Today, over lunch, togther with Christian Römmelsberger, we tried to understand Hamilton-Jacobi theory from a more geometric point of view.
The way this is usually presented (in the very end of a course on classical mechanics) is in terms of generating functions for canonical transformations such that in the new coordinates the Hamiltonian vanishes. Here I will rewrite this in the laguage of symplectic geometry.
As always, let us start with a 2N dimensional symplectic space M with symplectic form
![\omega](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vElYNPxaq-oigmqoMD3jc2OFTpeV_f-wgLKeWj2ANBDy29QnbQT1HHlAoDRuZqwZNBMJiMzfN8-D5-hxvrv7dQQ1-CnYF_ZlOUzslq6ge5uJZQpTycItKg13_oTA=s0-d)
. In addition, pick N functions
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
such that the submanifolds
![L(x^i)=\{m\in M|q^i(m)=x^i\}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uRv4zAkqgrXOWzFSsoSLL8kYh4J2wWmKrcuvGZjj7eWXxd3kMkHLSpmlKFAsShzaEKzm0FJRzp2aDzwTKPOA1hLLUtZz7A0zd-HUdXjIaMtZJ-1mZgI5CEeNlhTK5aOkcPN7Sl9o102Cs7-dZFUf0OIYnpJhJIzQkmRCbbUx6CRWy1PLI=s0-d)
are Lagrangian, that is
![T_mL(x^i)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ve8QCGMnghtrkMrdU1LULEbVJ6gPOwYKlLKT3VBbAxk46XKL96FJLSk-rjjboib-RZyfvR6MD5U1qqUJhTVfiwBCBuVbrjVAzJtsA2Ywsc5srty7jbqD5XeTTgNLPU0w=s0-d)
is a Lagrangian subspace of
![T_mM](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vwvb5NaUPNRIfWsvRcl3ymTMWmpretzcNlrtjMgwj5Lgg86EeBqti2PdKD9jq8rqWz6Tw8LypoXHldWB1xImQ9I604nf9UpNTE4zlUUtDPhdAaCYSv-rs_=s0-d)
(meaning that the symplectic form of any two tangent vectors of
![L(x^i)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s1V-6Cghze_2nilLMsZ0Cff9XfllKTqGdXuntkNeiO1qrQOoHmopsoJ4Mqp60yYoj43c2_bYB0PhQBrQ0f3RwAoBI17ZTKFKGxHtYA7oHOAeqwomNnc6OhHCxmoQ=s0-d)
vanishes). If this holds, the
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
can be regarded as position coordinates.
Starting from these Lagrangian submanifolds, we can locally find a family of 1-forms in the normal bundle
![\theta\in N^*L(x^i)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sfPYNI2dI0SWluh3wzUHscV84q00IHTSMcoP0PUJk_4Mp9zmwvRZutdR8lyUsCX99IB04USpB3Ck-1EO2Bz5qcq0U9wE7u7TWVUprVNulueGIWelDqcyqlF9DYZ9fp6t5BaJ8SILmkB8wzGHtEWg=s0-d)
such that
![\omega=d\theta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ux-HKyghVj-aJZHpKm0FfYGZytIum9Fbzh20WTr5uPbXuRrYQBe0EUYwdQzqsBkWwMYjwQdPERfO8RsROgviHlQL8B0u_ljBkXvGEHeOo4uw-PGCXM2hct0cXdwSHddTTk1UtaBUw=s0-d)
. You should think that
![\theta=p_idq^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vAZ3OQ7YCGmACq3JmPIzqcUzJy4hfPPzA56EF53dP7hWhlGUNPxYmUfDxPQv0i4iVupX7gPKDUsGRcGxxhUhWR3edUlGkdq6NjOrGc2p-o_r8ehvA2SNs0rYSuYuGnFKCB_1U42SY=s0-d)
for appropriate momentum coordinates
![p_i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u-4kBEM-MHzlUSPswTLvzVGIlq9u2ddomsTvfCqPVzDh-euSXpBDic4XiaDfr9cQYaqt1bMbAzGH3PH4qbiMt1tDmbgmZu3GnSuk-e6zPujpOwIhcrBAQ=s0-d)
on the Lagrangian leaves of constant
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
. But here, these are just coefficient funtions to make
![\theta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJpe2Eho3DUWDg9DvzcYE5s3iFpL7QM7xPrQ7qsNljpO3JUJrdxDp_Cn5HLGH7snorbOnYXkG38ji65Bw4HvTOwsDqxk-TAiJuAHTCgsRvt6oSpvBtYJZeEwCs=s0-d)
a potential for
![\omega](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vElYNPxaq-oigmqoMD3jc2OFTpeV_f-wgLKeWj2ANBDy29QnbQT1HHlAoDRuZqwZNBMJiMzfN8-D5-hxvrv7dQQ1-CnYF_ZlOUzslq6ge5uJZQpTycItKg13_oTA=s0-d)
.
Now we repeat this for another set of position coordinates
![Q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tC960DI7zfL0uYLi7T6y9wINRJPYdyFHc0hOt-Hta6uRxTzzcYVHCsmI51xkSjpPDR2zxu6fhNFCR9XiP48fOlyOOwMao5O30Or18aC9ZayOaaOV5EO5Rxxw=s0-d)
which we assume to be "sufficiently independent" of the
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
meaning that
![TM=TL(q^i)\oplus TL(Q^i)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tbARMJ1H8BD9QsJML5atQC3nNrpmvkHJZltmYCCsU9LKtBV5TLUDggQEtZeYLNgSa4CNtM3hO_TkhQIDDUjdFiDze29bzpnm56Q3-KZ_VOI3C93_sQ_1wchkUBJaNXrE6S0u8Ls0EUjtYt-13FVbtxJFY=s0-d)
. This implies that locally
![(q^i, Q^j)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_udG52oUkKpZuzszHebzU-FcD6lDCZ4-jSewwK00flXYGOWGt2MsKvVO0crt4wfdxclJLl5e1qr_Iam6RKXUyeelCdNMV_q5l9fLxLsSj4QRFvop2rNfmUiI0lhl9y1rjitZA=s0-d)
are coordinates on M. With the
![Q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tC960DI7zfL0uYLi7T6y9wINRJPYdyFHc0hOt-Hta6uRxTzzcYVHCsmI51xkSjpPDR2zxu6fhNFCR9XiP48fOlyOOwMao5O30Or18aC9ZayOaaOV5EO5Rxxw=s0-d)
comes another 1-form
![\Theta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uLOP_F0p033oxo2YzgaUecU1gRACNPJEI-GD9bo8Aaw1wMnN4LJ8UuxCSRtjasxnnGEfA3ViO_BCucd-qAhTn0YyXgCdrIHuykR39-nIfaxE9BWjf7Kx6ydSsf=s0-d)
and since
![d\Theta=\omega=d\theta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sq4Tf1PWgZmTYd2VQt9tWZ8UbC37GwSrrlfPI0I9utNT-uAyCi8-opaXPmvceyI_QB8NsO1ui2rWxlSCWlb_xi0xy1p5DXZtgKUSN8M-zFCoGfgDXdrfoLXLlHw1ppK9TRar5d1MZcO25W-7I8tJx3FWY=s0-d)
are locally related by a "gauge transformation". We have
![\theta = \Theta + dF](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u31-V7ixSHeO1YqA5yNvoXZbHBe3GWc8b4OVmSEokuaU8Axhcv1obe5Y6GVodN1QcfCjzEwrIif645UkCsWFoEEev3dEa_tvfr3hYSXyrliVXyv94Q4VlM119n10ayqeul2ynoqHppFWYci1M=s0-d)
for a function F.
Let's look at
![\theta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJpe2Eho3DUWDg9DvzcYE5s3iFpL7QM7xPrQ7qsNljpO3JUJrdxDp_Cn5HLGH7snorbOnYXkG38ji65Bw4HvTOwsDqxk-TAiJuAHTCgsRvt6oSpvBtYJZeEwCs=s0-d)
a little bit closer. A general normal 1-form would look like
![f_i(q^j,Q^k)dq^q](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_trkACZSsjP9I350UXDwGab1LRck76yQIFzptkEg6IM0lCretsIqCpp8H9uEb4MfzhaF8rAxcy5J639yoXND8zMSG_fosqZrEjlHHfovoNsv2ICmySLMEKm3t7YSel3uq7zp_OkluKrOD_G=s0-d)
. But since we started from Lagrangian leaves, there is no
![dq^i\wedge dq^j](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sWv3mfqIVYcrSV6e0i4RKyRcUoMLD3QrT1nnRj_pypBM1-_3y8Yy6OS4SZ55p5jkRFC52Ua_EpnkOrGPk78KnFVDSbP_SVC5aEq39hspFZcuUe0-w1bPkgSjq4rPdAb7gjFGKprsO4Cg=s0-d)
in
![\omega](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vElYNPxaq-oigmqoMD3jc2OFTpeV_f-wgLKeWj2ANBDy29QnbQT1HHlAoDRuZqwZNBMJiMzfN8-D5-hxvrv7dQQ1-CnYF_ZlOUzslq6ge5uJZQpTycItKg13_oTA=s0-d)
and thus
![\theta=p_i(Q^j)dq^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v-QMCI31d0m8xVWhGc72wgVXCqI-ce0vKAeUFPWHmj9MVkXmuqOFEVaPDWCZZRCQYANZwrtCuyAosEBJBHjFlBR71MgCWOA_Sw4um7jV9SToCsHB3ni3_p0xyF80fxukuDYmIIKg2c6y2Di-do=s0-d)
. But expressing this in coordinates yields
Comparing coefficients we find
![p_i = \frac{\partial F}{\partial q^i}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t2MARkSOHnJar4GFiraZoPfsah2WBbMKW7iLUUHblkYNGWeC8v8W-oTKCXzC0VWKPnb9Mo6orDvELNZoG1P9A7Ssep3yJlBKjh5LbxlvckS0q7IxozqJ68DedHHmsxkIVZfnjGGGpXRPwO9AYDg8V4ASKXnulK_xFCLaDZ__Xy381LsB2Q0ag5QQ=s0-d)
and
![P_i = -\frac{\partial F}{\partial Q^i}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_thGJ8XRz10Q2j4b0pGXDlkpFALu8-AwogdL1BGqlea5CHDce5m1NuP-wTKY6EQS2IZbtChmltydD0lj-fRT5fCcqm02B8T5zpm9yHyaOeZBWmbFKPMS7TLjtlLCa6BvrsS2XRincFb-ccmKy4gvBR7KdeFPjw7ZzCu3ZrvjE1SnHSzy8nZcS689AE=s0-d)
. You will recognize the expressions for momenta in terms of a "generating function".
What we have done was to take two Lagrangian foliations given in terms of
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
and
![Q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tC960DI7zfL0uYLi7T6y9wINRJPYdyFHc0hOt-Hta6uRxTzzcYVHCsmI51xkSjpPDR2zxu6fhNFCR9XiP48fOlyOOwMao5O30Or18aC9ZayOaaOV5EO5Rxxw=s0-d)
and compute a function
![F](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v7SWBWWNiCN7IEsiaSOqjrkR5FgPmzHnUrv-DvfGS7yrdBh6hRC6OJTTKCeTju9x8SxL1PAonfD53w9SG_50qEZ9z9ThLGl59B4vtjbLNsqaRwQJmN=s0-d)
from them. The trick is now to turn this procedure around: Given only the
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
and a function
![F(q^i,X^j)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tFwY_2oHnEX5gSs45QTpJx8VYFE6-AYhoMtUJu9nDQVEK-Z4nqOeDscqlmi4vH2IQKCIqo6-dTuMA4zw7afK33iDLMUIkeouuicKx-kLgiyRH6Yf5k6IQr8tI1Ipq8Bp9lSw=s0-d)
of these
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
and some N other variables
![X^j](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u9aTPyd2wJkQm-wayAxX7QlLhKujyg1rzkSa5RYsZCnMuq1BBMDbF0HhFiTArqFlWQ-4ZZJZafIY49e9QBmQX4fXobqZAk8KiKzIwpQMNxKqpINgkaoz4Myw=s0-d)
, one can compute the
![Q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tC960DI7zfL0uYLi7T6y9wINRJPYdyFHc0hOt-Hta6uRxTzzcYVHCsmI51xkSjpPDR2zxu6fhNFCR9XiP48fOlyOOwMao5O30Or18aC9ZayOaaOV5EO5Rxxw=s0-d)
as functions on M: Take a point
![m=(q,p)\in M](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tGesrqaeyUo7NLDYP2D1wKxGJA-YVDJGx6AUDAgZiFdvfvp2HCIkT7OKNYY3d3VpWKjbIuvm9WqVssL5w7a3nL4CNYvv0zMrLy35RA2CYaLKQMR70du9danSPFFhP_GGLUpg=s0-d)
and define
![Q^i(m)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s3D6L_Z-D6nW8mHGgbPk2uTMGgHfqSdrXkdwSbiyZ9Hcjd6YT9bXe5dxhXJLkOpfMwyO4c-pSVVSWbwVdAfCOhhtNsh8WPUeCOgBwUbygBxrqMzNxarCOhJyMN=s0-d)
by inverting
![p_i = \frac{\partial F(q,X=Q)}{\partial q^i}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tGV2QDcx28UW40VUGo_Skx1xiYDYbywxqRbnC0tBviqg-H81zrKXIkRvmZfVj0Geu6JTs_cng0ZiIBqo5-06N0DZywIKnWtW7pBSkyWdoLH_6VFgq6lX4XdhSLzLGELdphEToVAPzYeodsHm_S4pvCzDUUO7FfFq-3SWDb0lwScaOHV43PAyPYxdSisyfeNcRQyQ=s0-d)
. For this remember that
![p_i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u-4kBEM-MHzlUSPswTLvzVGIlq9u2ddomsTvfCqPVzDh-euSXpBDic4XiaDfr9cQYaqt1bMbAzGH3PH4qbiMt1tDmbgmZu3GnSuk-e6zPujpOwIhcrBAQ=s0-d)
was defined implicitly above: It is the coefficient of
![dq^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tJ4T5omV04mAQNfkk9mGQwzcksRYhZjBk3rLiwYjU9U8RIZohWppwo_UkeFzJ9hJsS3F5oC1993kQqixZzErkRN5_plj4eYT1MQtrpTdUcBjs93F_D95ukc9U=s0-d)
in
![\theta](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJpe2Eho3DUWDg9DvzcYE5s3iFpL7QM7xPrQ7qsNljpO3JUJrdxDp_Cn5HLGH7snorbOnYXkG38ji65Bw4HvTOwsDqxk-TAiJuAHTCgsRvt6oSpvBtYJZeEwCs=s0-d)
.
Up to here, we have only played symplectic games independent of any dynamics. Now specify this in addition in terms of a Hamilton function h. Then the Hamilton-Jacobi equations are nothing but the requirement to find a generating function
![F](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v7SWBWWNiCN7IEsiaSOqjrkR5FgPmzHnUrv-DvfGS7yrdBh6hRC6OJTTKCeTju9x8SxL1PAonfD53w9SG_50qEZ9z9ThLGl59B4vtjbLNsqaRwQJmN=s0-d)
such that the
![Q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tC960DI7zfL0uYLi7T6y9wINRJPYdyFHc0hOt-Hta6uRxTzzcYVHCsmI51xkSjpPDR2zxu6fhNFCR9XiP48fOlyOOwMao5O30Or18aC9ZayOaaOV5EO5Rxxw=s0-d)
are constants of motion.
Even better, by making everything (that is h and Q and F) explicitly time dependent, by the requirement that the action 1-form is invariant:
![\theta- hdt = \Theta - Hdt](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tc5_8BUa13CE6UiFt5Pc52FT13TN9wEBrIPIwG-jb1cYzUW5rAkL0e89KNmc5Bp47MYFW-JHUdx2rJg9-7oVsiuPa9NVPrG5m41_vQVZfNIkl4f5-Aip26hXEjMtH1cyKyis87TSWwMspLrfSTk48mceU=s0-d)
giving
![H = h +\frac Ft](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tFcLUoMLPXHiVuSfzRqWMrWxSbJSmaSaGDRf1aPGTWMqaP-8QNRM2qVg62KuduaGEJ6bgX7C2i3zC_CD5twPkOAulu1Wu-COMTAlVZlprpZBE33KwomboHoFGKM1ropOo7pKd8vg=s0-d)
we get a transforming Hamilonian and we can require this to vanish:
If we think of the Hamiltonian h given in terms of the coordinates
![(q^i,p_i)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v07hww9ZRXr1YqjotomhuCUevFbAcNy0mydFHKGa_Sux_f7tDeYuf-HJwbanIJ3204lnDWuewBLq1hWo_z5Fau7Mqt88Ni5csnaQ--K4kZsFYSODhpzy88nv3DYP5tfQ=s0-d)
this is now a PDE for F which has to hold for all
![m\in m](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v8rP7mAV_KA7WKCqH3xK1qLmmA_C9RcZsgcYVUBRyMrCWiKaxUh6l60xMAjfm-aoNzgqYjydlcJQBkKYxFrAcskZeTQlY_LgHNbbMJAxUOSRxQgnpM8-90ESg_dA=s0-d)
. That is, writing F as a function of
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
and
![X^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tzwE_f-9EjNBGeBIGksNPLjT0w-dNKFgYBmz1YoPQJj4bMaZ4yCaFwqfmLmjcm4kTYh_cRfi19HdR3cZPfJCxWjX5wg4yv-06-QjId-SCVO7poU2Z4uHZX-A=s0-d)
it has to hold for all (fixed)
![X^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tzwE_f-9EjNBGeBIGksNPLjT0w-dNKFgYBmz1YoPQJj4bMaZ4yCaFwqfmLmjcm4kTYh_cRfi19HdR3cZPfJCxWjX5wg4yv-06-QjId-SCVO7poU2Z4uHZX-A=s0-d)
as a PDE in the
![q^i](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t920bGya-Q5KgakFJAiiy8Dxb-3S2RJuwo6GxQe3j_oikHiJ5kc0PdM3okiqwd_lZJvks8ZP_vL5z3nxfSKl665hndHUEAOcUjjgOjAtBZRzBsQlQ6Jav2Wg=s0-d)
and t.
1 comment:
Nice, makes me want to take a look at my old course notes on theoretical mechanics again.
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