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Hamilton’s principle Variational Methods & Optimal Control lecture 25 Matthew Roughan <matthew.roughan@adelaide.edu.au> Discipline of Applied Mathematics School of Mathematical Sciences University of Adelaide We now have a group of equivalent methods ◮ Euler-Lagrange equations ◮ Hamilton’s equations ◮ Hamilton-Jacobi equation We saw earlier that these can give us other methods ◮ Hamilton’s principle ⇒ Newton’s laws of motion ◮ When L is not explicitly dependent on t, then the Hamiltonian H is constant in time. ⊲ conservation of energy ⊲ this is an illustration of a symmetry in the problem appearing in the Hamiltonian April 14, 2016 Variational Methods & Optimal Control: lecture 25 – p.1/29 Variational Methods & Optimal Control: lecture 25 – p.3/29 Conservation laws Given the functional Conservation Laws One of the more exciting things we can derive relates to fundamental physics laws: conservation of energy, momentum, and angular momentum. We can now derive all of these from an underlying principle: Noether’s theorem. F{y} = Z x1 f (x, y, y′ , . . . , y(n) ) dx x0 if there is a function φ(x, y, y′, . . . , y(k) ) such that d φ(x, y, y′ , . . . , y(k) ) = 0 dx for all extremals of F, then this is called a kth order conservation law ◮ use obvious extension for functionals of several dependent variables. Variational Methods & Optimal Control: lecture 25 – p.2/29 Variational Methods & Optimal Control: lecture 25 – p.4/29 Conservation law example Given the functional F{y} = Z x1 f (y, y′ ) dx x0 ◮ physically interesting ⊲ tell you about system of interest ◮ can simplify solution where f is not explicitly dependent on t, we know that the Hamiltonian H = y′ Conservation laws ∂f −f ∂y′ is constant, and so dH =0 dx is a first order conservation law for the system. ⊲ φ(x, y, y′ , . . . , y(k) ) = const is an order k DE, rather than E-L equations which are order 2n ◮ φ(x, y, y′ , . . . , y(k) ) = const is often called the first integral of the E-L equations ⊲ RHS is a constant of integration (determined by boundary conditions) ◮ how do we find conservation laws? ⊲ Noether’s theorem Variational Methods & Optimal Control: lecture 25 – p.5/29 Variational Methods & Optimal Control: lecture 25 – p.7/29 Several independent variables Variational symmetries For functionals of several independent variables, e.g. The key to finding conservation laws lies in finding symmetries in the problem. ◮ “symmetries” are the result of transformations under which the functional is invariant F{z} = ZZ Ω z(x, y) dx dy ◮ E.G. time invariance symmetry results in constant H the equivalent conservation law is ∇·φ = 0 For some function φ(x, y, z, z , . . . , z ). ′ (k) ◮ Results here can be extended to these cases, but we won’t look at them here. Variational Methods & Optimal Control: lecture 25 – p.6/29 ◮ more generally, take a parameterized family of smooth transforms where X = θ(x, y; ε), Y = φ(x, y; ε) x = θ(x, y; 0), y = φ(x, y; 0) e.g. we get the identity transform for ε = 0 ◮ examples translations and rotations Variational Methods & Optimal Control: lecture 25 – p.8/29 Example transformations Jacobian The Jacobian is ◮ translations (ε is the translation distance) θ θ x y J= = θx φy − θy φx φx φy X = x+ε Y = y ◮ smooth: if functions x and y have continuous partial derivatives. y ◮ non-singular: if Jacobian is non-zero (and hence an inverse transform exists) Now for ε = 0, we require the identity transform, so J = 1. Also, we require a smooth transform, so J is a smooth function of ε, and so for sufficiently small |ε|, the transform is non-singular. (X,Y)=(x+ ε,y) (x,y) ε x Variational Methods & Optimal Control: lecture 25 – p.11/29 Variational Methods & Optimal Control: lecture 25 – p.9/29 Example transformations Example transformations ◮ translations (ε is the translation distance) or X = x+ε X = x Y Y ◮ translations (ε is the translation distance) X = x+ε = y = y+ε both have Jacobian Y Y = y y J=1 and inverse transformations or x = X −ε x = X y = Y y = Y −ε ε (x,y) (X,Y)=(x+ ε,y) X Variational Methods & Optimal Control: lecture 25 – p.10/29 x Variational Methods & Optimal Control: lecture 25 – p.12/29 Example transformations Example transformations ◮ rotations (ε is the rotation angle) X = x cos ε + y sin ε ◮ rotations (ε is the rotation angle) Y = −x sin ε + y cos ε has Jacobian X = x cos ε + y sin ε Y = −x sin ε + y cos ε To derive this, change coordinates to polar coordinates J = cos ε + sin ε = 1 2 2 and inverse x = r cos(θ) and y = r sin(θ) Under a rotation by ε, the new coordinates (X,Y ) are x = X cos ε −Y sin ε X = r cos (θ − ε) y = X sin ε +Y cos ε and Y = r sin (θ − ε) Use trig. identities cos(u − v) = cos u cos v + sin u sin v and sin(u − v) = sin u cos v − cos u sin v, to get X Y = r cos(θ) cos(ε) + r sin(θ) sin(ε) = x cos(ε) + y sin(ε) = r sin(θ) cos(ε) − r cos(θ) sin(ε) = y cos(ε) − x sin(ε) Variational Methods & Optimal Control: lecture 25 – p.15/29 Variational Methods & Optimal Control: lecture 25 – p.13/29 Example transformations Transformation of a function ◮ rotations (ε is the rotation angle) X = x cos ε + y sin ε Y = −x sin ε + y cos ε Given a function y(x), we can rewrite Y (X) using the inverse transformation, e.g. φ−1 (X,Y (X); ε) = y(x) = y(θ−1 (X,Y ; ε)) y For example, taking the curve y = x under rotations Y X sin ε +Y cos ε = X cos ε −Y sin ε (x,y) which we rearrange to get x ε Y (X) = cos ε − sin ε X cos ε + sin ε Similarly we can derive Y ′ (X) X Variational Methods & Optimal Control: lecture 25 – p.14/29 Variational Methods & Optimal Control: lecture 25 – p.16/29 Examples Transform invariance If Z x1 f (x, y, y′ (x)) dx = x0 Z ◮ translations: X1 f (X,Y,Y ′ (X)) dX X0 for all smooth functions y(x) on [x0 , x1 ] then we say that the functional in invariant under the transformation. ◮ also called variational invariance ◮ The transform is called a variational symmetry ◮ Related to conservation laws or (X,Y ) = (x + ε, y) (X,Y ) = (x, y + ε) ◮ rotations: X = θ(x, y; ε) = x cos ε + y sin ε ξ = η = ∂θ = ∂ε ε=0 ∂φ = ∂ε Variational Methods & Optimal Control: lecture 25 – p.17/29 Infinitesimal generators Y = φ(x, y; ε) = −x sin ε + y cos ε ε=0 −x sin ε + y cos ε|ε=0 = y −x cos ε − y sin ε|ε=0 = −x Variational Methods & Optimal Control: lecture 25 – p.19/29 Emmy Noether For small ε we can use Taylor’s theorem to write ∂θ + O(ε2 ) X = θ(x, y; 0) + ε ∂ε (x,y;0) ∂φ Y = φ(x, y; 0) + ε + O(ε2 ) ∂ε (x,y;0) and then for small ε (ξ, η) = (1, 0) (ξ, η) = (0, 1) So Also note that the E-L equations are invariant under such a transform, e.g. they produce the same extremal curves. Define the infinitesimal generators ∂θ ξ(x, y) = ∂ε (x,y;0) ⇒ ⇒ ∂φ η(x, y) = ∂ε (x,y;0) X ≃ x + εξ Y ≃ y + εη Variational Methods & Optimal Control: lecture 25 – p.18/29 ◮ Amalie Emmy Noether, 23 March 1882 – 14 April 1935 ◮ Described by Einstein and many others as the most important woman in the history of mathematics. ◮ Most of her work was in algebra ◮ Worked at the Mathematical Institute of Erlangen without pay for seven years ◮ Invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophicqal faculty objected, however, and she spent four years lecturing under Hilbert’s name. Variational Methods & Optimal Control: lecture 25 – p.20/29 Noether’s theorem Example (ii) Suppose the f (x, y, y′ ) is variationally invariant on [x0 , x1 ] under a transform with infinitesimal generators ξ and η, then Invariance in translations in y, i.e. ηp − ξH = const (X,Y ) = (x, y + ε) (ξ, η) = (0, 1) along any extremal of F{y} = Z x1 ′ So, a system with such invariance has f (x, y, y ) dx x0 p = const which is what we showed earlier regarding functionals with no explicit dependence on y. Variational Methods & Optimal Control: lecture 25 – p.21/29 Variational Methods & Optimal Control: lecture 25 – p.23/29 Example (i) More than one dependent variable Invariance in translations in x, i.e. Transforms with more than one dependent variable (X,Y ) = (x + ε, y) (ξ, η) = (1, 0) T = θ(t, q; ε) Qk = φk (t, q; ε) and the infinitesimal generators are So, a system with such invariance has H = const which is what we showed earlier regarding functionals with no explicit dependence on x. Variational Methods & Optimal Control: lecture 25 – p.22/29 ξ = ηk = ∂θ ∂ε ε=0 ∂φk ∂ε ε=0 Variational Methods & Optimal Control: lecture 25 – p.24/29 More than one dependent variable . Noether’s theorem: Suppose L(t, q, q) is variationally invariant on [t ,t ] under a transform with infinitesimal generators ξ and ηk . Given n ∂L p= , H = ∑ pk qk − L ∂qk k=1 . . Then 0 1 n ∑ pk ηk − Hξ = const k=1 along any extremal of F{q} = Z Common symmetries . . . . 2 2 2 Given a system in 3D with Kinetic Energy T (q) = 21 m q1 + q2 + q3 , and Potential Energy V (t, q). ◮ invariance of L under time translations corresponds to conservation of Energy ◮ invariance of L under spatial translations corresponds to conservation of momentum ◮ invariance of L under rotations corresponds to conservation of angular momentum t1 . L(t, q, q) dt t0 Variational Methods & Optimal Control: lecture 25 – p.25/29 Variational Methods & Optimal Control: lecture 25 – p.27/29 Example: rotations Finding symmetries Invariance in rotations, i.e. Testing for non-trivial symmetries can be tricky. Useful result is the Rund-Trautman identity: It leads also to a simple proof of Noether’s theorem (T, Q1 , Q2 ) = (t, q1 cos ε + q2 sin ε, −q1 sin ε + q2 cos ε) (t, q1 , q2 ) = (T, Q1 cos ε − Q2 sin ε, Q1 sin ε + Q2 cos ε) The infinitesimal generators are ξ = 0 η1 = −q1 sin ε + q2 cos ε|ε=0 = q2 η2 = −q1 cos ε − q2 sin ε|ε=0 = −q1 So, a system with such invariance has 2 ∑ pi ηi − Hξ = p1 q2 − p2 q1 = const i=1 So angular momentum in conserved. Variational Methods & Optimal Control: lecture 25 – p.26/29 Variational Methods & Optimal Control: lecture 25 – p.28/29 More advanced cases ◮ Laplace-Runge-Lenz vector in planetary motion corresponds to rotations of 3D sphere in 4D ◮ symmetries in general relativity ◮ symmetries in quantum mechanics ◮ symmetries in fields Variational Methods & Optimal Control: lecture 25 – p.29/29