# Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability

@article{Kozlov2018LinearHS, title={Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability}, author={Valery V. Kozlov}, journal={Regular and Chaotic Dynamics}, year={2018}, volume={23}, pages={26-46} }

A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its…

## 16 Citations

### Linear Nonautonomous Systems of Differential Equations with a Quadratic Integral

- MathematicsDifferential Equations
- 2021

We consider nonautonomous linear systems of differential equations admitting a time-dependent first integral that is a quadratic form. The duality between mutually adjoint linear systems with…

### First integrals and asymptotic trajectories

- MathematicsSbornik: Mathematics
- 2020

We discuss the relationship between the singular points of an autonomous system of differential equations and the critical points of its first integrals. Applying the well-known Splitting Lemma, we…

### Quadratic conservation laws for equations of mathematical physics

- MathematicsRussian Mathematical Surveys
- 2020

Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the…

### Tensor invariants and integration of differential equations

- MathematicsRussian Mathematical Surveys
- 2019

The connection between tensor invariants of systems of differential equations and explicit integration of them is discussed. A general result on the integrability of dynamical systems admitting a…

### On the Ergodic Theory of Equations of Mathematical Physics

- Mathematics, Physics
- 2021

Abstract Linear evolution equations of mathematical physics admitting an invariant in the form of a positive quadratic form are considered. In particular, this includes the string vibration equation,…

### The Liouville Equation as a Hamiltonian System

- MathematicsMathematical Notes
- 2020

Smooth dynamical systems on closed manifolds with invariant measure are considered. The evolution of the density of a nonstationary invariant measure is described by the well-known Liouville…

### Complete Integrability of Quantum and Classical Dynamical Systems

- Physics, Mathematicsp-Adic Numbers, Ultrametric Analysis and Applications
- 2019

It is proved that the Schrödinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is…

### Partial integrals of ordinary differential systems.

- Mathematics
- 2018

Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of…

### A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness

- PhysicsRegular and Chaotic Dynamics
- 2019

This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is…

### Lax Pairs for Linear Hamiltonian Systems

- MathematicsSiberian Mathematical Journal
- 2019

We construct Lax pairs for linear Hamiltonian systems of differential equations. In particular, the Gröbner bases are used for computations. It is proved that the mappings in the construction of Lax…

## References

SHOWING 1-10 OF 28 REFERENCES

### Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization

- Mathematics
- 2005

We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear…

### On the Hamiltonian property of linear dynamical systems in Hilbert space

- Mathematics
- 2017

Conditions for the operator differential equation $$\dot x = Ax$$x˙=Ax possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional case, it suffices…

### On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems

- Mathematics
- 1936

Introduction. Let m be the number of degrees of freedom of a linear conservative dynamical systenm and let the point (q1, q2,9 * , q'Mn Pl p2, . . . p'mt) of the phase space be denoted by x = (xl,…

### Modern Aspects of Linear Algebra

- Mathematics
- 1998

Introduction: Euclidean linear spaces Orthogonal and unitary linear transformations Orthogonal and unitary transformations. Singular values Matrices of operators in the Euclidean space: Unitary…

### First integrals of ordinary linear differential systems

- Mathematics
- 2012

The spectral method for building first integrals of ordinary linear differential systems is elaborated. Using this method, we obtain bases of first integrals for linear differential systems with…

### Linear Hamiltonian systems are integrable with quadratics

- Mathematics
- 1982

A new proof of a theorem of Williamson on the complete integrability of time‐independent, real, linear Hamiltonian differential equations with quadratic integrals is given. The sets where these…

### On the mechanism of stability loss

- Mathematics
- 2009

We consider linear systems of differential equations admitting functions in the form of quadratic forms that do not increase along trajectories in the course of time. We find new relations between…