11.3 Stability with Conservative and Dissipative Forces
123
Thus, it turns out that in the considered example, the work of force performed
during the movement of the body depends on the method in which the movement
is performed. Forces with the latter property are called n o n - c o n s e r v a t i v e.
Forces whose work is determined only by the initial and final position of the system
are called c o n s e r v a t i v e. By definition, the potential (potential energy) of
such forces is the work done by them on the movements of the body from the final
position to the initial. In this case, the initial position can be arbitrarily selected. It
follows that the potential energy of the system is determined with an accuracy up to
a constant.
In addition to conservative and non-conservative forces in real mechanical
systems, there are always forces whose work during actual motion is negative
(for example, friction forces). Such forces are called d i s s i p a t i v e. They are
a special case of non-conservative forces.
11.3 Stability with Conservative and Dissipative Forces
Lagrange theorem 1 A system subject to the action of only conservative and
dissipative forces will be stable if in the equilibrium position the potential energy of
conservative forces has a minimum.
Proof The position of the system will be determined by the generalized coordinates
q 1 , q 2 , . . . , q n , where n is the number of degrees of freedom of the system. We
assume that in the equilibrium position q 1 = q 2 = . . . = q n = 0. The potential
of the conservative forces (q 1 , q 2 , . . . , q n ) in the equilibrium position is also
assumed to be zero, i.e. (0, 0, . . . , 0) = 0.
Suppose that the potential of conservative forces is monotonously increasing, but
this increase cannot be unlimited. After the potential reaches a certain value 1 ,
it can decrease. There may be several such values of 1 . The smaller of them is
denoted by E and is called potential barrier. The range of generalized coordinates
in the vicinity of the equilibrium under study, in which the potential of conservative
forces does not reach the potential barrier E, is called the minimum region of
the function , or potential well. Under the above assumptions, the function P i
vanishes only at the point (q 1 , q 2 , . . . , q n = 0), if the system does not go out of
the potential well.
We derive the system from the considered equilibrium position by communicating to its elements such velocities at which the kinetic energy (T ) does not reach the
potential barrier E. From the kinetic energy theorem [4] in this case, it follows that
the potential energy will always be less than E, i.e. the system cannot go beyond the
potential barrier and will be in a potential well. With an unlimited decrease in the
1 The wording given here is borrowed from [3] and is somewhat different from the generally
accepted ones.
123
Thus, it turns out that in the considered example, the work of force performed
during the movement of the body depends on the method in which the movement
is performed. Forces with the latter property are called n o n - c o n s e r v a t i v e.
Forces whose work is determined only by the initial and final position of the system
are called c o n s e r v a t i v e. By definition, the potential (potential energy) of
such forces is the work done by them on the movements of the body from the final
position to the initial. In this case, the initial position can be arbitrarily selected. It
follows that the potential energy of the system is determined with an accuracy up to
a constant.
In addition to conservative and non-conservative forces in real mechanical
systems, there are always forces whose work during actual motion is negative
(for example, friction forces). Such forces are called d i s s i p a t i v e. They are
a special case of non-conservative forces.
11.3 Stability with Conservative and Dissipative Forces
Lagrange theorem 1 A system subject to the action of only conservative and
dissipative forces will be stable if in the equilibrium position the potential energy of
conservative forces has a minimum.
Proof The position of the system will be determined by the generalized coordinates
q 1 , q 2 , . . . , q n , where n is the number of degrees of freedom of the system. We
assume that in the equilibrium position q 1 = q 2 = . . . = q n = 0. The potential
of the conservative forces (q 1 , q 2 , . . . , q n ) in the equilibrium position is also
assumed to be zero, i.e. (0, 0, . . . , 0) = 0.
Suppose that the potential of conservative forces is monotonously increasing, but
this increase cannot be unlimited. After the potential reaches a certain value 1 ,
it can decrease. There may be several such values of 1 . The smaller of them is
denoted by E and is called potential barrier. The range of generalized coordinates
in the vicinity of the equilibrium under study, in which the potential of conservative
forces does not reach the potential barrier E, is called the minimum region of
the function , or potential well. Under the above assumptions, the function P i
vanishes only at the point (q 1 , q 2 , . . . , q n = 0), if the system does not go out of
the potential well.
We derive the system from the considered equilibrium position by communicating to its elements such velocities at which the kinetic energy (T ) does not reach the
potential barrier E. From the kinetic energy theorem [4] in this case, it follows that
the potential energy will always be less than E, i.e. the system cannot go beyond the
potential barrier and will be in a potential well. With an unlimited decrease in the
1 The wording given here is borrowed from [3] and is somewhat different from the generally
accepted ones.