512
9 Variational Principles in Mechanics and Their Applications
Principles of mechanics
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
nonvariational
differential(such as D’Alembert principle)
integral(such as conservation of energy principle)
variational
differential(such as priciple of virtual displacement)
integral(such as Hamilton principle)
Of course, the of classification of the above mechanics principles is not the most
essential, because some mechanics principles can have different forms, for example,
the Hertz least curvature principle has both a differential form, but also has the
integral form.
This chapter presents the some most important principles in mechanics, such as
the principle of virtual displacement, the principle of minimum potential energy, the
principle of complementary virtual work, the principle of minimum complementary energy, the Hamilton principle, the Hellinger-Reissner generalized variational
principle, Hu Haichang-Kyuichiro Washizu generalized variational principle and
Maupertuis-Lagrange principle of least action etc., they are the differential variational principles and the integral variational principles respectively. These principles
associated with the variational method, or directly are the component part of the
variational method. Therefore, some fundamental concepts related to the mechanics
are first introduced.
9.1 Fundamental Conceptions in Mechanics
9.1.1 System of Mechanics
A whole composed of similar things according to certain relations is called the
system. The system composed of the material and subjected to the force action is
called the mechanical system. An object is a material entity which occupies a certain
space. The object composing system can be abstractly seen as a point or point system
in mathematical concept. In theoretical mechanics or analytical mechanics, the object
that has a certain mass but can ignore its size is called the particle or mass point. The
system composed of a group of particles with some connection is called a particle
system or system of particles. If each particle in the particle system can occupy
space anywhere and has an arbitrary velocity, namely various particles can freely
move in the space, then such a particle system is called the free particle system
or system of free particles, it is call the free system for short. If the particles in a
particle system are limited by some given conditions in advance, such that they can
not occupy space anywhere and have arbitrary speed, namely various particles can
not freely move in the space, then such a particle system is called the unfree particle
system or system of unfree particles, it is called the unfree system for short. The
particle system, free system and unfree system all belong to mechanical system, they
can all be called the system for short.
9 Variational Principles in Mechanics and Their Applications
Principles of mechanics
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
nonvariational
differential(such as D’Alembert principle)
integral(such as conservation of energy principle)
variational
differential(such as priciple of virtual displacement)
integral(such as Hamilton principle)
Of course, the of classification of the above mechanics principles is not the most
essential, because some mechanics principles can have different forms, for example,
the Hertz least curvature principle has both a differential form, but also has the
integral form.
This chapter presents the some most important principles in mechanics, such as
the principle of virtual displacement, the principle of minimum potential energy, the
principle of complementary virtual work, the principle of minimum complementary energy, the Hamilton principle, the Hellinger-Reissner generalized variational
principle, Hu Haichang-Kyuichiro Washizu generalized variational principle and
Maupertuis-Lagrange principle of least action etc., they are the differential variational principles and the integral variational principles respectively. These principles
associated with the variational method, or directly are the component part of the
variational method. Therefore, some fundamental concepts related to the mechanics
are first introduced.
9.1 Fundamental Conceptions in Mechanics
9.1.1 System of Mechanics
A whole composed of similar things according to certain relations is called the
system. The system composed of the material and subjected to the force action is
called the mechanical system. An object is a material entity which occupies a certain
space. The object composing system can be abstractly seen as a point or point system
in mathematical concept. In theoretical mechanics or analytical mechanics, the object
that has a certain mass but can ignore its size is called the particle or mass point. The
system composed of a group of particles with some connection is called a particle
system or system of particles. If each particle in the particle system can occupy
space anywhere and has an arbitrary velocity, namely various particles can freely
move in the space, then such a particle system is called the free particle system
or system of free particles, it is call the free system for short. If the particles in a
particle system are limited by some given conditions in advance, such that they can
not occupy space anywhere and have arbitrary speed, namely various particles can
not freely move in the space, then such a particle system is called the unfree particle
system or system of unfree particles, it is called the unfree system for short. The
particle system, free system and unfree system all belong to mechanical system, they
can all be called the system for short.