548
9 Variational Principles in Mechanics and Their Applications
is called the action for short. The action is the amount with dimension of energy
multiplied by time or momentum multiplied by the length. Hamilton’s principle
involves only two dynamics functions, namely the kinetic energy and potential energy
of the system.
If T, V and L are respectively regarded as the kinetic energy density (that is
the kinetic energy per unit volume) of a particle system at the moment t, potential
energy density (that is the potential energy per unit volume) and the Lagrange density
function, the Hamilton’s principle can be written in the following form
δ J = δ
t 1
t 0
˚
V
(T − V )dV dt = δ
t 1
t 0
˚
V
LdV dt = 0
(9.6.16)
where, V under the differential sign is the space domain occupied by the particle
system.
Using the Hamilton’s principle, the differential equation satisfied by the real
motion of a particle system can be derived. Below to illustrate the Hamilton principle
through several examples of conservative force field.
Consider the unconstrained particle system composed of the n particles, the masses
of the particles are m 1 , m 2 , …, m n respectively, in which the coordinates of the i-th
particle m i are (x i , y i , z i ), (i = 1, 2, …, n), the kinetic energy of the particle system
is
T =
1
2
n
i=1
m i [ ˙
x
2
i (t) + ˙
y
2
i (t) + ˙
z
2
i (t)]
(9.6.17)
The potential energy function of the i-th particle is expressed in terms of V =
V (t, x i , y i , z i ), the force of the potential function acting on the particle
F i = F x i i + F y i j + F z i k = −∇V
(9.6.18)
where
F x i = −
∂ V
∂ x i
, F y i = −
∂ V
∂ y i
, F z i = −
∂ V
∂z i
(i = 1, 2, . . . , n)
(9.6.19)
The motion of the particle system is described by Newton’s equations, namely
the motion equations of various particles
⎧
⎪ ⎨
⎪ ⎩
x i = x i (t)
y i = y i (t)
z i = z i (t)
(t 0 ≤ t ≤ t 1 , i = 1, 2, . . . , n)
(9.6.20)
The particle should satisfy the following system of differential equations
9 Variational Principles in Mechanics and Their Applications
is called the action for short. The action is the amount with dimension of energy
multiplied by time or momentum multiplied by the length. Hamilton’s principle
involves only two dynamics functions, namely the kinetic energy and potential energy
of the system.
If T, V and L are respectively regarded as the kinetic energy density (that is
the kinetic energy per unit volume) of a particle system at the moment t, potential
energy density (that is the potential energy per unit volume) and the Lagrange density
function, the Hamilton’s principle can be written in the following form
δ J = δ
t 1
t 0
˚
V
(T − V )dV dt = δ
t 1
t 0
˚
V
LdV dt = 0
(9.6.16)
where, V under the differential sign is the space domain occupied by the particle
system.
Using the Hamilton’s principle, the differential equation satisfied by the real
motion of a particle system can be derived. Below to illustrate the Hamilton principle
through several examples of conservative force field.
Consider the unconstrained particle system composed of the n particles, the masses
of the particles are m 1 , m 2 , …, m n respectively, in which the coordinates of the i-th
particle m i are (x i , y i , z i ), (i = 1, 2, …, n), the kinetic energy of the particle system
is
T =
1
2
n
i=1
m i [ ˙
x
2
i (t) + ˙
y
2
i (t) + ˙
z
2
i (t)]
(9.6.17)
The potential energy function of the i-th particle is expressed in terms of V =
V (t, x i , y i , z i ), the force of the potential function acting on the particle
F i = F x i i + F y i j + F z i k = −∇V
(9.6.18)
where
F x i = −
∂ V
∂ x i
, F y i = −
∂ V
∂ y i
, F z i = −
∂ V
∂z i
(i = 1, 2, . . . , n)
(9.6.19)
The motion of the particle system is described by Newton’s equations, namely
the motion equations of various particles
⎧
⎪ ⎨
⎪ ⎩
x i = x i (t)
y i = y i (t)
z i = z i (t)
(t 0 ≤ t ≤ t 1 , i = 1, 2, . . . , n)
(9.6.20)
The particle should satisfy the following system of differential equations