5.3 Space-Time Reference Based on Newtonian Mechanics
283
where r and r
stand for the position vectors of the particle at the K-system and
K’-system, respectively; u stands for the velocity of the coordinate origin of the
K’-system relative to the K-system, with an uniform motion.
When t = 0, the coordinate origins of the two inertial systems are coincident, the
coordinate axes of the K’-system are always parallel to the corresponding coordinate
axes of the K-system, that is, the K’-system is making the translational motion.
Similarly, the velocity and acceleration of the particle in the K’-system are expressed
respectively in vector form as
v
=
d r
dt
= v − u
a
=
d
2 r
dt 2 = a
.
(5.6)
In two inertial systems which move uniformly in a straight line, it can be seen
that the acceleration of a particle is an invariant.
In general, the origin O’ of the K’-system not only makes a curve motion relative
to the K-system, but also the K’-system itself rotates relative to the K-system. That is
to say, the coordinate axes between the two reference systems keep parallel no longer.
Thereby, it is needed to introduce a middle reference system K m , whose coordinate
origin O m coincides with the origin O’ of the K’-system, and the K m -system only
makes the translational motion relative to the K-system. In this way, the motion law
of the particle is first transformed from the inertial system K to the middle reference
system K m , and then from the K m -system to the K’-system. So, it is only needed
to consider the rotation transformation between two reference systems, and then
combining the translation relationship between the reference systems, the general
motion law of a particle can be gotten in different reference systems.
For single particle, its rotation motions between two different reference systems
have been discussed in the above. For a particle system, its fixed-point rotation is
investigated, which is the problem of fixed-point rotation of a rigid body, as described
in Sect. 4.7 of Chap. 4. In general, a free rigid body has six Degrees of Freedom
(DOF), which needs six generalized coordinates to describe its position, and the
position of the rigid body can be determined completely by the coordinates of any
three non-collinear points. When a point on the free rigid body is selected as a fixed
point, three constraints are added. It becomes the problem of fixed-point rotation
of a rigid body with three DOFs, and thus the motion of the rigid body can be
described by using the generalized coordinates composed of three Euler angles,
which is known as Euler’s kinematical equation. For the problem of fixed-point
motion of rigid body under external force moment, Euler’s dynamical equation is
used to express. Consequently, the nonlinear second-order differential equations can
be obtained, composed of Euler’s kinematical and dynamical equations for the fixedpoint motion of a rigid body [24]. When the external moment is known, the integral
calculation is carried out for the differential equations, and the relationship of three
Euler angles changing over time can be obtained, so as to master the fixed-point
motion law of the rigid body.
283
where r and r
stand for the position vectors of the particle at the K-system and
K’-system, respectively; u stands for the velocity of the coordinate origin of the
K’-system relative to the K-system, with an uniform motion.
When t = 0, the coordinate origins of the two inertial systems are coincident, the
coordinate axes of the K’-system are always parallel to the corresponding coordinate
axes of the K-system, that is, the K’-system is making the translational motion.
Similarly, the velocity and acceleration of the particle in the K’-system are expressed
respectively in vector form as
v
=
d r
dt
= v − u
a
=
d
2 r
dt 2 = a
.
(5.6)
In two inertial systems which move uniformly in a straight line, it can be seen
that the acceleration of a particle is an invariant.
In general, the origin O’ of the K’-system not only makes a curve motion relative
to the K-system, but also the K’-system itself rotates relative to the K-system. That is
to say, the coordinate axes between the two reference systems keep parallel no longer.
Thereby, it is needed to introduce a middle reference system K m , whose coordinate
origin O m coincides with the origin O’ of the K’-system, and the K m -system only
makes the translational motion relative to the K-system. In this way, the motion law
of the particle is first transformed from the inertial system K to the middle reference
system K m , and then from the K m -system to the K’-system. So, it is only needed
to consider the rotation transformation between two reference systems, and then
combining the translation relationship between the reference systems, the general
motion law of a particle can be gotten in different reference systems.
For single particle, its rotation motions between two different reference systems
have been discussed in the above. For a particle system, its fixed-point rotation is
investigated, which is the problem of fixed-point rotation of a rigid body, as described
in Sect. 4.7 of Chap. 4. In general, a free rigid body has six Degrees of Freedom
(DOF), which needs six generalized coordinates to describe its position, and the
position of the rigid body can be determined completely by the coordinates of any
three non-collinear points. When a point on the free rigid body is selected as a fixed
point, three constraints are added. It becomes the problem of fixed-point rotation
of a rigid body with three DOFs, and thus the motion of the rigid body can be
described by using the generalized coordinates composed of three Euler angles,
which is known as Euler’s kinematical equation. For the problem of fixed-point
motion of rigid body under external force moment, Euler’s dynamical equation is
used to express. Consequently, the nonlinear second-order differential equations can
be obtained, composed of Euler’s kinematical and dynamical equations for the fixedpoint motion of a rigid body [24]. When the external moment is known, the integral
calculation is carried out for the differential equations, and the relationship of three
Euler angles changing over time can be obtained, so as to master the fixed-point
motion law of the rigid body.