AP07_2-3.vp


In [1] we deal with a class of superintegrable Hamiltonian
systems on a 2n-dimensional phase space with arbitrary
natural n. This class is a natural generalization of the class
presented in [2].

Using the results of [3] on flat coordinates for the metric
tensors associated with the kinetic-energy parts of the
Hamiltonians under study we show that for the two subclasses
of the class in question there exist additional integrals of mo-
tion linear in momenta. In turn, the presence of these addi-
tional integrals enables us to solve the Hamilton-Jacobi and
Schrödinger equations for the systems in question for arbi-
trary sufficiently large n.

The general theory is illustrated by an example of the
Hamiltonian

H p p

q p p q

j n j
j

n

k

k

n

j n k j
j k

n

�

� � �

� �

�

�

�

� � �

� �

�

� �

1
2

1
2

1
1

1

1

1
1

3 2 1 2 1 3q q q� ( )

Acknowledgment
It is my great pleasure to thank Miloslav Znojil for the

warm hospitality and the stimulating atmosphere of the
microconferences.

The author would like to thank the Ministry of Educa-
tion, Youth and Sports of the Czech Republic for supporting
his participation at the microconference under grant MSM
4781305904.

References
[1] Sergyeyev, A.: Exact Solvability of Superintegrable

Benenti Systems, J. Math. Phys. Vol. 48 (2007), 052114;
arXiv: nlin.SI/0701015.

[2] Błaszak, M., Sergyeyev, A.: Maximal Superintegrability
of Benenti Systems, J. Phys. A: Math. Gen. Vol. 38, (2005),
L1–L5; arXiv: nlin.SI/0412018.

[3] Błaszak, M., Sergyeyev, A.: Natural Coordinates for a
Class of Benenti Systems, Phys. Lett. A Vol. 365 (2007)
p. 28–33; arXiv: nlin.SI/0604022.

Doc. RNDr. Artur Sergyeyev, Ph.D.
e-mail: Artur.Sergyeyev@math.slu.cz

Silesian University in Opava
Mathematical Institute
Na Rybníčku 1
746 01 Opava, Czech Republic

20 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 47 No. 2–3/2007

Flat Coordinates and Hidden Symmetry
for Superintegrable Benenti Systems
A. Sergyeyev

In this talk I present the results from my paper Exact solvability of superintegrable Benenti systems, J. Math. Phys. 48 (2007), 052114.

Keywords: superintegrability, Benenti system, Schrödinger equation, Hamilton-Jacobi equation, separation of variables.