Variational Calculus

Table of Contents

Sources

  • Most of these notes comes from the Variational Calculus taught by Prof. José Figueroa O'Farrill, University of Edinburgh

Overview

Notation

  • variational_calculus_ef47a3fbc7ec9272c1b760bcdd182f08bcf4987c.png denotes the space of possible paths (i.e. variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png curves) between points variational_calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png and variational_calculus_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png

Introduction

Precise analytical techniques to answer:

  • Shortest path between two given points on a surface
  • Curve between two given points in the place that yields a surface of revolution of minimum area when revolved around a given axis
  • Curve along which a bead will slide (under the effect of gravity) in the shortest time

Underpins much of modern mathematical physics, via Hamilton's principle of least action

Consider "standard" directional derivative of variational_calculus_da065fc7c2b914075249f4d59f87eda6913a9d60.png, with variational_calculus_71c05083089d4bc2c376637af84e09d0fa106ce0.png, at variational_calculus_c462c980a45481116745a8647094b2b1d245df0f.png along some vector variational_calculus_d2efb2e527666807e62cae95366f963774f95bd6.png:

variational_calculus_e3b078ea250bd39e2c058f73c69229730be25e37.png

where variational_calculus_c462c980a45481116745a8647094b2b1d245df0f.png is a critical point of variational_calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png, i.e.

variational_calculus_b6ed652c81417cf44797f968b0f5ae36891a1c72.png

(since we know that variational_calculus_1e0e726a8e1a78630a970cfff5dd8e49ab9c4c81.png form a basis in variational_calculus_d480cdf1ef59f8f40028e11f330ff16e2cca1824.png, so by lin. indep. we have the above!)

Stuff

Let variational_calculus_29bd389ce89cf22d73f867f8f8e72e71ad7f2fc1.png be a continuous function which satisfies

variational_calculus_23c50131a58e23f106bdf500877553481020e0a8.png

for all variational_calculus_e4b55b5b77820fba319b5ff126b313508647c428.png with

variational_calculus_3bbeec508636f08d3c584628e90c6051540bae70.png

Then variational_calculus_ef4ff0ef7784783ce7d106b2ac7f6de8a32823be.png.

Observe that if we only consider variational_calculus_3e67b91c275ca363f9ef19a10395e1bb74f0cccc.png, then we can simply genearlize to arbitrary variational_calculus_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png since integration is a linear operation.

Let variational_calculus_1fa8d4c033d2dd96584709fd15964d2600ee9380.png which obeys

variational_calculus_e9585b70c2d6b5a31a8e7246a10d2db6bc6a888b.png

Assume, for the sake of contradiction, that variational_calculus_48df7d96b8a1511ba907f092b85b04731470adac.png, i.e.

variational_calculus_e4f8b23395a96840b81529fb84fdfd49f255a44c.png

Let, w.l.o.g., variational_calculus_4316c95d6a9ddce7219d40b55dd34206247c4e1a.png.

Since variational_calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous, there is some interval variational_calculus_068561ccdec6b7d026a615dc9460ef5ce86dfe0d.png such that variational_calculus_f40d90e43a086d633baf828f072025cea15a8cda.png and some variational_calculus_cceff0a93740f9b1b736f7e926efed7d24be194b.png such that

variational_calculus_9a8b1931fbd20ef2bf0f16f92bac625c113f0601.png

Suppose for the moment that there exists some variational_calculus_5b0726ea94af1eb87ed06f3ff8b4b752029e0911.png such that

  1. variational_calculus_cd6b5e433a7f6c9b23e406c201e9917624d3882b.png for all variational_calculus_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png outside variational_calculus_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png
  2. variational_calculus_e1bca4169cee7bf22907ab34b60054aa2f2732c1.png

Then observe that

variational_calculus_6423379b53c457aaab6b909877a0d0f246e0f526.png

This is clearly a contradiction with our initial assumption, hence variational_calculus_3e1037db78289e4ee2c8b1fa0f565a0a779069aa.png such that variational_calculus_f061ea164a1c80bdd06ab1ac07e0c3e2475f5521.png. Hence, by continuity,

variational_calculus_6cf260bf9ebe444bab7711a677da4623e8a53c32.png

Now we just prove that there exists such a function variational_calculus_618fe9be653e96ddd9af4019330c6b09b069e53a.png which satisfies the properties we described earlier.

Let

variational_calculus_8b525de4029ec88d1e1a7862cbf52c83b5e5b2e9.png

which is a smooth function. Then let

variational_calculus_80246871ab0113cbec2854b3b40bd278473ce548.png

which is a smooth function, since it's a product of smooth functions.

To make this vanish outside of variational_calculus_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png, we have

variational_calculus_0d6676411b8ecf7bc927aeaf03f5d5fb5bc10a29.png

This function is clearly always positive, hence,

variational_calculus_a7be2615e8b15b8310ed3df89a96b75f0f19862a.png

Hence, letting variational_calculus_1983325907f693973b0f9d08d72df305b662bd16.png we get a function used in the proof above!

This concludes our proof of Fundamental Lemma of the Calculus of Variations

General variations

Suppose we want te shortest path in variational_calculus_e4f375c26796781f71b7ae3026445db617a6e78b.png between variational_calculus_a9090c77ce9916955c745920bf8f134a0932d59d.png and a curve variational_calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png on variational_calculus_e4f375c26796781f71b7ae3026445db617a6e78b.png.

We assume that variational_calculus_7f4f51e1f853f075144ce7a00dad665b1e5e64f9.png with variational_calculus_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png differentiable, such that variational_calculus_762062ca8572782393f2a81893b4e616d5030a03.png.

Then observe that variational_calculus_52844f5fe6eb562de8d53444a68c55809d5090c2.png, then

variational_calculus_b4d8bda0851baee304c33b2f1839f3f42a8c5b85.png

Then

variational_calculus_20d4138ac0fd4f89daf847c687d0591fffe56a75.png

where we've used the fact that

variational_calculus_9f43ceba69734ecb3a08e27ee5279dcd315b0644.png

We cannot just drop the endpoint-terms anymore, since these are now not necessarily vanishing, as we had in the previous case.

Then, by our earlier assumption, we have

variational_calculus_393b32e340fdd8631d9460b92360bd66b52d175c.png

which implies that

variational_calculus_4128605c11d508ea5738c7e5bc7fc035bed981e6.png

Hence,

variational_calculus_7ef1f1701040d7b4507a9a302338c1ed91f8d288.png

must hold for all variational_calculus_600d5e86fe2eeb91a0099bcef6fbc9d6785d778f.png and variational_calculus_b40de3f849cacfacc718c4a535ad31aa799aafd1.png and variational_calculus_64b79c83671ad924b0cab335b0788feec92a431b.png

In particular, variational_calculus_4c7e970348bb43f2b388e83c714bfe5a14170a41.png, then

variational_calculus_383dbd8a0e58ad0077ad88d994bf01fb406c3d84.png

and

variational_calculus_44222ea93d8afa716dfc21a012396f3ab9422866.png

i.e. variational_calculus_743b57d75a51a9ef7cfdaddc1733f629efebd8db.png is normal to variational_calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png at the point where it intersects with variational_calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png.

Euler-Lagrange equations

Notation

Endpoint-fixed variations

Let variational_calculus_ef47a3fbc7ec9272c1b760bcdd182f08bcf4987c.png be the space of variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png curves variational_calculus_dc51a789bf4b58f46b5e0f9d8029fc146d5b4dad.png with

variational_calculus_4211d8cf1dff5ee4df1de1898a59c7deb6f8a210.png

The Lagrangian is defined

variational_calculus_a04c2c1ac0b268e434f72d02511ba986e89e15d4.png

where variational_calculus_f5af0b45fad6dbafb39023bc971a47eb47a5bd2e.png and variational_calculus_570462f792cbe96a48effcfae87b005cefc599cb.png. Let variational_calculus_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png be "sufficiently" differentiable (usually taken to be smooth in applications).

Then the function variational_calculus_0cd6b95ab662e490dd58c4c9346a036db233e7bf.png, called the action, is defined

variational_calculus_a545d3b349f97026408fc087196c6cb5ef188e8f.png

A path variational_calculus_7a84c9a383f9772338016d101ccc096be06af784.png is a critical point for the action if, for all endpoint-fixed variations variational_calculus_224ae917d74dc2133d4403064c971bf562d4db50.png, we have

variational_calculus_db2258f3735aebea7209fc63c61220d61a80b464.png

Bringing the differentiation into to the integral, we have

variational_calculus_f8e72bdc84557d46a177b62189644a9ededff2ba.png

Properties

  1. If variational_calculus_667701ca05327a4396fe789942be647710fde98b.png, then the "energy" given by

    variational_calculus_6e5136289d9c7bf646a860cf86d57b1d8a20cc98.png

    is constant along extremals of the Lagrangian. Then observe that

    variational_calculus_7502e773dbf2637cd8ad0d52cd5e109c3ca45763.png

    where we've used the fact that variational_calculus_94730a24dba93c790a1a9b108554b058d1e8ec0d.png in the thrid equality. Thus,

    variational_calculus_a4969583f23a10ce00e78334d5249d0e89040447.png

    and,

    variational_calculus_f17bc9b2393fa9992eef107c9f385d7bb59a0ba8.png

    Hence,

    variational_calculus_21a61db372012ef0504c646bef0267d76c2c9fe4.png

    i.e time invariance! This is an instance of Noether's Theorem.

If variational_calculus_667701ca05327a4396fe789942be647710fde98b.png, so that the lagrangian does not depend explicitly on variational_calculus_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, then the energy

variational_calculus_e8091a3b3f207f426b32fbbff943bf1aa6ca046a.png

is constant.

This is known as the Beltrami's idenity.

Euler-Lagrange

Let variational_calculus_ef47a3fbc7ec9272c1b760bcdd182f08bcf4987c.png be the space of variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png curves variational_calculus_dc51a789bf4b58f46b5e0f9d8029fc146d5b4dad.png with

variational_calculus_4211d8cf1dff5ee4df1de1898a59c7deb6f8a210.png

Let

variational_calculus_a04c2c1ac0b268e434f72d02511ba986e89e15d4.png

where variational_calculus_f5af0b45fad6dbafb39023bc971a47eb47a5bd2e.png and variational_calculus_570462f792cbe96a48effcfae87b005cefc599cb.png, be sufficiently differentiable (typically smooth in applications) and let us consider the function variational_calculus_0cd6b95ab662e490dd58c4c9346a036db233e7bf.png defined by

variational_calculus_68ccfd297e806cc75ceffada66d5d97eab91846b.png

Then he extremals must satisfy the Euler-Lagrange equations:

variational_calculus_d339a5db7c3c44817f7ec9e1a86dec99e9134d2a.png

Newtonian mechanics

Notation

  • worldlines refer to the trajectory of a particle:

    variational_calculus_17b3e5cb8079e91e8dd77b39423272a71f234fad.png

    with variational_calculus_f7700aa9398c2c8d10d2a57c277cac4931dd7c8d.png

Galilean relativity

  • Affine transformations (don't assume a basis)
  • Relativity group: group of transformations on the universe preserving whichever structure we've endowed the universe with

The subgroup of affine transformations of variational_calculus_e6a60e21999924c5f25946405600f9b55a69c2eb.png which leave invariant the time interval between events and the distance between simultaneous events is called the Galilean group.

That is, the Galilean group consists of affine transformations of the form

variational_calculus_b9f81f28998526c4e8272638ebd38318ccfb641d.png

These transformations can be written uniquely as a composition of three elementary galilean transformations:

  1. translations in space and time:

    variational_calculus_ac9ea709fa887414e13ef0c0cefb4d65fd0f0e5e.png

  2. orthogonal transformations in space:

    variational_calculus_f77adf85849be70947089fe0699e5a04dca89fac.png

  3. and galilean boosts:

    variational_calculus_eb69d47665f58f3872b8cd4ee6a72f1b4f160f42.png

Observe that if choose the action

variational_calculus_a6d1b343bc0b8896988a16213bfe1c285c6bc8cf.png

which has Lagrangian

variational_calculus_60a0a84c4e4bceac0b4a6d0c62f64a70efd0b361.png

then we observe that the minimizing path variational_calculus_7a84c9a383f9772338016d101ccc096be06af784.png should satisfy

variational_calculus_34bbc081e3bc4bda34868af0df6d05e5f4daa487.png

which is

variational_calculus_27fa64678eda96213f32a13dc0c38ab95186e414.png

where variational_calculus_b2db59aeb94cbc1050079892ff07b21b493513b7.png denotes the force. Further,

variational_calculus_0047e00f9429a4617a1b224eb0ce7502c78447c3.png

which is the momentum! Then,

variational_calculus_d01120d9f5cc579097eb3d8f19afe18efa929910.png

Hence we're left with Newton's 2nd law.

Noether's Theorem

Notation

  • variational_calculus_b4f08b3dd875a63f4a2867507c894225e6885ccb.png of variational_calculus_abfc81685b43058ceaa81aa83ec4f0a65a5358d8.png functions called one-parameter subgroup of variational_calculus_abfc81685b43058ceaa81aa83ec4f0a65a5358d8.png diffeomorphisms , which are differentiable wrt. variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png
  • variational_calculus_2e8427caaac53f80f45e5f9d2874c6a4d4df75ab.png and variational_calculus_fe4a117954236dfc50dac7063e017c1092d1f06f.png are defined by

    variational_calculus_0126dd1f7e6aa59a80479309fd964cab08a478ad.png

    and

    variational_calculus_012a2a216f6a5d23412aaf8655dcd9db412a1575.png

  • variational_calculus_fd5c69feb04c2d89885a64a8375703142946ac67.png
  • variational_calculus_452799f8ae7f0455fa63df725b0e84ca0638b28a.png

Stuff

We've seen the following continuous symmetries this far:

  1. Momentum is conserved:

    variational_calculus_5cd35ff1d14ecd528b8368b70ad09aa96a129693.png

  2. Energy is conserved:

    variational_calculus_024983b9962e2a69d6fcc1ff4db56f1907a23fef.png

We say that variational_calculus_078b85cd3478400338e3a1ee425c2a468644be7e.png is a symmetry of the Lagrangian variational_calculus_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png if

variational_calculus_a49712ef202a8f3ed9d5110d2f6ace3a12e0f86f.png

where

Equivalently, one says that variational_calculus_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is invariant under variational_calculus_078b85cd3478400338e3a1ee425c2a468644be7e.png.

Let variational_calculus_b4f08b3dd875a63f4a2867507c894225e6885ccb.png of variational_calculus_abfc81685b43058ceaa81aa83ec4f0a65a5358d8.png functions, defined for all variational_calculus_d3891705174fad008f05f86fdc2bab3159fd35f3.png and depending differentiably on variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png.

Moreover, let this family satisfies the following properties:

  1. variational_calculus_75e1f18f778e978aca992b850a484c81f17dfc23.png for all variational_calculus_e64e42fb2e86f13facccbb0c709164958fdce248.png
  2. variational_calculus_3fabb42764d35ee1f09bb6f4ab00b89f5db7af27.png for all variational_calculus_0001f3693b48e56ae6718c4676281ad7fa447989.png

Then the family variational_calculus_0c01771224fb4bb6a7db0b39916648ebbc5f9174.png is called a one-parameter subgroup of variational_calculus_abfc81685b43058ceaa81aa83ec4f0a65a5358d8.png diffeomorphisms on variational_calculus_004097ff73cb85a0f596c8a3b60218ece0e16be1.png.

Let variational_calculus_3d6b024a763668e8bc7c78a6a0741519f70cd8e8.png be an action for curves variational_calculus_dc51a789bf4b58f46b5e0f9d8029fc146d5b4dad.png, and let variational_calculus_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png be invariant under a one-parameter group of diffeomorphisms variational_calculus_0c01771224fb4bb6a7db0b39916648ebbc5f9174.png.

Then the Noether charge variational_calculus_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png, defined by

variational_calculus_1f1c578c62d695db12a3d9bfb7229388c959000a.png

is conserved; that is, variational_calculus_2da0b1fad3f70c2c7f029b5b6b2bdd51ada2b065.png along physical trajectories.

Consider functions which are defined by Lagrangians variational_calculus_3b99e5f3995767c93a73e592760d2563e780bac6.png and such that they are invariant under one-parameter family of diffeomorphisms of variational_calculus_05e826243668e46473a9ff4a62b7a2b9bbc510dd.png such that

variational_calculus_0126dd1f7e6aa59a80479309fd964cab08a478ad.png

This means, in particular, that

variational_calculus_012a2a216f6a5d23412aaf8655dcd9db412a1575.png

where

  • variational_calculus_fd5c69feb04c2d89885a64a8375703142946ac67.png
  • variational_calculus_452799f8ae7f0455fa63df725b0e84ca0638b28a.png

Then the Noether charge

variational_calculus_bee17e39e01938c9b04593fc9e4cfbf799a31be1.png

is conserved along extremals; that is, along curves which obey the Euler-Lagrange equation.

Hamilton's canonical formalism

Notation

  • Considering

    variational_calculus_c4bd8b492bcd0c04077d45587cbaef5781d03e49.png

    for variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png curves variational_calculus_dc51a789bf4b58f46b5e0f9d8029fc146d5b4dad.png

  • Differentiable function variational_calculus_18d2d61778524354c84f1be3a60509ddc6db938a.png
  • variational_calculus_14a40f189f2ce698341c03cd5c099a337431c49f.png denotes the Hamiltonian

Canonical form of Euler-Lagrange equation

  • Can convert E-L into equivalent first-order ODE:

    variational_calculus_d339a5db7c3c44817f7ec9e1a86dec99e9134d2a.png

    is equivalent to system

    variational_calculus_fd1aadc056aab2174d465e14e5379a647fd6cf60.png

    for the variables variational_calculus_0b8ed86b429c1df696ad4de7546253cf1d4496cc.png.

  • Consider

    variational_calculus_0da670f4e90e9948cb075a209c8cb4a822d1a6ec.png

    Then letting

    variational_calculus_2f67c06397db469d5185f4fc49ff9dc7d6e92a0e.png

    the above system becomes

    variational_calculus_d9a43dbfbf69eb2caa518726cca69851dbea558f.png

  • Hamiltonian becomes

    variational_calculus_993fdc5828a3e0d35cea15cdd03a269b4cbe516f.png

    which has the property

    variational_calculus_4dbe4e0ff595b0375fb7120424446186002f12e7.png

    known as Hamilton's equations.

  • Observe that

    variational_calculus_818bd0f1045144180a2dfbcec36b604ebdaf0adb.png

    where the coefficient matrix, let's call it variational_calculus_d9c991e572f71067e0f8c2e2021ee9a8771ac2f1.png, can be thought of as a bilinear form on variational_calculus_e4f375c26796781f71b7ae3026445db617a6e78b.png, which defines a symplectic structure.

  • In general case, existence of solution set to the variational_calculus_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png equations is guaranteed by the implicit function theorem in the case where the Hessian variational_calculus_dbefa664fb20f9145402794c8c245b7edb9b6bae.png is invertible
    • variational_calculus_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is then said to be regular (or non-degenerate)

General case:

  • Hamiltonian

    variational_calculus_215f83ddb0f19ea76fc55523392f32e889daee2c.png

  • Total derivative of variational_calculus_14a40f189f2ce698341c03cd5c099a337431c49f.png (or as we recognize, the exterior derivative of a continuous function)

    variational_calculus_bf86482b6a4823c7c5d97ef6ed8b27baebc53a9b.png

    where we have used that variational_calculus_49eeb555bb2e29d01bba158bf96d7552f52f4924.png.

    • Give us

      variational_calculus_6d08e8aa7402d993ec7a4461cd4dc2d6006a0ed6.png

    • First-order version of Euler-Lagrange equations in canonical (or hamiltonian) form:

      variational_calculus_4ac00d1350f0aaa7b690f6845fcd6bcfa21a3bff.png

      which we call Hamilton's equations.

In general, the Hamiltonian variational_calculus_18a7a25950729ba88ebeab5cff558fd95f2a3d3b.png is given by

variational_calculus_215f83ddb0f19ea76fc55523392f32e889daee2c.png

Taking the total derivative, we're left with

variational_calculus_60f3696658cc13a166312000002cafb7e7039707.png

where we've used variational_calculus_49eeb555bb2e29d01bba158bf96d7552f52f4924.png.

This gives us

variational_calculus_6d08e8aa7402d993ec7a4461cd4dc2d6006a0ed6.png

Conserved quantity using Poisson brackets

Consider energy conserved, i.e. variational_calculus_4ef59f1e3f8dcdba20be45b21f3e2adfbb8e2024.png, and a differentiable function variational_calculus_18d2d61778524354c84f1be3a60509ddc6db938a.png:

variational_calculus_2e34d4f2e4d1321ea8abfae58a6ed7b3135d5856.png

where we have introduced the Poisson bracket

variational_calculus_d7a70c8ca78c142670cb91679ab5f4ebde637f9b.png

for any two differentiable functions variational_calculus_27886bf638af4b4e2dfe9b601583dbc014ae9ac0.png of variational_calculus_2c3f4c69b2df3052c3ed79fde1f1c270b66145ed.png.

Hence

variational_calculus_2a75ed63775c1b94b1e90e93fc5cf2da5cc1bcdd.png

If variational_calculus_05594e8ce8f52728892c4c673d8bc28149b4b24d.png depend explicity on variational_calculus_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, then the same calculation as above would show that

variational_calculus_8f03f190472ff5c75fcd9ea378637e88c8f63528.png

In this case variational_calculus_05594e8ce8f52728892c4c673d8bc28149b4b24d.png could still be conserved if

variational_calculus_fa0691bd17528ff105c706514294e2ff4d1db873.png

Given two conserved quantities variational_calculus_765a999359197ca992dce562a1e707fc89051a36.png and variational_calculus_32b098c3172d47aeb9b7d945b0237687149d4437.png, i.e. Poisson-commute with the hamiltonian variational_calculus_14a40f189f2ce698341c03cd5c099a337431c49f.png.

Then we can generate new conserved quantities from old using the Jacobi identity:

variational_calculus_d303e1984f8942308eee43e73594bd00cf4a5a5d.png

Therefore variational_calculus_5d9e161fd26d25cf305f3d25ea0840a09d3cc5db.png is a conserved quantity.

Can we associate some * to an conserved charge?

Consider a conserved quantity variational_calculus_18d2d61778524354c84f1be3a60509ddc6db938a.png which satisfy

variational_calculus_fbd030ec4d7f2dfaaec8d0529e609244fe1b784d.png

Then

variational_calculus_338e5dbd0f80853afe1cce2892678c8113535125.png

defines a vector field on the phase space variational_calculus_759d91cbd6055f733f91560e09ed06f5e5296dea.png which we may integrate to find through every point a solution to the differential equation

variational_calculus_997844e2a35e084108d847961f6fc0cac8e2e540.png

Then, by existence and uniqueness of solutions to IVPs on some open interval variational_calculus_37cfd5efbb6156137f735745e145d03795b9394c.png as given above, gives us the unique solutions variational_calculus_0e3d5dfd7757e62c359d0d5e8e348cfc7b5a917c.png and variational_calculus_f4109efed68d81b960065af8c33f868c9f95b598.png for some initial values variational_calculus_8c381df6f98957e62deb6ce911da030611d4c244.png and variational_calculus_3064020f7ac6878d8f7821b767502456981f22fb.png.

Therefore,

variational_calculus_ab8266c57f6abcff1ee11db012a53ccfe5bfb318.png

Thus we have a continuous symmetry of Hamilton's equations, which takes solutions to solutions.

That is, these solutions variational_calculus_00efff0fb11452f5aebb175aecf510e6feb99795.png which are in some sense generated from variational_calculus_05594e8ce8f52728892c4c673d8bc28149b4b24d.png and variational_calculus_14a40f189f2ce698341c03cd5c099a337431c49f.png contain symmetries!

E.g. solution to the system of ODEs above could for example be a linear combination of variational_calculus_f745886878b7a69982a7d449c8c0f7b704086913.png and variational_calculus_d307f3c032533d09e3046e36cfd82c213b4fb519.png, in which case we would then understand that "symmetry" generated by variational_calculus_05594e8ce8f52728892c4c673d8bc28149b4b24d.png is rotational symmetry.

The caveat is that this may not actually extend to a one-parameter family of diffeomorphisms, as we used earlier in Noether's theorem as the invariant functions or symmetries.

Example:

  • Lagrangian:

    variational_calculus_d792da09a3db8bb97acdf29eb5afbcaa4f931446.png

  • Exists

    variational_calculus_d0592788cba2aa1c68aa5bea533c7b6d0f8bcc78.png

TODO Integrability

A set of functions which Poisson-commute among themselves are said to be in involution.

Liouville's theorem says that if a hamiltonian variational_calculus_5410c5a1f5d45606181997cf1a786363ad54b5d2.png on variational_calculus_759d91cbd6055f733f91560e09ed06f5e5296dea.png admits variational_calculus_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png independent conserved quantities in involution, then there is a canonical transformation to so called action / angle variables variational_calculus_5a4e65a53530c63056b0ae29c88ddb72dc44f6be.png such that Hamilton's equations imply that

variational_calculus_d7d2760af0d5fa7c97ffe89f24cb1c9febe45b2a.png

Such a system is said to be integrable.

Constrained problems

Isoperimetric problems

Notation

  • Typically we talk about the constrained optimization problem:

    variational_calculus_1eaffb8735901190b63f420c890e760a06abfdc1.png

    for variational_calculus_6c827f1028825031f9fe7705c6efb628ef4bc1da.png.

Stuff

  • Extremise a functional subject to a functional constraint

Consider a closed loop variational_calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png enclosing some area variational_calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

The Dido's problem or (original) isoperimetric problem is the problem of maximizing the area of variational_calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png while keeping the length of variational_calculus_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png constant.

Consider variational_calculus_59f0371243be89cfb62836b504366108ed6089a2.png for variational_calculus_b534646e7618c1bde5e1cd0fc9bb549add242934.png with initial conditions

variational_calculus_bb6da1fea85a8bff34ce4971ff27c7d6d1614d68.png

Using Green's theorem, we have

variational_calculus_cd1f0ae715854eaa12f3b8b8bf610de2d1b953f7.png

and length

variational_calculus_689bdc800a07975d51fdbe685d5f40669f7790c0.png

The problem is then that we want to extremise variational_calculus_adb20a6952631c1eb965d9ea21484958ceedc0f9.png wrt. the constraint variational_calculus_ead7493dbd015e60df3be5e4ef893b9e4466934d.png.

More generally, given functions for variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png

variational_calculus_c8ece6d843a24360534328c2eb7afc0f9a940e0d.png

being endpoint fixed, e.g. variational_calculus_ab414e54b83f44b4f5a6654c3588c1f12e113716.png and variational_calculus_a1e25b7cb95103c4752d28a3635b049f54f4b281.png for some variational_calculus_95ee299416118560918dfdcbcbfd73b4b5bdf060.png.

Given the functionals

variational_calculus_f5b2c4dcc01e11389c39528e2d89c5d8d3bb8267.png

we want to extremise variational_calculus_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png subject to variational_calculus_ce08eb658eeeefed51b91d0b010a0cd720e6b0f1.png.

Let variational_calculus_cbe1f941a4c62754bf4d583c33b8302e2fed526c.png and let variational_calculus_7764516a30eeeb0b11482305c3a21e72d92cdbb1.png be an extremum of variational_calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png subject to variational_calculus_aca5f96b06a711f3752551c212f0066d7607e623.png.

If variational_calculus_e68879b8c8146c6ecb047752ef0edc6e6c86a47a.png (i.e. variational_calculus_c462c980a45481116745a8647094b2b1d245df0f.png is not a critical point), then variational_calculus_b47ca8eefc631cf0a173ea7506b664832f72250a.png (called a Lagrange multiplier) s.t. variational_calculus_63afd0487e287da0e11d6a3d0c99687353e0e162.png is a critical point of the function variational_calculus_2f61d83ffd0606da29b3f9f1fab6964d12723cd1.png defined by

variational_calculus_cd005fe475877598a55f840bfdae77b590bbd947.png

Supose variational_calculus_eb83d466c7d035356e9f39998f357cee73da1e26.png is an extremal of variational_calculus_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png in the space of variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png with variational_calculus_85f5022d3ea5bd4baa28c66d0890eb66b69b6f11.png. Then

variational_calculus_774bc03471065df6af628c6d2cab9167f6b15c71.png

for small variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png, where variational_calculus_e34f86b6c321ea230612d8eeb5254f861dfac16b.png.

This might constrain

variational_calculus_2aee1c4112149de117d281bdeea23b81995ba6cc.png

and prevent use of the Fundamental Lemma of Calculus of Variations.

Idea:

  • Consider variational_calculus_dffd871dfd64bedb2e9c767722dd2328fa0a3dae.png and variational_calculus_f81c84464cf4fbbeb50c98c9f2149dea68aea2ea.png for all variational_calculus_d81b42d161cdcad0243173490517752a20a9d7ae.png near variational_calculus_96f53e8f2667720f54bd85623f46cbf545733989.png, defined variational_calculus_c8300640e0ee5e36eae049ab982964eadce1dbae.png.
  • Then express one of the variations as the other, allowing us to eliminate one.

Let variational_calculus_3f70e72d19a510e9e4047b758d2f34847cefd98d.png be defined

variational_calculus_a7fd8a6033900f5489892f594e26f3d8e1af0774.png

Assume now that variational_calculus_9f3d61ffd9367a482f616bf576d6925f0ddb811c.png and variational_calculus_622b0f23422187e4114b41cae0f6b5121e9b537f.png.

If we specifically consider (wlog) variational_calculus_7c71051f0a4a40d2762051d7d660879eacb37251.png, then IFT implies that

variational_calculus_71461a80a38bd66e7fae1d490ecd5741c3ae5fc5.png

for "small" variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png. Therefore,

variational_calculus_ced7f6048c6b6486fcf5da83d07be79460bbdff3.png

is a

Let

variational_calculus_d61cd535336d82b11a33f701f83a1a60fc97c577.png

be functionals of functions variational_calculus_fc9c2afcd4f065824121d3378427c407c9dd70d8.png subject to BCs

variational_calculus_e9da60290fb718e835586a0cebbd25c647bde3e3.png

Suppose that variational_calculus_6d82e966445a565af1e609bffa0808e04ef6299b.png is an extremal of variational_calculus_d9c991e572f71067e0f8c2e2021ee9a8771ac2f1.png subject to the isoperimetric constraint variational_calculus_5a8556fcea83ac9905fe9c9a48ed3d63e85fc405.png. Then if variational_calculus_eb83d466c7d035356e9f39998f357cee73da1e26.png is not an extremal of variational_calculus_8adf8dbafca41879a81d94a89f336dc7aaa1c5ed.png, there is a constant variational_calculus_da9eb912c1f040970a272b334e2197c291627041.png so that variational_calculus_eb83d466c7d035356e9f39998f357cee73da1e26.png is an extremal of variational_calculus_25c0f38f6c3ecb72435711834baac78f68ee0a06.png. That is,

variational_calculus_9e03849b0d1aa7d51175d36fe04e190be43e9bce.png

Method of Lagrange multipliers for functionals

Suppose that we wish to extremise a functional

variational_calculus_1eaffb8735901190b63f420c890e760a06abfdc1.png

for variational_calculus_6c827f1028825031f9fe7705c6efb628ef4bc1da.png.

Then the method consists of the following steps:

  1. Ensure that variational_calculus_8adf8dbafca41879a81d94a89f336dc7aaa1c5ed.png has no extremals satisfying variational_calculus_5a8556fcea83ac9905fe9c9a48ed3d63e85fc405.png.
  2. Solve EL-equation for

    variational_calculus_895d95305ff3ef4d91dfd12a640f42c193c48f23.png

    which is second-order ODE for variational_calculus_6d82e966445a565af1e609bffa0808e04ef6299b.png.

  3. Fix constants of integration from the BCs
  4. Fix the value of variational_calculus_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png using variational_calculus_5a8556fcea83ac9905fe9c9a48ed3d63e85fc405.png.

Classical isoperimetric problem

The catenary

Notation
  • variational_calculus_058c565d6cf273aa3406a878a8de7dda2bd05d9a.png denotes height as a function of arclength variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png
Catenary
  • Uniform chain og length variational_calculus_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png hangs under its own weight from two poles of height variational_calculus_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png a distance variational_calculus_11c189267da2e02cdcbf1a07a43d0c478436c517.png apart
  • Potential energy is given by variational_calculus_baa757b11196483adc7a557882e85cf608221e99.png, which we can parametrize

    variational_calculus_a65201071efdb3a253fa16cc78d709cff18fe0b8.png

    giving us

    variational_calculus_ba084eaa9d20a2b081d033c6883d69b3b18cf690.png

  • Observation:
    • All extremals of arclength are straight lines
    • Extremal to constrained problem has non-zero gradient of the constraint
    • → straight lines are not the solutions
  • Consider lagrangian

    variational_calculus_4af8dfe9d822d0c71a85c2f36ece2e808d49740f.png

  • Using Beltrami's identity:

    variational_calculus_d77f35a5037883ed386de7b727189ea1605a3043.png

    rewritten to

    variational_calculus_9480e1e7151dcc60de0d6cc84ff2f7afc36c40b8.png

    which, given the BCs variational_calculus_df8000bbc98ba44eddb384259089764d39d4f187.png we get the solution

    variational_calculus_1c7f8ce50316409169d49c1a91e1336993724125.png

    which follows from taking the derivative of both sides and then solving.

  • Impose the isoperimetric condition:

    variational_calculus_d0df1dfe0d97c9aae7350f411396d23ccc57b0ce.png

  • Introducing variational_calculus_e4d020039957ceae0e3511faa1362fd48aaded00.png, we find the following transcendental equation

    variational_calculus_242d757fd17ffdd63b6b4c6c56b9e8000fb97024.png

    for which

    • variational_calculus_2f25c7956c73eaee508a7f812f75501ebf0763b7.png is the trivial solution
    • variational_calculus_5f12655585df2d0fd769908f9a5fa984dcb395c4.png small, together with the condition variational_calculus_f71d0bea0fac6ac0ce46f8d9f9f7fd3a9ade6543.png ensures that variational_calculus_cfe2a0fe4c98b38c9518246f43fcd6eec2c2c3ae.png
      • But for variational_calculus_be8b148475def07d0d6bfafb1e286f5517937061.png the exponential term dominates

        variational_calculus_92736102f5f128069a2b83ba4e6db5f48eb96593.png

        by continuity.

Holonomic and nonholonomic constraints

We say a constraint is scleronomic if the constraint does not depend explicitly on variational_calculus_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, and rheonomic if it does.

We say a constraint is holonomic if it does not depend explicitly on variational_calculus_d198c0d06b0da5525f9a966222601066509ac01d.png, and nonholonomic if it does.

In the case where nonholonomic constraints are at most linear in variational_calculus_743b57d75a51a9ef7cfdaddc1733f629efebd8db.png, then we say that the constraints are pfaffian constraints.

Typical usecases

  • Finding geodesics on a surface as defined as the zero locus of a function, which are scleronomic and holonomic.
  • Reducing higher-order lagrangians to first-order lagriangians
  • Mechanical problems: e.g. "rolling without sliding", which are typically nonholonomic

Holonomic constraints

Let variational_calculus_dc51a789bf4b58f46b5e0f9d8029fc146d5b4dad.png be a smooth extremal for

variational_calculus_80451cd3f907c1e62a6d6157150884c038c1f8bd.png

with variational_calculus_ea1b0cb3aec565a9466dd0501b3f1b44034a310b.png for all variational_calculus_b534646e7618c1bde5e1cd0fc9bb549add242934.png.

Then there exists variational_calculus_3907f48f13fdd58f911eae69041370a0ab4773fe.png such that variational_calculus_7a84c9a383f9772338016d101ccc096be06af784.png obeys the EL equations of

variational_calculus_b32b824cd2045a4042fe13b523c45c4c8ff1a2e6.png

variational_calculus_6db3dcb7183928c90917445e35f06c9b9fa5a2a4.png is the gradient wrt. variational_calculus_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png for all variational_calculus_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png, NOT variational_calculus_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png.

Let variational_calculus_e093086826fdb1bea5d0688ac237bbeca2d73ab6.png be admissible variation of variational_calculus_7730e2ad24baf39fefb8984dbfeb6445d5a14518.png.

Then the constraint variational_calculus_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png requires

variational_calculus_d291f232670bea73bdd10d8342f727df37492afa.png

Let

variational_calculus_a1968f8df8367046cb036906fa49d6b64a0a6a07.png

Then the contraint above implies that

variational_calculus_642309bc843c8bd007200c0531a832756de118c8.png

i.e. if variational_calculus_0264d457a79bf4db510b9d548fc9bd6f44d19bec.png then variational_calculus_224ae917d74dc2133d4403064c971bf562d4db50.png is tangent to the implicitly defined surface variational_calculus_36c1582aa697900e91a6ea208160ad06f69f1c2c.png.

Consider variational_calculus_42818c8ce1e42b57391f6ca31faab3ca421e7d43.png, then the above gives us

variational_calculus_1c6962b0f4483a4d67250e29fdf14320573ccc9f.png

We therefore suppose that

variational_calculus_6151c92138df3d62aa6b487c5d847fd4e2061e28.png

The implicit function then implies that we can solve variational_calculus_7bf0e4ff3bcc6f50982e77e297f26b2d936cc1a9.png for one component. Then the above gives us

variational_calculus_4af87111feecfe111db4e615f86171bf63e49c70.png

i.e. variational_calculus_36618a8478f59fd497ff1c3ae5edfb9a30bda857.png is arbitrary and variational_calculus_e19e93deb6a0704da92433902b100d2327223167.png is fully determined by variational_calculus_36618a8478f59fd497ff1c3ae5edfb9a30bda857.png in this "small" neighborhood variational_calculus_78ebf6fb1550e2700b0a582d651eabed602ed0eb.png.

Then

variational_calculus_01c37ddb9bcccfb99b3e2c9e335ec192e6ef585b.png

Since variational_calculus_36618a8478f59fd497ff1c3ae5edfb9a30bda857.png is arbitrary, FTCV implies that

variational_calculus_a5e4e1855ccbcc0ef0f25a0f3c4fb919238e52f3.png

Which is jus the E-L equation for a lagrangian given by

variational_calculus_94deb3679b7f9e8e293160afc43510a3d79c55b8.png

Above we use the implicit function theorem to prove the existence of such extremals, but one can actually prove this using something called "smooth partion of unity".

In that case we will basically do the above for a bunch of different neighborhoods, and the sum them together to give us (apparently) the same answer!

Nonholonomic constraints

Examples

(Non-holonomic) Higher order lagrangians

variational_calculus_9fc1dbc182981620358541021e10be9ee8b1c8f4.png can be replaced by variational_calculus_a0dec78b6f6739fedb04c0954ff679067431b70a.png

variational_calculus_e2bfacd534e3fec84c12d32c799e4f2b661f2fe9.png

which are scleronomic and nonholonomic.

So we consider the Lagriangian

variational_calculus_978e216aa94bd009c946e37985819622e3c519a6.png

Then

variational_calculus_a5a04124776d7f3b67db2fb3e6b24cc040a8ac20.png

So the E-L equations gives us

variational_calculus_27ccd761419b8c47695ac73bc38de8679ec5d2bf.png

which gives us

variational_calculus_5d491dde7a11f02495e30775ff8da0b62fc085e9.png

where we've used

variational_calculus_d268b151582972a61e0af52c5a4d278e9be5cf38.png

Variational PDEs

Notation

  • variational_calculus_acb958fd5cfc048a9cdaa2197f00e12d182cf3b4.png be a variational_calculus_abfc81685b43058ceaa81aa83ec4f0a65a5358d8.png function on the set variational_calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png
  • Lagrangian over a surface variational_calculus_f37531bd1f253d239ad77df5db01208c978ccf73.png with corresponding action

    variational_calculus_d754a935ac9afd62a8b288a874b6b87e4bf90bc2.png

    where variational_calculus_186e3e9d90efcdfee8537c008e6397df204260d3.png and variational_calculus_8ff3aa450a42db35b51b8a3fb2dfe7b33c136219.png denote collectively the variational_calculus_ea678c796505a80071595b15988a423ca4da3c76.png partial derivatives variational_calculus_c9897e07e94aab678228a4191c96b0ef4761d420.png and variational_calculus_60f9e34004aae0626565ce79a640e10abb3e4083.png for variational_calculus_b04e17015e8dba88fcf02612546124f7929636b2.png

  • BCs are given by

    variational_calculus_2a0da1f20d318fa4b0f483734a48bc8b82c0746f.png

    where variational_calculus_cf788d00275d0e49c22d0ffce8e1411c7edc6e68.png is given

  • Variations are variational_calculus_f4b5d8e2f52c1949a165d5632709a0bc237a58ed.png functions variational_calculus_2bb92918f7e87c39f1798c576d9d381eec2c5961.png such that

    variational_calculus_f14c286fd8955ef5f5d6ec56a48d70571db91867.png

Stuff

variational_calculus_d754a935ac9afd62a8b288a874b6b87e4bf90bc2.png

Then

variational_calculus_ffd6e986f29f7a5e37342fabc7974df95d88ea2f.png

where we've used integration by parts in the last equality.

The Divergence (or rather, Stokes') theorem allows us to rewrite the last integral as

variational_calculus_59cc15c528273e2f2af2af806eb7563bd419e6bc.png

where variational_calculus_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png is the arclength. And since variational_calculus_219f9ddacd857cf4db1187f15d74c6feac195943.png this vanishes.

Generalisation of the FLCV the first integral term must vanish and so we get

variational_calculus_afbde94f380f2d15c3bf5ce8b85b220dc893d45f.png

We can generalise this to more than just 2D!

Multidimensional Euler-Lagrange equations

Let

  • variational_calculus_1e9f4955959fbffb135472ac9a33daae01e37922.png be a bounded region with (piecewise) smooth boundary.
  • variational_calculus_e4629acebd9e992b697ceaca162c0ad3f45f4830.png denote the coordinates for variational_calculus_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png
  • variational_calculus_b97a17a4deba7678548f9bc62ff768aa6b24cf52.png be the Lagrangian for maps variational_calculus_f5be69a14f3dee1e584b604cd759c6df0ef15ae9.png where variational_calculus_5c949f8a367efe9905f144c2012cf5f94f261755.png denotes collectively the variational_calculus_15322ad1f6dcb92909f253f1625bdc79548e8788.png partial derivatives

    variational_calculus_ac8273328c028b78068e084424fa26dd3922af4b.png

Then the general Euler-Lagrange equations are given by

variational_calculus_bc66258f682d7b5e8da894bdcf7b0afef624e618.png

using Einstein summation.

Notice that we are treating variational_calculus_1232201903c8d9bbb7aed799bb21ebaa03cccf4c.png as a function of the x's and differentiate wrt. variational_calculus_ce70902ffb03450d510daa52f96fe692783c3885.png keeping all other x's fixed!

(This is really Stokes' theorem)

Let

  • variational_calculus_1e9f4955959fbffb135472ac9a33daae01e37922.png be bounded open set with (piecewise) smooth boundary variational_calculus_c33799d5ac4c860cc786049110a86841d538e94b.png
  • variational_calculus_996475e0fef0f7fa9527695f945d6c28af091c09.png be a smooth vector field defined on variational_calculus_6cea4d4bfb66ba3899f48fc1e02a119e8f7a9053.png
  • variational_calculus_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png be the unit outward-pointing normal of variational_calculus_c33799d5ac4c860cc786049110a86841d538e94b.png

Then,

variational_calculus_d3283adb207f12a187a70bcca4be94eea631822b.png

(using Einstein summation) where variational_calculus_2921b52e8f7c71aec10c79258e2a445b2a2b8fb2.png is the volume element in variational_calculus_e04952b75c7e1b1f43954670321ab89bd8d2e93b.png and variational_calculus_ab16cb534bc30ec44aeae384814052b32ed7a1be.png is the area element in variational_calculus_c33799d5ac4c860cc786049110a86841d538e94b.png and variational_calculus_84d9d552378b49d6862906448ec3cab2d0be627c.png denotes the Euclidean inner product in variational_calculus_e04952b75c7e1b1f43954670321ab89bd8d2e93b.png.

Let

  • variational_calculus_1e9f4955959fbffb135472ac9a33daae01e37922.png be bounded open with (piecewise) smooth boundary variational_calculus_c33799d5ac4c860cc786049110a86841d538e94b.png
  • variational_calculus_1914f85d6fab75e0d16a289824d046439b27624c.png be a continuous function which obeys

    variational_calculus_963c03c440256691465a0d1e6f813c87550305d2.png

    for all variational_calculus_142e99eaf45f1872ae84813baea3d90f924b5333.png functions variational_calculus_6209d57f4fff6db156abb1f5a2772b790e21892d.png vanishing on variational_calculus_c33799d5ac4c860cc786049110a86841d538e94b.png.

Then variational_calculus_ef4ff0ef7784783ce7d106b2ac7f6de8a32823be.png.

Noether's theorem for multidimesional Lagrangians

Notation

  • variational_calculus_1e9f4955959fbffb135472ac9a33daae01e37922.png
  • Lagrangian

    variational_calculus_893630b152f8d55d0fcad1ca542af51ec1360af6.png

    where

    variational_calculus_8560ba7bb0ee353e60fbfd0219c7780ca30c54af.png

  • Use the notation

    variational_calculus_bcf8a73ad26eb011a59de3a58e08b25f725634e4.png

  • Conserved now refers to "divergenceless", that is, variational_calculus_d9c991e572f71067e0f8c2e2021ee9a8771ac2f1.png is a conserved quantity if

    variational_calculus_24199a8eadea72b8fc9587d352a5b42f11b4a157.png

    where we're using Einstein summation.

  • variational_calculus_2a87bc0e3e3cce4abf1521d043973e7d8e1a3141.png for variational_calculus_d3891705174fad008f05f86fdc2bab3159fd35f3.png denotes a one-parameter group of diffeomorphisms
  • variational_calculus_103a84cb2338542a6076188139c5c68014f25f9f.png is defined

    variational_calculus_ea3760921c7fa3b1475c453f583a00c24d2a3dba.png

Stuff

  • We say it's a conserved "current" because

    variational_calculus_00254dd0a421f608eeaaf73bb2f02329e87f9701.png

    where variational_calculus_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png denotes the normal to the boundary

Consider

variational_calculus_90223a002b77be1e4955d4f68a4219d553cab17d.png

And let

variational_calculus_02bbbd5e7c2aaa4b002a65dde61971204b30dd1f.png

such that

variational_calculus_7a7cb5329260ea113f869b65eb024cd416565e7d.png

We suppose that the action is invariant, so that the Lagrangian obeys

variational_calculus_4e9db85272683e27a5f077e04710321d2ae072bd.png

or equivalently, one can (usually more easily) check that the following is true

variational_calculus_90b188832045fa301f7299c5d9418c3a55afdce0.png

The Noether current is then given by

variational_calculus_47a9fdda1f35e9f31820ca46b7a19c4525297637.png

variational_calculus_454c2b200f93192b6cefa01dbfc5cf0445d993af.png

since the RHS is independent of variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png. The LHS on the other hand is given by

variational_calculus_716e93035205f95d17155a3e85533b66717594ff.png

where variational_calculus_f11ce59a731ad899c68bea8661c7d34a3d0647b7.png.

Now we observe the following:

variational_calculus_47591142dc90fc5a40ac33e602f0b544e67a29db.png

and

variational_calculus_30802beb827f9520a5b6663700507d21243fbd43.png

where we've simply taking the derivatives of the Taylor expansions. Hence, we are left with

variational_calculus_0f3ccbc5c90bb9ff72914fe51aff7905e26d57eb.png

Now we need to evaluate variational_calculus_2c0b90d66bd58ed8ed95b0358f6c87801e2673b4.png and it's derivative wrt. variational_calculus_aa1698cb8ee1665238ec3e91824191643c62ee93.png. First we notice that

variational_calculus_5c5944cb9743c7d9361747fded12c9041578d3d4.png

We now compute variational_calculus_af0b3023bd468e70dda18e0ecb376f5e4b538b4e.png. We first have

variational_calculus_eee4bb33b05abff611704277c5ee11f5d45ff454.png

Finally using the fact that if a we have some matrix variational_calculus_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png given by

variational_calculus_e42382e696f2fcfbbe4927de2b4bb81f2f1bad54.png

we have

variational_calculus_ecf2cc19cbd8eb79933ef6235b201b57c44a6ed0.png

Finally we need to compute variational_calculus_3cf0457d94b2e4a786be6364a69c9284bf7d5151.png, which one will find to be

AND I NEED TO PRACTICE FOR MY EXAM INSTEAD OF DOING THIS. General ideas are the above, and then just find an expression for the missing part. Then, you do some nice manipulation, botain an expression which vanish due to the EL equations being satisfied by the non-transformed variational_calculus_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png, and you end up with the Noether's current for the multi-dimensional case.

Examples

Minimal surface

Let variational_calculus_0aff32cb2918305c8f1b50f35ab9299de8b3fcde.png be a twice differentiable function.

The grahp variational_calculus_103b1420072afa000514c1f12d0d30ff4f5095f9.png defines a surface variational_calculus_6c901fbc4cd9fb30cb7dc6db72ee435613d74c78.png. The area of this surface is the functional of variational_calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png given by

variational_calculus_84e4692754d31c2e0c9777afed3c90624399f7bd.png

If variational_calculus_93369077affb352dbdb97e8b3182fd50784f2b14.png is an extremal of this function, we say that variational_calculus_017948b866be67b1a8e56a6b5f8848f823410c34.png is a minimal surface.

In this case the Lagrangian is

variational_calculus_2691f987c7b9ea8fe430f007c16838fabffaa5ed.png

with the EL-equations

variational_calculus_1cddef653a4713299fb5f6f90fae7cf04446e9b8.png

where

variational_calculus_6c9e69657c78fa8ca44140d7850457f24cade838.png

Therefore,

variational_calculus_9db326ecb5ce7aac03c219268b66c5a91b469d49.png

and similarily for variational_calculus_eb83d466c7d035356e9f39998f357cee73da1e26.png. When combined, and multiplied by variational_calculus_aca200dfb5c9acd05e7cedee0722f936df888864.png since the combination equal zero anyways, we're left with

variational_calculus_69ccc46a07bc227a7586e85d2788d0b376af9f2f.png

where we've used the fact that variational_calculus_d60df569da3b7915c2ef87c918be7d6f4a5f057d.png.

This is then the equation which must be satisfied by a minimal surface.

variational_calculus_e0ac87dd85f167fe86f65516d0bbb3caa5c866b6.png model

variational_calculus_1893bd523dc52e1ed0de11d230141092c6ba904c.png

then

variational_calculus_832307d5276765df84568bff7be97395a39b9ebc.png

Then

variational_calculus_bf448dedcccae91b8021950977d9e5aea703147d.png

Then the Noether's current variational_calculus_d9c991e572f71067e0f8c2e2021ee9a8771ac2f1.png is

variational_calculus_f013312619ea6e54ad141924ebb352b612918e37.png

which explicitly in this case is

variational_calculus_f1cd848a1a1e93c1c90f385b4d3bff2f9eafa391.png

Then

variational_calculus_b00629275d0e1ac80725d75c4630990bdab90a30.png

Noether's current for multidimensional Lagrangian

Classical Field Theory

Notation

  • variational_calculus_8a0205fed64a0a405dbac72eb0a1a02d148db620.png denotes a field written variational_calculus_ce33476b15d1116b2d97ea42417959b16f428af4.png
  • variational_calculus_efe71591afc4e44f5c259adff77c9bfcd4a96caf.png
  • Concerned with action functionals of the form

    variational_calculus_f576f5562dd345c4c8069f41b997a4b79743c01c.png

    where variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png is called a Lagrangian density and variational_calculus_c1d7d6357e16149f504542458ac45d558c803cab.png, i.e.

    variational_calculus_19f659c29d203b502db958c6a92b759eff661d09.png

  • variational_calculus_a14fffc5af6718ab08dc4ea0d940594e3397751e.png for variational_calculus_95491f345e8114759e9af07020841db8b0a6ead4.png for some variational_calculus_3d136f0fc4860468633907421c098b9feb0eef24.png and

    variational_calculus_996b6baa943fa5da72ccb91797d3858a20b75d48.png

  • variational_calculus_ad52d259865a7b699571fe42e46448c9625ad3f1.png denotes a "cylindrical" region

    variational_calculus_e88644bfeb79df18dd51592a01341aa8707d6148.png

  • variational_calculus_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png is the outward normal to the boundary variational_calculus_ad52d259865a7b699571fe42e46448c9625ad3f1.png

Stuff

Lagrangian density variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png is just used to refer to the fact that we are now looking at a variation of the form

variational_calculus_a73d86ad885250fc746810480c346ee68efe36ac.png

So it's like the variational_calculus_16114146bd11e95b6055d55fde0bcb2c74a12444.png is the Lagrangian now, and the "inner" functional is the a Lagrangian density.

Klein-Gordon equation in variational_calculus_8c06529d7bf2008e2a7467207a3fff55abe292c9.png (i.e. variational_calculus_30e9a33763263ddcf6414890e892d9384c32bf30.png) is given by

variational_calculus_288054cb24ee8f7ccba2e625e2e43dce760ddef4.png

where variational_calculus_4174cb28db875db895768f049bab0e4942ebfd79.png is called the mass.

If variational_calculus_8015f9ec50140ce02143b9ef0580fa0f4302e44e.png, then this is the wave equation, whence the Klein-Gordon equation is a sort of massive wave equation.

More succinctly, introducing the matrix

variational_calculus_d747f66df9f88d52af6ef0d57a8882137c5a4593.png

then the Klein-Gordon equation can be written

variational_calculus_106857b295f349af946c09c61c653cd670de3baf.png

Note: you sometimes might see this written

variational_calculus_32f3295a8c68a49c47ed8b759ea9f7b7e2fdad25.png

where they use the notation variational_calculus_62efc4e1e1287b85c4ae135c6b3c4a6765a94dd3.png so we have sort of "summed out" the variational_calculus_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png.

Calculus of variations with improper integrals

Noether's Theorem for improper actions

  • Consider action function for a classical field variational_calculus_4a8351fa9026579beaa2bd5a8dae0b3eda94bf53.png which is invariant under continuous one-parameter symmetry with Noether current variational_calculus_f82dbc8624c79219ceafcf1ea2361687bbb76faa.png
  • Integrate (zero) divergence of the current on a "cylindrical region" variational_calculus_441efb3f16fa3990ec4a20b4a24001490e9b6c5c.png and apply the Divergence Theorem

    variational_calculus_405bd56f455cd7c1a51103c236f28042d76e053e.png

  • variational_calculus_cb9e59ff4fff6a9e86eae883b5500bc809b0d6f1.png consists of "sides" variational_calculus_7d45a88ddbfa72b4d1b4fbbc711de93ac8664ba1.png of the "cylinder", where
    • variational_calculus_54e7d12ced5e21df31c9ea823fd792662e7d691e.png is the m-sphere of radius variational_calculus_3d136f0fc4860468633907421c098b9feb0eef24.png
    • top cap variational_calculus_50052648b9eeff9406e634c74c94a3c75cf8545c.png
    • bottom cap variational_calculus_7397088f7bfb65ba9131f15033e021bc1754f895.png
  • Can rewrite the above as

    variational_calculus_99d2480be6b4665558e352bf2935964811190cb7.png

    using the fact that variational_calculus_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png points outward at the bottom cap → negative variational_calculus_541d445ce1ef3ec6dbc06faa68e8cf78c2b10b32.png axis

    • Last term vanishes due to BCs on the field implies variational_calculus_38c12f4db45267398f1e65596ee08751809c17e1.png on variational_calculus_7d45a88ddbfa72b4d1b4fbbc711de93ac8664ba1.png as variational_calculus_b94245b89d479b8150bfcbf7246a726254e3f773.png
  • Since variational_calculus_01a878c23dd20c1b291712c8a1306f1fb3e8976a.png arbitrary, we have

    variational_calculus_12dfdb6cff8b9ba06595aad77f00fd737653a12b.png

    is conserved, i.e. we have a Noether's charge for the improper case!

Maxwell equations

Notation

  • variational_calculus_8d0b618205a8cc8d6e21fd38f550c9fd73d9998e.png is the magnetic field
  • variational_calculus_af75e3aee9a5048ccb379313f6cd15dcd718cb1c.png is the electric field
  • variational_calculus_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is the electric charge density
  • variational_calculus_516c45ae557d7e629e472fedef5068157b27516d.png is the electric current density
  • variational_calculus_5621442eb84314df5b6d4f90193ccefd86434bb0.png is the magnetic potential
  • Let

    variational_calculus_02a74101e8240c7c965af9e580cb5354ba1e1907.png

Stuff

  • Maxwell's equations

    variational_calculus_4ebf106a1a437fda73067741c76286862c8d348d.png

  • Observe that variational_calculus_f473861c81eaf056cd09f90792d2c101bac377de.png can be solved by writing

    variational_calculus_02a74101e8240c7c965af9e580cb5354ba1e1907.png

    • Does not determine variational_calculus_8d0b618205a8cc8d6e21fd38f550c9fd73d9998e.png uniquely since

      variational_calculus_d6387d49d95a00a995b3cfeb80cd0ab9c35d37f6.png

      leaves variational_calculus_8d0b618205a8cc8d6e21fd38f550c9fd73d9998e.png unchanged (since variational_calculus_04d63d664f8d7b4bece6a178e5b174de0bdc6c07.png), and is called a gauge transformation

  • Substituting variational_calculus_e2f4b8c492a28bedad0b878e1de67e732f1500d3.png into Maxwell's equations:

    variational_calculus_7161179997a4eac263b2bdf374b09314590304d4.png

  • Thus there exists a function variational_calculus_d21892f3bdfae9ec08a78f3061484c467c1030ec.png (again since variational_calculus_04d63d664f8d7b4bece6a178e5b174de0bdc6c07.png), called the electric potential, such that

    variational_calculus_e6190b27f090da5bc28af5107d6052397b1222bf.png

  • Performing gauge transformation → changes variational_calculus_5621442eb84314df5b6d4f90193ccefd86434bb0.png and variational_calculus_af75e3aee9a5048ccb379313f6cd15dcd718cb1c.png unless also transform

    variational_calculus_102d058801aa664e6a9127438921f135d0497885.png

  • In summary, two of Maxwell's equations can be solved by

    variational_calculus_a656ca5402c5835af8de70e8e55b47cc03610e2f.png

    where variational_calculus_5621442eb84314df5b6d4f90193ccefd86434bb0.png and variational_calculus_d21892f3bdfae9ec08a78f3061484c467c1030ec.png are defined up to gauge transformations

    variational_calculus_8116cf59b0885d8eb7d954fa4d8e8f12b06b9935.png

    for some function variational_calculus_025f7afcfed9615d1f65a005e078e9bb9a811ef6.png

    • We can fix the "gauge freedom" (i.e. limit the space of functions variational_calculus_7bec91615d02a5342b34270ae4a669947491ecaf.png) by imposing restrictions on variational_calculus_7bec91615d02a5342b34270ae4a669947491ecaf.png, which often referred to as a choice of gauge, e.g. Lorenz gauge

The ambiguity in the definition of variational_calculus_5621442eb84314df5b6d4f90193ccefd86434bb0.png and variational_calculus_d21892f3bdfae9ec08a78f3061484c467c1030ec.png in the Maxwell's equations can be exploited to impose the Lorenz gauge condition:

variational_calculus_5e65a12f6fbb9e750bac7332c5ca365a801b288e.png

In which case the remaining two Maxwell equations become wave equations with "sources":

variational_calculus_b1d57848089edf0ddaf9254cc46e28586aca2c77.png

From these wave-equations we get electromagnetic waves!

Maxwell's equations are variational

  • Let variational_calculus_55bcb50ac4b0cc12e0164b5e1ee7f3bf00ccf210.png and variational_calculus_0106768838479ca9a89131a9bb83786aa4b7bba4.png at first
  • Consider Lagrangian density

    variational_calculus_c0b1edddf5456854a310c36a8c6fe39147437254.png

    as functions of variational_calculus_5621442eb84314df5b6d4f90193ccefd86434bb0.png and variational_calculus_d21892f3bdfae9ec08a78f3061484c467c1030ec.png, i.e.

    variational_calculus_899d0b0e496b2d50083d2f9cd9d296c276ccef51.png

  • Observe that

    variational_calculus_9808de75a96b24da25b0d1316f5bc161db364d32.png

  • variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png does not depend explicitly on variational_calculus_5621442eb84314df5b6d4f90193ccefd86434bb0.png or variational_calculus_d21892f3bdfae9ec08a78f3061484c467c1030ec.png, only on their derivatives, so E-L are

    variational_calculus_a314f739d28eb815a6d722e1f4b4feb931150a26.png

    and

    variational_calculus_1cd5f21b97b388dd37ceb247266c042ffb0748c2.png

    which are precisely the two remaining Maxwell equations when variational_calculus_55bcb50ac4b0cc12e0164b5e1ee7f3bf00ccf210.png and variational_calculus_0106768838479ca9a89131a9bb83786aa4b7bba4.png.

We can obtain the Maxwell equations with variational_calculus_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png and variational_calculus_516c45ae557d7e629e472fedef5068157b27516d.png nonzero by modifying variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png:

variational_calculus_ccfc70b15aa5c74c64411b47a12684d500244845.png

We can rewrite variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png by introducing the electromagnetic 4-potential

variational_calculus_167b095d33328dd9076d39cfea9b4d387e388acf.png

with variational_calculus_46d4366722a48a3a9e219cfb0f6dab417ccb60ea.png so that variational_calculus_5396c4e1bfa7fff4f253cafd4afd2ef6b190ba86.png.

The electromagnetic 4-current is defined

variational_calculus_6cb31269c91484fd859009fac282f7cc9134741f.png

so that variational_calculus_66ab9f8cbe06df6c246152a11408f1827436a0d2.png.

We define the fieldstrength

variational_calculus_64e8695ef99ea8af094e371d5bcb3818cd4cca2a.png

which obeys variational_calculus_d7c0e7517d5cc9e0fb0beed3c1e0e77b71f8b5f0.png.

We can think of variational_calculus_6f7e81759a3d23b41d3a6ddf629263f67f98a0d8.png as entries of the variational_calculus_f5e74029134a733a377bd9a29ecf0454844aaa46.png antisymmetric matrix

variational_calculus_615871afba859c42da27a5b6a167769b14fd8da9.png

where we have used that

variational_calculus_957b562f3d22b59e606bca49226d8a4dbb582eed.png

In the terms of the fieldstrength variational_calculus_6f7e81759a3d23b41d3a6ddf629263f67f98a0d8.png we can write Maxwell's equations as

variational_calculus_05e37d417e9ff5fb07515b81f60cc193bc05b34c.png

where we have used the "raised indices" of variational_calculus_6f7e81759a3d23b41d3a6ddf629263f67f98a0d8.png with variational_calculus_34d8f49000772bd2630294b5f3ef7fc35e27ad9b.png as follows:

variational_calculus_db4aa8259a69cd78712156547c01353cfe6c40a6.png

The Euler-Lagrange equations of variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png are given by

variational_calculus_fc3476672c5db5fc4e3cb6e8f112da93c5cd60fa.png

and the gauge transformations are

variational_calculus_6c6671912aeacc73603093ff3aaad37a5f882ef1.png

under which variational_calculus_6f7e81759a3d23b41d3a6ddf629263f67f98a0d8.png are invariant.

In the absence of sources, so when variational_calculus_cd37b5afaf24d2aae92d0ba2d0a02a1a4e704171.png, variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png is gauge invariant.

Let variational_calculus_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png denote the action corresponding to variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png, then

variational_calculus_37ef7f6742786e6f0432db05a0e9fcfe9a9a1a8b.png

where

variational_calculus_40ff57fe626692d4b5c5fcfceb1c62c7a431e195.png

and

variational_calculus_1c377a0ed5590507e578cb789f1b216982629a2b.png

variational_calculus_7677942ed3a368583de4e3d2f570bf23a628a949.png

Therefore,

variational_calculus_87598508df4b3cfd564202f5cf8feff8ef219d0f.png

Substituing this into our E-L equations from above, we (apparently) get

variational_calculus_55866ae5070a6f4bed331115f6556fd507b8c423.png

In the absence of sources, so when variational_calculus_cd37b5afaf24d2aae92d0ba2d0a02a1a4e704171.png, variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png is gauge invariant. This is seen by only considering the second-order of the transformation

  1. First show that variational_calculus_ea79abafc5317bf8aa5cdab40cae77f94d9e75f4.png is invariant under the following

Consider

variational_calculus_c8af376e984f200be2c26d624dfe3bf45b7a75a7.png

where

variational_calculus_b3aa1fbf6ef3eddadb6c6100b794372dffbd82c1.png

and

variational_calculus_980508663f2ea73f812f70f541c370f834a6351b.png

then

variational_calculus_448dedc1f83fdbfe920304c7c837708b237f5c38.png

  1. Find the Nother currents variational_calculus_97deda59aa89ca5660f8b5df63ea681db5f5395e.png and variational_calculus_cd2c57dc3a6506c09160e6672cb6b248cedb2b22.png

Examples

The Kepler Problem

  • Illustrates Noether's Theorem and some techniques for the calculation of Poisson brackets
  • Will set up problem both from a Lagrangian and a Hamiltonian point of view and show how to solve the system by exploiting conserved quantities

Notation

  • Two particles of masses variational_calculus_937ac307652c479877aab0f9607fea79ea25f7d0.png and variational_calculus_921df09a52c87777e1ac388adf65f0cce2ab0929.png moving in variational_calculus_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png, with variational_calculus_868de2d73f553ca62e999768ad7c2950547e7867.png and variational_calculus_3d42d41e2ac965ca0b43bc50570bfa5d441b6fae.png denoting the corresponding positions
  • Assuming particles cannot occupy same position at same time, i.e. variational_calculus_65286941425c5b970766c3813af941fec541d018.png for all variational_calculus_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png.
  • We then have the total kinetic energy of the system given by

    variational_calculus_8221e2aad58b36bf6e57f49c308a8d321c75d734.png

    and potential energy

    variational_calculus_37f4b5b4adad64898171a753d4d8a3dd05055c7d.png

Lagrangian description

  • Lagrangian is, as usual given by

    variational_calculus_f37068a064a2788955b783b6c5ef1f04c8c926b3.png

  • variational_calculus_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is invariant under the diagonal action of the Euclidean group of variational_calculus_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png on the configuration space, i.e. if variational_calculus_c1bab56697b041aa6e5c6a8ba9245c63662b39fb.png is an orthonormal transformation and variational_calculus_5c9f7d532635801426664d6c9f3dacee1ea4c4de.png, then

    variational_calculus_305f195aa064ebda814ef43fa4dcc7b1947b2c5d.png

    leaves the Lagrangian invariant.