Integrals of motion

In mechanics function $I=I(q,{\dot {q}}),$ ${\ displaystyle I = I (q, {\ dot {q}}),}$ $I = I (q, \ dot q),$ Where $q$ ${\ displaystyle q}$ $q$ - generalized coordinates ${\dot {q}}$ ${\ displaystyle {\ dot {q}}}$ $\ dot q$ - the generalized velocity of a system, is called the integral of motion (of this system), if $I(q,{\dot {q}})=\mathrm {const}$ ${\ displaystyle I (q, {\ dot {q}}) = \ mathrm {const}}$ $I (q, \ dot q) = \ mathrm {const}$ on each trajectory $q(t)$ ${\ displaystyle q (t)}$ $q (t)$ this system but the function $I(q,{\dot {q}})$ ${\ displaystyle I (q, {\ dot {q}})}$ $I (q, \ dot q)$ is not identically constant.

Integrals of motion with additivity or asymptotic additivity are called conservation laws .

Integrals of motion in classical mechanics

In classical mechanics for a closed system of $N$ ${\ displaystyle N}$ $N$ particles in three-dimensional space, between which there are no rigid connections, it is possible to form $6N-1$ ${\ displaystyle 6N-1}$ $6N-1$ independent integrals of motion are the first integrals of the corresponding system of Hamilton equations . Three of them are additive: energy , impulse , angular momentum ^[1] .

Application

Integrals of motion are useful because some properties of this motion can be learned even without integrating the equations of motion. In the most successful cases, the trajectories of motion represent the intersection of the isosurfaces of the corresponding integrals of motion. For example, the construction of Poinsot shows that without a torque, the rotation of a rigid body is the intersection of a sphere (the preservation of the total angular momentum) and an ellipsoid (energy conservation) - a trajectory that is difficult to draw and visualize. Therefore, finding the integrals of motion is an important goal in mechanics .

Methods for finding integrals of motion

There are several methods for finding the integrals of motion:

The simplest, but also the least rigorous method is an intuitive approach, often based on experimental data and subsequent mathematical proof of the conservation of magnitude.

The Hamilton-Jacobi equation offers a strict and direct method for finding the integrals of motion, especially if the Hamiltonian takes on a familiar functional form in orthogonal coordinates .

Another approach is to compare the conserved magnitude and some Lagrangian symmetry . Noether's theorem gives a systematic way to derive such quantities from symmetries. For example, the energy conservation law is a result of the fact that the Lagrangian does not change with respect to the time shift, the law of conservation of momentum is equivalent to the invariance of the Lagrangian with respect to the shift of the origin of coordinates in space ( translational symmetry ) and the law of conservation of angular momentum follows from the isotropy of space (the Lagrangian does not change coordinates). The reverse is also true: each symmetry of the Lagrangian corresponds to an integral of motion.

Magnitude $A$ ${\ displaystyle A}$ is preserved if it does not explicitly depend on time and its Poisson bracket with the Hamiltonian of the system is zero

{\frac {dA}{dt}}={\frac {\partial A}{\partial t}}+[A,H]=0

{\ displaystyle {\ frac {dA} {dt}} = {\ frac {\ partial A} {\ partial t}} + [A, H] = 0}

Another useful result is known as the Poisson theorem , which states that if there are two integrals of motion $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ then the Poisson brackets $[A,B]$ ${\ displaystyle [A, B]}$ of these two quantities is also an integral of motion, subject to obtaining an expression independent of the integrals.

A system with n degrees of freedom and n integrals of motion such that the Poisson brackets of any pair of integrals are zero is known as a fully integrable system . Such a set of integrals of motion is said to be in involution with each other.

In fluid dynamics

With a free (without external forces) motion of an ideal (no dissipation, no viscosity) incompressible (the volume of any part is preserved), the following values are stored:

kinetic energy $\int \limits _{V}{\vec {v}}^{2}\,dV$ ${\ displaystyle \ int \ limits _ {V} {\ vec {v}} ^ {2} \, dV}$ (see also the Bernoulli integral )
hydrodynamic helicity $\int \limits _{V}{\vec {v}}\cdot \operatorname {rot} \,{\vec {v}}\,dV$ ${\ displaystyle \ int \ limits _ {V} {\ vec {v}} \ cdot \ operatorname {rot} \, {\ vec {v}} \, dV}$

If the motion is two-dimensional, then the enstrophy also remains $\int _{S}\left(\nabla \times {\vec {v}}\right)^{2}dS$ ${\ displaystyle \ int _ {S} \ left (\ nabla \ times {\ vec {v}} \ right) ^ {2} dS}$ .

In ideal magnetic hydrodynamics, the first integral (total energy, as the sum of the kinetic energy of a liquid and the energy of a magnetic field) is preserved, the second (hydrodynamic helicity) disappears, but two other integrals of motion appear:

Magnetic field coil - $\int \limits _{V}{\vec {A}}\cdot \operatorname {rot} \,{\vec {A}}$ ${\ displaystyle \ int \ limits _ {V} {\ vec {A}} \ cdot \ operatorname {rot} \, {\ vec {A}}}$ where ${\vec {A}}$ ${\ displaystyle {\ vec {A}}}$ - vector magnetic potential .
Cross Helicity - $\int \limits _{V}{\vec {v}}\cdot {\vec {B}}$ ${\ displaystyle \ int \ limits _ {V} {\ vec {v}} \ cdot {\ vec {B}}}$

In quantum mechanics

The observed value of Q is preserved if it commutes with the Hamiltonian H , which does not explicitly depend on time. therefore

{\frac {d}{dt}}\langle \psi |{\hat {Q}}|\psi \rangle ={\frac {i}{\hbar }}\langle \psi |\left[{\hat {\mathcal {H}}},{\hat {Q}}\right]|\psi \rangle +\langle \psi |{\frac {\partial {\hat {Q}}}{\partial t}}|\psi \rangle

{\ displaystyle {\ frac {d} {dt}} \ langle \ psi | {\ hat {Q}} | \ psi \ rangle = {\ frac {i} {\ hbar}} \ langle \ psi | \ left [ {\ hat {\ mathcal {H}}}, {\ hat {Q}} \ right] | \ psi \ rangle + \ langle \ psi | {\ frac {\ partial {\ hat {Q}}} {\ partial t}} | \ psi \ rangle}

where switching relation is used

[{\hat {\mathcal {H}}},{\hat {Q}}]={\hat {\mathcal {H}}}{\hat {Q}}-{\hat {Q}}{\hat {\mathcal {H}}}

{\ displaystyle [{\ hat {\ mathcal {H}}}, {\ hat {Q}}] = {\ hat {\ mathcal {H}}} {\ hat {Q}} - {\ hat {Q} } {\ hat {\ mathcal {H}}}}

.

Conclusion

Let there be some observable Q , which depends on the coordinate, momentum and time

Q=Q(x,p,t)

{\ displaystyle Q = Q (x, p, t)}

and there is also a wave function , which is a solution to the corresponding Schrödinger equation

i\hbar {\frac {\partial \psi }{\partial t}}={\hat {\mathcal {H}}}\psi .

{\ displaystyle i \ hbar {\ frac {\ partial \ psi} {\ partial t}} = {\ hat {\ mathcal {H}}} \ psi.}

To calculate the time derivative of the average value of observable Q , the rule of product differentiation is used, and the result after some manipulations is given below.

{\frac {d}{dt}}\langle Q\rangle \,={\frac {d}{dt}}\langle \psi |{\hat {Q}}|\psi \rangle \,=

{\ displaystyle {\ frac {d} {dt}} \ langle Q \ rangle \, = {\ frac {d} {dt}} \ langle \ psi | {\ hat {Q}} | \ psi \ rangle \, =}

=\langle {\frac {\partial \psi }{\partial t}}|{\hat {Q}}|\psi \rangle +\langle \psi |{\frac {\partial {\hat {Q}}}{\partial t}}|\psi \rangle +\langle \psi |{\hat {Q}}|{\frac {\partial \psi }{\partial t}}\rangle \,=

{\ displaystyle = \ langle {\ frac {\ partial \ psi} {\ partial t}} | {\ hat {Q}} | \ psi \ rangle + \ langle \ psi | {\ frac {\ partial {\ hat { Q}}} {\ partial t}} | \ psi \ rangle + \ langle \ psi | {\ hat {Q}} | {\ frac {\ partial \ psi} {\ partial t}} \ rangle \, =}

={\frac {i}{\hbar }}\langle {\hat {\mathcal {H}}}\psi |{\hat {Q}}|\psi \rangle +\langle \psi |{\frac {\partial {\hat {Q}}}{\partial t}}|\psi \rangle -{\frac {i}{\hbar }}\langle \psi |{\hat {Q}}|{\hat {\mathcal {H}}}\psi \rangle \,=

{\ displaystyle = {\ frac {i} {\ hbar}} \ langle {\ hat {\ mathcal {H}}} \ psi | {\ hat {Q}} | \ psi \ rangle + \ langle \ psi | { \ frac {\ partial {\ hat {Q}}} {\ partial t}} | \ psi \ rangle - {\ frac {i} {\ hbar}} \ langle \ psi | {\ hat {Q}} | { \ hat {\ mathcal {H}}} \ psi \ rangle \, =}

={\frac {i}{\hbar }}\langle \psi |{\hat {\mathcal {H}}}{\hat {Q}}|\psi \rangle +\langle \psi |{\frac {\partial {\hat {Q}}}{\partial t}}|\psi \rangle -{\frac {i}{\hbar }}\langle \psi |{\hat {Q}}{\hat {\mathcal {H}}}|\psi \rangle \,=

{\ displaystyle = {\ frac {i} {\ hbar}} \ langle \ psi | {\ hat {\ mathcal {H}}} {\ hat {Q}} | \ psi \ rangle + \ langle \ psi | { \ frac {\ partial {\ hat {Q}}} {\ partial t}} | \ psi \ rangle - {\ frac {i} {\ hbar}} \ langle \ psi | {\ hat {Q}} {\ hat {\ mathcal {H}}} | \ psi \ rangle \, =}

={\frac {i}{\hbar }}\langle \psi |\left[{\hat {\mathcal {H}}},{\hat {Q}}\right]|\psi \rangle +\langle \psi |{\frac {\partial {\hat {Q}}}{\partial t}}|\psi \rangle \,=

{\ displaystyle = {\ frac {i} {\ hbar}} \ langle \ psi | \ left [{\ hat {\ mathcal {H}}}, {\ hat {Q}} \ right] | \ psi \ rangle + \ langle \ psi | {\ frac {\ partial {\ hat {Q}}} {\ partial t}} | \ psi \ rangle \, =}

As a result, we get

{\frac {d}{dt}}{\hat {Q}}={\frac {i}{\hbar }}\left[{\hat {\mathcal {H}}},{\hat {Q}}\right]+{\frac {\partial {\hat {Q}}}{\partial t}}

{\ displaystyle {\ frac {d} {dt}} {\ hat {Q}} = {\ frac {i} {\ hbar}} \ left [{hat {\ mathcal {H}}}, {\ hat {Q}} \ right] + {\ frac {\ partial {\ hat {Q}}} {\ partial t}}}

Relation to quantum chaos and quantum integrability

In classical mechanics there is a Liouville theorem , according to which a system in which the number of integrals of motion in an involution coincides with the number of degrees of freedom $n$ ${\ displaystyle n}$ , can be fully integrated (solved) by the method of separation of variables in the Hamilton – Jacobi equation. Such a system is an integrable system . The trajectory of such a system in $2n$ ${\ displaystyle 2n}$ -dimensional phase space can be represented in suitable variables ( action-angle variables ) as winding on $n$ ${\ displaystyle n}$ -dimensional torus. The system, the number of integrals in which is less than the number of degrees of freedom, exhibits a chaotic behavior , that is, trajectories in the phase space with close initial conditions can diverge exponentially. With a small deformation of the integrable system into non-integrable $n$ ${\ displaystyle n}$ -dimensional torus $2n$ ${\ displaystyle 2n}$ -dimensional phase space is destroyed ("blurred"), turning, for example, into a strange attractor .

The quantum analogue of the Liouville theorem is unknown, however, in the quantum case, the systems can be divided into integrable and nonintegrable. In this case, integrable means systems that allow an exact solution, in the sense of being able to find all eigenvalues and eigenfunctions of the Hamiltonian in a reasonable form. The quantum analogue of the method of separation of variables is known, but its application is not so universal in classical cases. Famous examples show that in quantum integrable systems, as well as in classical ones, there are $n$ ${\ displaystyle n}$ integrals of motion commuting with each other. However availability $n$ ${\ displaystyle n}$ integrals of motion, apparently, does not yet guarantee quantum integrability. The quantization problem for integrable systems is the search for such a quantum system that would allow an exact solution and would give a given classical system in the classical limit. There are also examples of integrable quantum systems that have no integrable classical analogs. This happens if the system can be solved with special values of the parameters of the quantum Hamiltonian , or when the system does not allow the classical description (such as the system of spins ).

All other quantum systems exhibit signs of quantum chaos to one degree or another. Classical chaotic systems allow quantization in the sense that their state space and Hamiltonian can be correctly defined, however, like classical chaotic systems , quantum systems , apparently, do not allow an exact solution. They can be investigated by approximate methods, such as perturbation theory and the variational method , and also investigated numerically by molecular dynamics methods in the classical case or in the numerical diagonalization of the Hamiltonian in the quantum case.

Notes

↑ Saveliev, 1987 , p. 74.

Literature

Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). - Prentice Hall, 2004. - ISBN ISBN 0-13-805326-X .
Landau LD , Lifshits E.M. Mechanics. - Edition 4th, revised. - M .: Science , 1988. - 215 p. - (“ Theoretical Physics ”, Volume I). - ISBN 5-02-013850-9 .
Arnold V.I. “Mathematical methods of classical mechanics”, from. 5th, M .: Editorial URSS, 2003, ISBN 5-354-00341-5
Saveliev I.V. The course of general physics. T. 1. Mechanics. Molecular physics. - M .: Science, 1987. - 432 p.

[_6bcf5dabfaf3d9aa-1] Saveliev, 1987 , p. 74.