This paper deals with the evolution dynamical processes in nonlinear mechanical Lagrangian systems. To understand clearly the nature of the governing evolution equations, we introduce the Hamiltonian function, where. Analogously, the degenerated Hamiltonian yields the linearized equations. The amplitudes of the linear field solution (interpreted as integration constants at) should thus satisfy the following relation, where stands for the Lie-Poisson brackets with appropriate definition of the functional derivatives. In turn, at, the complex amplitudes are slowly varying functions such that. This means that
(1) and,
where the difference can be interpreted as the free energy of the system. So that, if the scalar, then the nonlinear dynamical structure can be spontaneous one, otherwise the system requires some portion of energy to create a structure at, while represents some indifferent case.
Note that the set (1) can be formally rewritten as
(2),
where is a vector function. Using the polar coordinates, eqs. (2) read the following standard form
(3);,
where. In most practical problems the vector function appears as a power series in. This allows one to apply procedures of the normal transformations and the asymptotic methods of investigations.
Parametric approach
As an illustrative example we consider the so-called Bernoulli-Euler model governing the motion of a thin bar, according the following equations [3]:
(4)
with the boundary conditions
В
By scaling the sought variables: and, eqs. (4) are reduced to a standard form (0). p> Notice that the validity range of the model is associated with the wave velocities that should not exceed at least the characteristic speed. In the case of infinitesimal oscillations this set represents two uncoupled linear differential equations. Let, then the linearized equation for longitudinal displacements possesses a simple wave solution
,
where the frequencies are coupled with the wave numbers through the dispersion relation. Notice that. In turn, the linearized equation for bending oscillations reads
(5).
As one can see the right-hand term in eq. (5) contains a spatio-temporal parameter in the form of a standing wave. Allowances for the this wave-like parametric excitation become principal, if the typical velocity of longitudinal waves is comparable with the group velocities of bending waves, otherwise one can restrict consideration, formally assuming that or, to the following simplest model:
(6),
which takes into account the temporal parametric excitation only.
We can look for solutions to eq. (5), using the Bubnov-Galerkin procedure:
,
where denote the wave numbers of bending wav...
