A Primer on Geometric Mechanics Variational principles and Hamiltonian Mechanics Alex L. Castro, PUC Rio de Janeiro Henry O. Jacobs, CMS, Caltech Christian Lessig, CMS, Caltech Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 1 / 43
Outline Overview Variational mechanics The Hamiltonian Picture Bibliography
Course Outline Overview Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 3 / 43
Outline Overview Variational mechanics The Hamiltonian Picture Bibliography
Variational mechanics The principle of least action Feynman s lectures on Physics, vol. I Lecture 19 Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 5 / 43
Variational mechanics Lagrange s equations Notation: q, q R d and q(t) is a smooth path in R d. Given a Lagrangian L(q, q), Lagrange s equation of motion is d dt ql(q, q) q L(q, q) = 0. This equation is the Euler-Lagrange equation minimizing the action integral (functional) S[q(t)] := t 1 t 0 L(q(t), q(t))dt. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 6 / 43
Variational mechanics Euler-Lagrange equations = S(q(t) + h(t)) S(q(t)) = t1 t 0 (L(q + h, q + ḣ) L(q, q)dt = ( L q ḣ + L q h)dt + O(h2 ) = ( d L dt q + L q )hdt + boundary term + O(h 2 ) δs = 0 d dt L q + L q = 0 Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 7 / 43
Variational mechanics Hamilton s principle Most famous action integral from classical mechanics is S = (T U)dt, where T = kinetic energy U =potential energy For a particle of mass m in a constant gravitational field gˆk, S = t2 t 1 [ 1 2 m(dq dt )2 mgq], where q is the height measured from ground level. E.-L. eqn. s: q = g/m. Hamilton stated his principle in 1834-35. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 8 / 43
Example/exercise: Variational mechanics Consider a particle moving in a constant force field (e.g. gravity near earth, gˆk) and starting at (x 1, y 1 ) (rest) and descending to some lower point (x 2, y 2 ). Find the path that allows the particle to accomplish the transit in the least possible time. Hint. Compute the Euler-Lagrange equations for the transit time functional given by time = x2 x 1 (1 + y 2 )/2gxdx. Can you describe the solution curves geometrically? Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 9 / 43
Variational mechanics Calculus of variations For us, calculus of variations = calculus with functionals A functional is a scalar field whose domain is a certain space of functions (e.g. C k paths γ(t) on [0, 1] plus bdry. conditions). E.g. (calculus): arc length, area, time to travel etc. s x 2 + y 2 s = b a 1 + y (x)dx. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 10 / 43
Variational mechanics An important remark The condition that q(t) be an extremal of a functional does not depend on the choice of a coordinate system. For example, arc length of q(t) is given in different coordinates by different formulas s = s = t1 t 0 t1 t 0 ẋ1 2 + ẋ 2 2 dt (cartesian), ṙ 2 + r 2 φ 2 dt (polar). However, extremals are the same: straight lines in the plane. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 11 / 43
Variational mechanics Modern take on Variational Calculus Hamilton s principle has been generalized to various nonlinear/curved contexts (e.g. constraints, optimal control, Lie groups (matrix groups), field theories etc.). Focus later on will be on motion on Lie Groups (Henry Jacobs). Dynamics on Lie groups: tops, fluids, plasma, Maxwell-Vlasov equations, Maxwell s equations etc. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 12 / 43
Variational mechanics Variational Calculus on Manifolds Let (Q n, g ij ) be a Riemannian manifold. Let V(t) = q(t), A(t) = D V(t) and dt J(t) = D A(t). Examples of dt functionals on a Riemannian manifold: 1. S 1 = t1 g q(t) ( q(t), q(t))dt t 0 E-L: D2 q dt 2 = 0 (geodesic motion). 2. S 2 = t1 t 0 g q(t) ( q(t), q(t))dt E-L: DJ R(V, A)V = 0. dt Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 13 / 43
Variational mechanics Motion on a potential field We can generalize geodesic motion to include potentials V : Q R. The action functional is now S = t1 t 0 ( 1 2 g q(t)( q(t), q(t))dt V (q(t))dt E.L. : D2 q(t) gradv (q(t)) = 0. dt Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 14 / 43
Variational mechanics Reduced variational principles: Euler-Poincaré I X x R(t) Ṙ(t) = d dɛ ɛ=0r(t + ɛ) SO(3) = orthogonal matrices TSO(3) = tangent bundle reference configuration spatial configuration motion gen. velocity configuration space state space Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 15 / 43
Variational mechanics Reduced variational principles: Euler-Poincaré II x(t) = R(t)X, X is a point on the reference configuration. Therefore, ẋ = ṘR 1 x(t). Exercise. ṘR 1 = ˆω is an anti-symmetric matrix. The kinetic energy of the body is: 1 L(R, Ṙ) = kinetic = body 2 dm ẋ 2 = 1 2 body m Ω X 2 dx = 1 IΩ, Ω := l(ω). 2 Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 16 / 43
Variational mechanics Reduced variational principles: Euler-Poincaré III Theorem (Poincaré(1901-02): Geometric Mechanics is born) Hamilton s principle for rigid body action δs = δ t 1 t 0 L(R, Ṙ)dt = 0 is equivalent to t1 δs red = δ l(ω)dt = 0, t 0 with Ω R 3 and for variations of the form δω = Σ + Ω Σ, and bdry. conditions Σ(a) = Σ(b) = 0. How do they look like for the rigid body equation? Reduced Lagrange s equations are called Euler-Poincaré equations. Euler-Poincaré equations occur for many systems: fluids, plasma dynamics etc. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 17 / 43
Variational mechanics What s next? Lagrangian Reduction and other bargains. Kummer equations, Lagrange-Poincaré equations etc. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 18 / 43
Variational mechanics Example/exercise: discrete variational mechanics Consider the Lagrangian function L(q, q) and the action integral S[q(t)] := t1 t 0 L(q(t), q(t))dt. We replace the integral by a finite sum (discrete action) S dis [{q n }] = n L(q n, qn+1 q n ) t t and find the local minimizer from the condition q n S[{qn }] = 0. What numerical scheme do you obtain by explicitly evaluating the previous formula for a density L(q, q) = q 2 /2 V (q)? This derivation is a simple example of a simple discrete variational principle. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 19 / 43
Outline Overview Variational mechanics The Hamiltonian Picture Bibliography
The Hamiltonian Picture The Legendre Transform; Hamiltonian mechanics I Let y = f (x) be convex. Define g(p) = max(p x f (x)). x Exercise. Experiment to compute the Legendre transform of a convex function that s a broken line. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 21 / 43
The Hamiltonian Picture The Legendre Transform; Hamiltonian mechanics II Take the Legendre transform w.r.t. v = q of L(q, v) and obtain H(q, p) called Hamiltonian function. After passing to the Hamiltonian side of the picture (on S-S. Chern s word s: The sophisticated side. ) we obtain that Lagrange s equations become: q = ṗ = H(q, p), p H(q, p). q q q p := L L q q generalized coordinates generalized velocities generalized momentum generalized force field Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 22 / 43
The Hamiltonian Picture Canonical and non-canonical Hamiltonian structures I Let z = (q, p) T R d where d = no. of D.O.F. The space of positions and generalized momenta is called phase space. It often has the[ structure of ] a cotangent bundle. 0d I Define J := d. The block-matrices 0 I d 0 d, I d are d d d matrices. A canonical Hamiltonian system is an O.D.E. system of the form ż = J z H(z). For a mechanical system with Lagrangian L(q, v), the Hamiltonian function is H(q, p) = (p, v) L(q, v) Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 23 / 43
The Hamiltonian Picture Canonical and non-canonical Hamiltonian structures II and v = v(p) by the Legendre transform. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 24 / 43
The Hamiltonian Picture Canonical and non-canonical Hamiltonian structures III Example 1. H = (z T Lz)/2 ż = (JL)z. Matrices of the form JL w/ L symmetric are called Hamiltonian matrices. They generate the algebra of infinitesimally symplectic matrices. (More about symplectic transformations later on.) Take for example [ the] harmonic [ ] oscillator Hamiltonian ω H = 1[q, p] 2 0 q = p 2 0 1 p 2 /2 + ω 2 q 2 /2. Re-scaling: p = ˆp ω and q = ˆq/ ω we can re-write Hamiltonian as, Ĥ = ω(ˆp 2 + ˆq 2 )/2. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 25 / 43
The Hamiltonian Picture Canonical and non-canonical Hamiltonian structures IV In the new (canonical) coordinates: [ ] [ d ˆq 0 ω = dt ˆp ω 1 and exponentiating [ ] ˆq(t) = exp(t ˆp(t) [ 0 ω ω 1 ] [ ˆq0 ) ˆp 0 and Φ ωt is a rotation matrix in the plane. ] [ ˆq ˆp ] ] = Φ ωt [ ˆq0 ˆp 0 ], Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 26 / 43
The Hamiltonian Picture Spectral structure of Hamiltonian matrices λ eigenvalue λ, λ, λ are also eigenvalues. Proof: JLv = λv JL(Jw) = λjw L(Jw) = λw, but (LJ) = (JL) T and therefore λ is eigenvalue of the transposed Hamiltonian matrix. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 27 / 43
Examples I The Hamiltonian Picture (1) One D.O.F. problems. (2) Central forces. (3) Charged particle in a magnetic field (non-canonical Hamiltonian system): [ ] [ ] [ d q 0d I d q = dt p I d ˆb p non-canonical Hamiltonian structure. For this non canonical structure, the charged particle Hamiltonian is written as: ], the matrix Jˆb is an example of a H(q, p) = 1 2m p 2 γ/ q. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 28 / 43
The Hamiltonian Picture Examples II (4) N-body problems. F ij = φ ij(r ij ) r ij and H = 1 N 2 i=1 p i 2 /m i + N 1 N i=1 j=i+1 φ(r ij) = T + U. (5) Rigid-body motion, other types of Lie-Poisson dynamics etc. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 29 / 43
The Hamiltonian Picture First Integrals and Poisson Brackets I A first integral (or integral of motion) is a function G : R 2d R such that G(z(t; z 0 ) = G(z 0 ). This physical quantities are conserved along the trajectories (solution curves) of a Hamiltonian system ż = J z H(z). First integrals usually lead to geometric reduction of the problem: solution curves will live in {G = constant}. Problem: find a practical way to determine a function is an integral of motion. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 30 / 43
The Hamiltonian Picture First Integrals and Poisson Brackets II Use the chain rule: d dt (G(z(t; z 0)) = z G(z(t; z 0 )) T d dt z(t; z 0), but d dt z(t; z 0) = J z H(z(t; z 0 )) and therefore d dt (G(z(t; z 0)) = z G(z(t; z 0 )) T J z H(z(t; z 0 )). This leads us to introduce the following bilinear operation on scalar fields defined on phase space: {F, G}(z) := z F (z) T J z G(z) (Poisson bracket). Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 31 / 43
The Hamiltonian Picture First Integrals and Poisson Brackets III 1. antisymmetry {F, G} = {G, F }. 2. A fundamental property of the Poisson bracket is the Jacobi identity: {H, {F, G}} = {G, {H, F }} {F, {G, H}}. Exercise. Check that the components of the vector m = q p is a conserved quantity for the system with Hamiltonian H(p, q) = p 2 1. 2 q The matrix J does not need to be constant. Non-canonical structures are very common all over physics and mechanics. Take for instance the following poisson structure in R 3 : {F (M), G(M)} EP := F (M), C(M) G(M), Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 32 / 43
The Hamiltonian Picture First Integrals and Poisson Brackets IV C = (M 2 1 + M 2 2 + M 2 3 )/2 = M 2 /2. Take H = M, I 1 M /2 (rigid body kinetic energy), therefore Ṁ = C H = M I 1 M. Our non-constant Poisson structure is 0 M 3 M 2 J EP := M 3 0 M 1, M 2 M 1 0 and the Poisson bracket becomes {F (M), G(M)} EP = F T J EP G. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 33 / 43
The Hamiltonian Picture First Integrals and Poisson Brackets V The inertia tensor can be made diagonal by a orthogonal change of basis, and I = diag(i 1, I 2, I 3 ), I 1 > I 2 > I 3. We need to check that surfaces {H = const} and {C = const} are invariant manifolds. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 34 / 43
The Hamiltonian Picture First Integrals and Poisson Brackets VI By intersecting different ellipsoids The famous picture from JEM s book cover. {H = M 2 1 I 1 + M 2 2 I 2 + M 2 3 I 3 = const} and the sphere {C = M 2 1 + M 2 2 + M 2 3 = const} we obtain the reduced solution curves depicted in the blue sphere. Using reconstruction formulas we can compute the associated motion in SO(3). Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 35 / 43
The Hamiltonian Picture Applications of Poisson structures Lie-Poisson integrators, Lie-transformation methods in bifurcation theory, field theories, constrained Hamiltonian systems etc. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 36 / 43
The Hamiltonian Picture Hamiltonian flows I Flows generated by Hamiltonian vector fields possess many useful geometric properties. The Hamiltonian vector field X H (z) = J H(z) generates a flow on the manifold M 2d, often T Q with coordinates (q, p). For example, consider a free particle moving in space q = 0. Its equations of motion in Hamiltonian form are q = p, ṗ = 0, and the corresponding Hamiltonian is Hpart(q, p) = p 2 /2. The flow map is Φ t H(q 0, p 0 ) = (q 0 + tp 0, p 0 ). Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 37 / 43
The Hamiltonian Picture Hamiltonian flows II The mapping Φ t H is a 1-parameter family of transformations of R 2d. Another useful example is the harmonic oscillator, Hosc = p 2 /2 + ω 2 q 2 /2. The flow generated by the Hamiltonian vector field in this case is [ ] [ ] cos(ωt) ω Φ t H(p 0, q 0 ) = 1 sin(ωt) q0, ω sin(ωt) sin(ωt) which is conjugate to a rotation matrix in the plane. An important property of Hamiltonian flows is that they infinitesimally preserve the symplectic (resp. Poisson) structure. p 0 Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 38 / 43
The Hamiltonian Picture Hamiltonian flows III This means that Φ z (z)j(z)φ z (z) T = J(z) Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 39 / 43
Examples/exercises The Hamiltonian Picture 1. A particle in a central field and since m = 1, p = q. L = 1 2 q 2 ( 1/ q ), 2. A charged particle in a magnetic field L = m 2 q 2 ( γ/ q 1 B(q, q)), 2 B is an anti-symmetric matrix representing a constant magnetic field. In this case, p = m q 1 B(q). Example 2 of a non-canonical hamiltonian system. More later. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 40 / 43
The Hamiltonian Picture The braquistocrhone problem: Pontryagin Principle Our next goal is to connect the calculus of variations with Hamiltonian mechanics. Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 41 / 43
Outline Overview Variational mechanics The Hamiltonian Picture Bibliography
Bibliography Bibliography Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 43 / 43