mu} \dot p_s {\mkern 1mu} = {\mkern 1mu} - \frac{{\partial H_0 }} {{\partial q_s }}{\mkern 1mu} \left( {s{\mkern 1mu} = {\mkern 1mu} 1, \ldots ,{\mkern 1mu} n} \right).$$

(11.1.2)

A. I. Lurie

12. Variational principles in mechanics

Abstract

The basic statements of dynamics, which are Newton’s axioms and D’Alembert’s principle, allows us to formulate the laws of motion in the form of differential equations of motion. However they do not exhaust all the ways of representing the laws governing the motion of material bodies. An alternative is variational statements dealing with the stationary properties of certain values and enabling the complete replacement of the above statements.

A. I. Lurie

Backmatter

Titel: Analytical Mechanics
verfasst von: A. I. Lurie
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-45677-3
Print ISBN: 978-3-642-53650-2
DOI: https://doi.org/10.1007/978-3-540-45677-3

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Basic definitions

2. Rigid body kinematics — basic knowledge

3. Theory of finite rotations of rigid bodies

4. Basic dynamic quantities

5. Work and potential energy

6. The fundamental equation of dynamics. Analytical statics

7. Lagrange’s differential equations

8. Other forms of differential equations of motion

9. Dynamics of relative motion

10. Canonical equations and Jacobi’s theorem

11. Perturbation theory

12. Variational principles in mechanics

Backmatter