2,293 research outputs found
Nonclassical symmetries as special solutions of heir-equations
In (Nucci M.C. 1994, Physica D 78 p.124), we have found that iterations of
the nonclassical symmetries method give rise to new nonlinear equations, which
inherit the Lie point symmetry algebra of the given equation. In the present
paper, we show that special solutions of the right-order heir-equation
correspond to classical and nonclassical symmetries of the original equations.
An infinite number of nonlinear equations which possess nonclassical symmetries
are derived
Lorenz integrable system moves \`a la Poinsot
A transformation is derived which takes Lorenz integrable system into the
well-known Euler equations of a free-torque rigid body with a fixed point, i.e.
the famous motion \`a la Poinsot. The proof is based on Lie group analysis
applied to two third order ordinary differential equations admitting the same
two-dimensional Lie symmetry algebra. Lie's classification of two-dimensional
symmetry algebra in the plane is used. If the same transformation is applied to
Lorenz system with any value of parameters, then one obtains Euler equations of
a rigid body with a fixed point subjected to a torsion depending on time and
angular velocity. The numerical solution of this system yields a
three-dimensional picture which looks like a "tornado" whose cross-section has
a butterfly-shape. Thus, Lorenz's {\em butterfly} has been transformed into a
{\em tornado}.Comment: 14 pages, 6 figure
Noether symmetries and the quantization of a Lienard-type nonlinear oscillator
The classical quantization of a Lienard-type nonlinear oscillator is achieved
by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011)
that preserves the Noether point symmetries of the underlying Lagrangian in
order to construct the Schr\"odinger equation. This method straightforwardly
yields the correct Schr\"odinger equation in the momentum space (V. Chithiika
Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002,
2012), and sheds light into the apparently remarkable connection with the
linear harmonic oscillator.Comment: 18 page
Lie point symmetries and first integrals: the Kowalevsky top
We show how the Lie group analysis method can be used in order to obtain
first integrals of any system of ordinary differential equations.
The method of reduction/increase of order developed by Nucci (J. Math. Phys.
37, 1772-1775 (1996)) is essential. Noether's theorem is neither necessary nor
considered. The most striking example we present is the relationship between
Lie group analysis and the famous first integral of the Kowalevski top.Comment: 23 page
From Lagrangian to Quantum Mechanics with Symmetries
We present an old and regretfully forgotten method by Jacobi which allows one
to find many Lagrangians of simple classical models and also of nonconservative
systems. We underline that the knowledge of Lie symmetries generates Jacobi
last multipliers and each of the latter yields a Lagrangian. Then it is shown
that Noether's theorem can identify among those Lagrangians the physical
Lagrangian(s) that will successfully lead to quantization. The preservation of
the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger
equation is the key that takes classical mechanics into quantum mechanics.
Some examples are presented.Comment: To appear in: Proceedings of Symmetries in Science XV, Journal of
Physics: Conference Series, (2012
Gauge Variant Symmetries for the Schr\"odinger Equation
The last multiplier of Jacobi provides a route for the determination of
families of Lagrangians for a given system. We show that the members of a
family are equivalent in that they differ by a total time derivative. We derive
the Schr\"odinger equation for a one-degree-of-freedom system with a constant
multiplier. In the sequel we consider the particular example of the simple
harmonic oscillator. In the case of the general equation for the simple
harmonic oscillator which contains an arbitrary function we show that all
Schr\"odinger equations possess the same number of Lie point symmetries with
the same algebra. From the symmetries we construct the solutions of the
Schr\"odinger equation and find that they differ only by a phase determined by
the gauge.Comment: 12 page
- …