Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Action-angle coordinates
open-in-new
<h2 id="derivation">Derivation</h2> <p>Action angles result from a <a href="/facts/Canonical_transformation/J8vfkj37">type-2</a> <a href="/facts/Canonical_transformation/J8vfkj37">canonical transformation</a> where the generating function is <a href="/facts/Hamilton%25E2%2580%2593Jacobi_equation/XM0R7nzD">Hamilton's characteristic function</a> W ( q ) {\displaystyle W(\mathbf {q} )} (<i>not</i> Hamilton's principal function S {\displaystyle S} ). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian K ( w , J ) {\displaystyle K(\mathbf {w} ,\mathbf {J} )} is merely the old Hamiltonian H ( q , p ) {\displaystyle H(\mathbf {q} ,\mathbf {p} )} expressed in terms of the new <a href="/facts/Canonical_coordinates/9g2SFFRz">canonical coordinates</a>, which we denote as w {\displaystyle \mathbf {w} } (the action angles, which are the <a href="/facts/Generalized_coordinates/xrhApGk2">generalized coordinates</a>) and their new generalized momenta J {\displaystyle \mathbf {J} } . We will not need to solve here for the generating function W {\displaystyle W} itself; instead, we will use it merely as a vehicle for relating the new and old <a href="/facts/Canonical_coordinates/9g2SFFRz">canonical coordinates</a>. </p><p>Rather than defining the action angles w {\displaystyle \mathbf {w} } directly, we define instead their generalized momenta, which resemble the <a href="/facts/Action_(physics)/NhWS2Mts">classical action</a> for each original <a href="/facts/Generalized_coordinate/xrhApGk2">generalized coordinate</a> </p> J k ≡ ∮ p k d q k {\displaystyle J_{k}\equiv \oint p_{k}\,\mathrm {d} q_{k}} <p>where the integration path is implicitly given by the constant energy function E = E ( q k , p k ) {\displaystyle E=E(q_{k},p_{k})} . Since the actual motion is not involved in this integration, these generalized momenta J k {\displaystyle J_{k}} are constants of the motion, implying that the transformed Hamiltonian K {\displaystyle K} does not depend on the conjugate <a href="/facts/Generalized_coordinates/xrhApGk2">generalized coordinates</a> w k {\displaystyle w_{k}} </p> d d t J k = 0 = ∂ K ∂ w k {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}J_{k}=0={\frac {\partial K}{\partial w_{k}}}} <p>where the w k {\displaystyle w_{k}} are given by the typical equation for a type-2 <a href="/facts/Canonical_transformation/J8vfkj37">canonical transformation</a> </p> w k ≡ ∂ W ∂ J k {\displaystyle w_{k}\equiv {\frac {\partial W}{\partial J_{k}}}} <p>Hence, the new Hamiltonian K = K ( J ) {\displaystyle K=K(\mathbf {J} )} depends only on the new generalized momenta J {\displaystyle \mathbf {J} } . </p><p>The dynamics of the action angles is given by <a href="/facts/Hamilton%2527s_equations/2ReerlNA">Hamilton's equations</a> </p> d d t w k = ∂ K ∂ J k ≡ ν k ( J ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}w_{k}={\frac {\partial K}{\partial J_{k}}}\equiv \nu _{k}(\mathbf {J} )} <p>The right-hand side is a constant of the motion (since all the J {\displaystyle J} s are). Hence, the solution is given by </p> w k = ν k ( J ) t + β k {\displaystyle w_{k}=\nu _{k}(\mathbf {J} )t+\beta _{k}} <p>where β k {\displaystyle \beta _{k}} is a constant of integration. In particular, if the original <a href="/facts/Generalized_coordinate/xrhApGk2">generalized coordinate</a> undergoes an oscillation or rotation of period T {\displaystyle T} , the corresponding action angle w k {\displaystyle w_{k}} changes by Δ w k = ν k ( J ) T {\displaystyle \Delta w_{k}=\nu _{k}(\mathbf {J} )T} . </p><p>These ν k ( J ) {\displaystyle \nu _{k}(\mathbf {J} )} are the frequencies of oscillation/rotation for the original <a href="/facts/Generalized_coordinate/xrhApGk2">generalized coordinates</a> q k {\displaystyle q_{k}} . To show this, we integrate the net change in the action angle w k {\displaystyle w_{k}} over exactly one complete variation (i.e., oscillation or rotation) of its <a href="/facts/Generalized_coordinate/xrhApGk2">generalized coordinates</a> q k {\displaystyle q_{k}} </p> Δ w k ≡ ∮ ∂ w k ∂ q k d q k = ∮ ∂ 2 W ∂ J k ∂ q k d q k = d d J k ∮ ∂ W ∂ q k d q k = d d J k ∮ p k d q k = d J k d J k = 1 {\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} <p>Setting the two expressions for Δ w k {\displaystyle \Delta w_{k}} equal, we obtain the desired equation </p> ν k ( J ) = 1 T {\displaystyle \nu _{k}(\mathbf {J} )={\frac {1}{T}}} <p>The action angles w {\displaystyle \mathbf {w} } are an independent set of <a href="/facts/Generalized_coordinates/xrhApGk2">generalized coordinates</a>. Thus, in the general case, each original generalized coordinate q k {\displaystyle q_{k}} can be expressed as a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> in <i>all</i> the action angles </p> q k = ∑ s 1 = − ∞ ∞ ∑ s 2 = − ∞ ∞ ⋯ ∑ s N = − ∞ ∞ A s 1 , s 2 , … , s N k e i 2 π s 1 w 1 e i 2 π s 2 w 2 ⋯ e i 2 π s N w N {\displaystyle q_{k}=\sum _{s_{1}=-\infty }^{\infty }\sum _{s_{2}=-\infty }^{\infty }\cdots \sum _{s_{N}=-\infty }^{\infty }A_{s_{1},s_{2},\ldots ,s_{N}}^{k}e^{i2\pi s_{1}w_{1}}e^{i2\pi s_{2}w_{2}}\cdots e^{i2\pi s_{N}w_{N}}} <p>where A s 1 , s 2 , … , s N k {\displaystyle A_{s_{1},s_{2},\ldots ,s_{N}}^{k}} is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate q k {\displaystyle q_{k}} will be expressible as a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> in only its own action angles w k {\displaystyle w_{k}} </p> q k = ∑ s k = − ∞ ∞ A s k k e i 2 π s k w k {\displaystyle q_{k}=\sum _{s_{k}=-\infty }^{\infty }A_{s_{k}}^{k}e^{i2\pi s_{k}w_{k}}} <h2 id="summary-of-basic-protocol">Summary of basic protocol</h2> <p>The general procedure has three steps: </p> <ol><li>Calculate the new generalized momenta J k {\displaystyle J_{k}} </li> <li>Express the original Hamiltonian entirely in terms of these variables.</li> <li>Take the derivatives of the Hamiltonian with respect to these momenta to obtain the frequencies ν k {\displaystyle \nu _{k}} </li></ol> <h2 id="degeneracy">Degeneracy</h2> <p>In some cases, the frequencies of two different <a href="/facts/Generalized_coordinate/xrhApGk2">generalized coordinates</a> are identical, i.e., ν k = ν l {\displaystyle \nu _{k}=\nu _{l}} for k ≠ l {\displaystyle k\neq l} . In such cases, the motion is called degenerate. </p><p>Degenerate motion signals that there are additional general conserved quantities; for example, the frequencies of the <a href="/facts/Kepler_problem/oxyp49JD">Kepler problem</a> are degenerate, corresponding to the conservation of the <a href="/facts/Laplace%25E2%2580%2593Runge%25E2%2580%2593Lenz_vector/1gdBtfNj">Laplace–Runge–Lenz vector</a>. </p><p>Degenerate motion also signals that the <a href="/facts/Hamilton%25E2%2580%2593Jacobi_equation/XM0R7nzD">Hamilton–Jacobi equations</a> are completely separable in more than one coordinate system; for example, the Kepler problem is completely separable in both <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a> and <a href="/facts/Parabolic_coordinates/MLledsNu">parabolic coordinates</a>. </p> <h2 id="see-also">See also</h2> <ul> <li>Physics portal</li></ul> <ul><li><a href="/facts/Bohr%25E2%2580%2593Sommerfeld_model/pfkUmzUP">Bohr–Sommerfeld model</a></li> <li><a href="/facts/Integrable_system/Crgzjw6G">Integrable system</a></li> <li><a href="/facts/Einstein%25E2%2580%2593Brillouin%25E2%2580%2593Keller_method/nrQAhAVh">Einstein–Brillouin–Keller method</a></li> <li><a href="/facts/Superintegrable_Hamiltonian_system/XnpdUfJY">Superintegrable Hamiltonian system</a></li> <li><a href="/facts/Tautological_one-form/FH8WKyxV">Tautological one-form</a></li></ul> <ul><li>Landau, L. D.; Lifshitz, E. M. (1976), <i>Mechanics</i> (3rd ed.), Pergamon Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-08-021022-8 (hardcover) and <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-08-029141-4 (softcover)</li> <li>Goldstein, H. (1980), <i>Classical Mechanics</i> (2nd ed.), Addison-Wesley, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-02918-9</li> <li><a href="/facts/Gennadi_Sardanashvily/EDrGkc9r">Sardanashvily, G.</a> (2015), <i>Handbook of Integrable Hamiltonian Systems</i>, URSS, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-5-396-00687-4</li> <li>Previato, Emma (2003), <i>Dictionary of Applied Math for Engineers and Scientists</i>, <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2003dame.book.....P">2003dame.book.....P</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-58488-053-0</li></ul>