Action angle coordinates

Action angle coordinates , also action angle variables , are a set of canonically conjugate coordinates with which dynamic systems can be simplified. With the transformation to action angle coordinates, natural frequencies of oscillators can be determined without having to solve the system's equations of motion .

Action angle coordinates are particularly suitable if the Hamilton-Jacobi equations are separable . The Hamilton function then does not depend explicitly on time, so that the total energy of the system is preserved.

The action angle coordinates define invariant tori in phase space . Their surfaces are surfaces of constant effect .

application areas

According to the quantization conditions for Bohr-Sommerfeld's atomic model , the effect must be an integral multiple of Planck's quantum of action , and even in modern quantum mechanics , difficulties in quantizing non-integrable systems can be expressed by action angle coordinates.

Action angle coordinates are also useful in perturbation theory of Hamiltonian mechanics , especially to determine adiabatic invariants. One of the first results of chaos theory for nonlinear disturbances of dynamic systems is the KAM theorem , which makes statements about the stability of the above. invariant tori hits.

Action angle coordinates are used for the solution of the Toda lattice , the definition of Lax pairs , or the idea of ​​the isospectral development of systems.

Definition and derivation

The angles of action can be derived by a canonical transformation of the second kind, in which the generating function is the time-independent characteristic Hamilton function ( not the Hamiltonian action function ). Since the original Hamilton function does not depend explicitly on time, the new Hamilton function is nothing other than the old one, expressed in new canonical coordinates. The new coordinates consist of the angles of action which correspond to the generalized coordinates and the coordinates which correspond to the generalized impulses . (The generating function is only used here to link the new and old coordinates; the explicit form will not be discussed further.) ${\displaystyle {\vec {w}}}$ ${\displaystyle W({\vec {q}})}$ ${\displaystyle S}$ ${\displaystyle H({\vec {q}},{\vec {p}})}$ ${\displaystyle K({\vec {w}},{\vec {J}})}$ ${\displaystyle {\vec {w}}}$ ${\displaystyle q}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle p}$ ${\displaystyle W}$

Instead of defining the angles of action directly, it is easier to first determine their generalized impulses . These are defined as ${\displaystyle {\vec {J}}}$

${\displaystyle J_{k}:=\oint p_{k}\,\mathrm {d} q_{k}}$

where the integration path is implicitly given by the condition of constant energy . Since the actual motion is not needed for the integration, these generalized impulses are preserved , provided that the transformed Hamilton function does not depend on the generalized coordinates : ${\displaystyle E=E(q_{k},p_{k})}$ ${\displaystyle J_{k}}$ ${\displaystyle K}$ ${\displaystyle w_{k}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}J_{k}=0={\frac {\partial K}{\partial w_{k}}}}$

in which

${\displaystyle w_{k}:={\frac {\partial W}{\partial J_{k}}}}$

is given by the canonical transformation. Therefore the new Hamilton function depends only on the new generalized impulses . ${\displaystyle K=K({\vec {J}})}$ ${\displaystyle {\vec {J}}}$

properties

The equations of motion of the system in the new coordinates are obtained from Hamilton's equations

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}w_{k}={\frac {\partial K}{\partial J_{k}}}\equiv \nu _{k}({\vec {J}})}$

Since all have been preserved, the right side has also been preserved. The solution is therefore ${\displaystyle J_{k}}$

${\displaystyle w_{k}=\nu _{k}({\vec {J}})\,t+\beta _{k}}$

where is a corresponding constant of integration. In particular for an oscillation or a circular movement in the original coordinates with a period , the angle of action changes by . ${\displaystyle \beta _{k}}$ ${\displaystyle T}$ ${\displaystyle w_{k}}$ ${\displaystyle \Delta w_{k}=\nu _{k}({\vec {J}})\,T}$

They are therefore the frequencies of the oscillation of the original coordinates . This can be shown by integrating the change in the angle of action over a period in the original coordinates ${\displaystyle \nu _{k}({\vec {J}})}$ ${\displaystyle q_{k}}$ ${\displaystyle q_{k}}$

${\displaystyle \Delta w_{k}\equiv \oint {\frac {\partial w_{k}}{\partial q_{k}}}\,\mathrm {d} q_{k}=\oint {\frac {\partial ^{2}W}{\partial J_{k}\,\partial q_{k}}}\,\mathrm {d} q_{k}={\frac {\mathrm {d} }{\mathrm {d} J_{k}}}\oint {\frac {\partial W}{\partial q_{k}}}\,\mathrm {d} q_{k}={\frac {\mathrm {d} }{\mathrm {d} J_{k}}}\oint p_{k}\,\mathrm {d} q_{k}={\frac {\mathrm {d} J_{k}}{\mathrm {d} J_{k}}}=1}$

If you set both expressions to be equal, you get the desired equation ${\displaystyle \Delta w_{k}}$

${\displaystyle \nu _{k}({\vec {J}})={\frac {1}{T}}}$

literature

• L. D Landau, E. M Lifshitz: Mechanics . 32591126th edition. Pergamon Press, Oxford / New York 1976, ISBN 0-08-021022-8 .
• Herbert Goldstein: Classical mechanics . 2nd Edition. Addison-Wesley Pub. Co, Reading, Mass 1980, ISBN 0-201-02918-9 .
• G. A. Sardanashvili: Handbook of integrable hamiltonian systems . URSS, Moscow 2015, ISBN 978-5-396-00687-4 .

Individual evidence

1. ↑ Edwin Kreuzer: Numerical investigation of nonlinear dynamic systems . Springer-Verlag, 2013, ISBN 3-642-82968-6 , pp. 54 f . ( limited preview in Google Book search).