96
3 Computer Simulation of Dynamic Systems
3.2.2 Variational Principles
Along with the fundamental laws for constructing models, variational principles are
used that are comparable with them in terms of breadth of application and universality
[8]. Variational principles are based on the consideration of rather general statements
about the object under study, which state that only those that satisfy a certain condition
are selected from all possible variants of its behavior. Usually, the condition indicates
that the value associated with the object should reach an extreme value when passing
from one state to another. These include, for example, the principle of possible
movements and the principle of least action.
Let us explain the application of variational principles by the example of the
Hamilton principle. To do this, make a brief description of it.
Let there be a mechanical system, the formal and strict definition of which we will
not give yet, bearing in mind, however, that all interactions between the elements of
such a system are determined by the laws of mechanics. We introduce the concept of
generalized coordinates Q(t) that completely determine the position of a mechanical system in space. The quantity Q(t) can be a Cartesian coordinate (e.g., the r
coordinate in the “load on spring” system), radius vector, angular coordinate, a set of
coordinates of material points that make up the system, etc. It is natural to call dQ/dt
the generalized velocity of a mechanical system at time t. The set of quantities Q(t)
and dQ/dt determines the state of the mechanical system at all instants of time.
To describe the mechanical system, the Lagrange function is introduced. In the
simplest cases, the Lagrange function has a clear meaning and is written as
L(Q, dQ/dt) = E k − E p ,
where E k , E p —kinetic and potential energies of the system, respectively. For the
purposes of this problem, there is no need to give a general definition of the quantities
E k , E p , since in the examples considered they are calculated in an obvious way.
We introduce the quantity S[Q], called the action:
S[Q] =
t 2
t 1
L
Q,
dQ
dt
dt
The last integral, obviously, is a functional of the generalized coordinate Q(t), i.e.,
of the function Q(t) defined on the interval [t 1 , t 2 ], he associates a certain number S
(action).
The Hamilton principle for a mechanical system states: If the system moves
according to the laws of mechanics, then Q(t) is a stationary function for S[Q],
or
d
dε
S[Q + εϕ] = 0, ε = 0
3 Computer Simulation of Dynamic Systems
3.2.2 Variational Principles
Along with the fundamental laws for constructing models, variational principles are
used that are comparable with them in terms of breadth of application and universality
[8]. Variational principles are based on the consideration of rather general statements
about the object under study, which state that only those that satisfy a certain condition
are selected from all possible variants of its behavior. Usually, the condition indicates
that the value associated with the object should reach an extreme value when passing
from one state to another. These include, for example, the principle of possible
movements and the principle of least action.
Let us explain the application of variational principles by the example of the
Hamilton principle. To do this, make a brief description of it.
Let there be a mechanical system, the formal and strict definition of which we will
not give yet, bearing in mind, however, that all interactions between the elements of
such a system are determined by the laws of mechanics. We introduce the concept of
generalized coordinates Q(t) that completely determine the position of a mechanical system in space. The quantity Q(t) can be a Cartesian coordinate (e.g., the r
coordinate in the “load on spring” system), radius vector, angular coordinate, a set of
coordinates of material points that make up the system, etc. It is natural to call dQ/dt
the generalized velocity of a mechanical system at time t. The set of quantities Q(t)
and dQ/dt determines the state of the mechanical system at all instants of time.
To describe the mechanical system, the Lagrange function is introduced. In the
simplest cases, the Lagrange function has a clear meaning and is written as
L(Q, dQ/dt) = E k − E p ,
where E k , E p —kinetic and potential energies of the system, respectively. For the
purposes of this problem, there is no need to give a general definition of the quantities
E k , E p , since in the examples considered they are calculated in an obvious way.
We introduce the quantity S[Q], called the action:
S[Q] =
t 2
t 1
L
Q,
dQ
dt
dt
The last integral, obviously, is a functional of the generalized coordinate Q(t), i.e.,
of the function Q(t) defined on the interval [t 1 , t 2 ], he associates a certain number S
(action).
The Hamilton principle for a mechanical system states: If the system moves
according to the laws of mechanics, then Q(t) is a stationary function for S[Q],
or
d
dε
S[Q + εϕ] = 0, ε = 0