Chapter 9
Variational Principles in Mechanics
and Their Applications
The most basic and the most common law in mechanics is called the principle
of mechanics, mechanical principle or mechanics principle. That the variational
method in mathematics is applied to a mechanical principle is called a variational
principle of mechanics or variational principle in mechanics. Since the functional describing variational principle of mechanics often relates to the energy of the
mechanical system, the variational principle of mechanics is also called the energy
principle, the corresponding various variational methods of soling are called energy
method. The principles of mechanics in the form of the mathematical expression can
be divided into two categories, the nonvariational principles and variational principles, each category is divided into two forms of differential and integral. The correctness of these principles of mechanics is the consistency of the same objective fact
described by them, is the consistency with the experimental facts of the various rules
and the derived conclusions described by them.
The mechanics principle of the universal law of directly describing the real movement of a system is called the nonvariational principle of mechanics or nonvariational principle in mechanics, it is called the nonvariational principle for short.
If a mechanics principle does not directly describe the universal law of the real
motion of a system, but to provide a criteria, according to the criteria, the real motion
under the action of force of the system may be distinguished from the possible
other motions allowed by the constraints, thereby the real motion of the system
can be determined, the mechanics principle is called the variational principle of
mechanics or variational principle in mechanics, it is called variational principle
or variation principle for short. For some variational principles, the criteria is: For
a real motion, a physical quantity (namely functional) composed of the motion state
parameters of the system has extremum or stationary value. If a mechanics principle
only reflects a transient motion law of the system, then the mechanics principle is
called the differential variational principle. If a mechanics principle can describe
the motion process of a limited time, then the mechanics principle is called the
integral variational principle. Thus, the mechanics principle can be classified as
follows
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_9
511
Variational Principles in Mechanics
and Their Applications
The most basic and the most common law in mechanics is called the principle
of mechanics, mechanical principle or mechanics principle. That the variational
method in mathematics is applied to a mechanical principle is called a variational
principle of mechanics or variational principle in mechanics. Since the functional describing variational principle of mechanics often relates to the energy of the
mechanical system, the variational principle of mechanics is also called the energy
principle, the corresponding various variational methods of soling are called energy
method. The principles of mechanics in the form of the mathematical expression can
be divided into two categories, the nonvariational principles and variational principles, each category is divided into two forms of differential and integral. The correctness of these principles of mechanics is the consistency of the same objective fact
described by them, is the consistency with the experimental facts of the various rules
and the derived conclusions described by them.
The mechanics principle of the universal law of directly describing the real movement of a system is called the nonvariational principle of mechanics or nonvariational principle in mechanics, it is called the nonvariational principle for short.
If a mechanics principle does not directly describe the universal law of the real
motion of a system, but to provide a criteria, according to the criteria, the real motion
under the action of force of the system may be distinguished from the possible
other motions allowed by the constraints, thereby the real motion of the system
can be determined, the mechanics principle is called the variational principle of
mechanics or variational principle in mechanics, it is called variational principle
or variation principle for short. For some variational principles, the criteria is: For
a real motion, a physical quantity (namely functional) composed of the motion state
parameters of the system has extremum or stationary value. If a mechanics principle
only reflects a transient motion law of the system, then the mechanics principle is
called the differential variational principle. If a mechanics principle can describe
the motion process of a limited time, then the mechanics principle is called the
integral variational principle. Thus, the mechanics principle can be classified as
follows
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_9
511