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Appendix: All Solutions to the Problems
Chapter 9: Solutions to the Problems in Variational
Principles in Mechanics and Their Applications
9.1 Let a particle of mass m free fall in the uniform gravitational field, find the
equation of motion of the particle with the Hamilton’s principle.
Solution Choosing the y axis vertically upward, the Lagrange function is
L = T − V =
1
2
m ˙
y
2
− mgy
The Hamilton’s action can be expressed as the following functional
J =
t 1
t 0
Ldt =
t 1
t 0
1
2
m ˙
y
2
− mgy
dt
The Euler equation of the functional is
−mg − m ¨
y = 0
or
¨
y + g = 0
9.2 Let a particle of mass m move in the conservative force field, find its motion
equation in spherical coordinates. Assume that the expression V (r, θ, ϕ) of potential
energy V in spherical coordinates is known.
Solution In the spherical coordinate system, the velocities of the r , θ , ϕ three
directions are respectively
v r = r
, v θ = (r sin ϕ)θ
, v ϕ = r ϕ
The kinetic energy and potential energy of the particle is
T =
1
2
m[r
2
+ (r sin ϕθ
)
2
+ (r ϕ
)
2
], V = V (r, θ, ϕ)
The Lagrange function is
L = T − V =
1
2
m[r
2
+ (r sin ϕθ
)
2
+ (r ϕ
)
2
] − V (r, θ, ϕ)
The Hamilton’s action can be expressed as the following functional