9.6 The Hamilton Principles and Their Applications
545
then the constraint is called the idea constraint or idealized constraint. Where, R i
is the reactive force of constraint acting on any particle i in the system; δr i is an
arbitrary virtual displacement of the particle.
Let the system composed of n particles be subjected to the idea constraint and be
in a state of motion, in which the resultant force of the active forces acting on the i
particle P i is F i , the resultant force of constraint reactive forces is R i , the mass is
m i , the acceleration is a i , and at the initial time t 0 and final time t 1 , the actual path
and nearby path of the system are in the same position. According to the Newton
second law, at any instant, there is
m i a i = F i + R i (i = 1, 2, . . . , n)
(9.6.2)
Equation (9.6.2) can be rewritten as
F i − m i a i + R i = 0 (i = 1, 2, . . . , n)
(9.6.3)
Equation (9.6.3) shows that at any instant, the active force F i , constraint reactive
force R i and inertia force −m i a i acting on every particle in the particle system
constitute an equilibrium force system, which is called the D’Alembert(’s) principle
of a particle system. At the instant, giving the system an arbitrary virtual displacement
δr i (i = 1, 2, . . . , n) and summarising, because the system is subjected to the ideal
constraints, there is
n
i=1
(F i − m i a i ) · δr i =
n
i=1
F i · δr i +
n
i=1
(−m i a i ) · δr i = 0
(9.6.4)
Equation (9.6.4) is called the general equation of dynamics, or is called the
D’Alembert-Lagrange(’s) equation, sometimes it is called the Lagrangian form
of D’Alembert(’s) principle. This equation can be expressed as: When a particle
system with ideal constraints moves, at any moment, the sum of elementary
works done by the active force and inertia force on an arbitrary virtual displacement is zero. All the theorems and equations of dynamics can be derived by the
general equation of dynamics. Obviously, in the general equation of dynamics, the
constraint reactive force of the idea constraint does not appear.
The first term of Eq. (9.6.4) can be written as
δ
W =
n
i=1
F i · δr i
(9.6.5)
where, δ
W is the elementary work done by the given system of forces on the virtual
displacement namely virtual work. It should be pointed out that the resultant force
F i of the active forces acting on the i-th particle may be dependent on r i , ˙
r i and
t, but the expression of virtual work does not include the term with δ ˙
r i . Hence the
virtual work δ
W in general does not mean the variation of the total work W, That is
545
then the constraint is called the idea constraint or idealized constraint. Where, R i
is the reactive force of constraint acting on any particle i in the system; δr i is an
arbitrary virtual displacement of the particle.
Let the system composed of n particles be subjected to the idea constraint and be
in a state of motion, in which the resultant force of the active forces acting on the i
particle P i is F i , the resultant force of constraint reactive forces is R i , the mass is
m i , the acceleration is a i , and at the initial time t 0 and final time t 1 , the actual path
and nearby path of the system are in the same position. According to the Newton
second law, at any instant, there is
m i a i = F i + R i (i = 1, 2, . . . , n)
(9.6.2)
Equation (9.6.2) can be rewritten as
F i − m i a i + R i = 0 (i = 1, 2, . . . , n)
(9.6.3)
Equation (9.6.3) shows that at any instant, the active force F i , constraint reactive
force R i and inertia force −m i a i acting on every particle in the particle system
constitute an equilibrium force system, which is called the D’Alembert(’s) principle
of a particle system. At the instant, giving the system an arbitrary virtual displacement
δr i (i = 1, 2, . . . , n) and summarising, because the system is subjected to the ideal
constraints, there is
n
i=1
(F i − m i a i ) · δr i =
n
i=1
F i · δr i +
n
i=1
(−m i a i ) · δr i = 0
(9.6.4)
Equation (9.6.4) is called the general equation of dynamics, or is called the
D’Alembert-Lagrange(’s) equation, sometimes it is called the Lagrangian form
of D’Alembert(’s) principle. This equation can be expressed as: When a particle
system with ideal constraints moves, at any moment, the sum of elementary
works done by the active force and inertia force on an arbitrary virtual displacement is zero. All the theorems and equations of dynamics can be derived by the
general equation of dynamics. Obviously, in the general equation of dynamics, the
constraint reactive force of the idea constraint does not appear.
The first term of Eq. (9.6.4) can be written as
δ
W =
n
i=1
F i · δr i
(9.6.5)
where, δ
W is the elementary work done by the given system of forces on the virtual
displacement namely virtual work. It should be pointed out that the resultant force
F i of the active forces acting on the i-th particle may be dependent on r i , ˙
r i and
t, but the expression of virtual work does not include the term with δ ˙
r i . Hence the
virtual work δ
W in general does not mean the variation of the total work W, That is