Seminar Geometrical aspects of theoretical mechanics

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1 Seminar Geometrical aspects of theoretical mechanics Topics 1. Manifolds Gisela Baños-Ros 2. Vector fields and Alexander Holm and Matthias Sievers 3. Differential forms , and Stanislav Katzmin, Enrico Lohmann and Carl Suckfüll 5. Symplectic Geometry and Danilo Uzhogov and Benjamin Schlesinger 6. Hamiltonian Mechanics and Ecaterina Bodnariuc and Richard Busch 7. Application: the force-free rigid body and Richard Busch Time 16:00 17:00 at SR 210, Institute for Theoretical Physics, Brüderstraße 16 Tutor Matthias Schmidt Institute for Theoretical Physics, Brüderstraße 16, room nr. 317B Tel , matthias.schmidtitp.uni-leipzig.de

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3 1 Manifolds 1 talk of approx. 1 hour Notion of submanifold of R n with examples (e.g. the unit spheres in R 2, R 3 ). Charts and atlases. Motivation: manifolds are necessary for the description of systems with holonomic scleronomic constraints, like the spherical pendulum. Level set theorem: If f : R n R m is differentiable and if y R m is a regular value, then the level set f 1 (y) is a submanifold of R n. Sketch the proof (use the inverse functions theorem). Examples (e.g. the spheres in R 2, R 3, ellipsoids, hyperboloids,...) I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 3 B. T. Bröcker: Analysis, Band II, II.3, II:4

4 2 Vector fields 2 talks of approx. 1 hour Notions of tangent vector and of tangent space T m M of a submanifold M of R n at a point m M. Try to visualize how the tangent space changes from point to point. Construction of the tangent bundle TM = {(x,x) R n R n : x M,X tangent to M at x}. Charts on M induce charts on TM. If M = f 1 (c) is a level set of the mapping f, then T ( f 1 (c) ) = {(x,x) R n R n : f(x) = c,f (x)x = 0}. Give examples, like the unit spheres in R 2, R 3. Notion of tangent mapping (aka differential, derivative) of a differentiable mapping. Definition of vector fields as C -mappings which assign to every point m M a tangent vector X m T m M. Interpretation of vector fields as first order differential operators and thus as derivations of the algebra C (M) of C -functions on M. Commutator of vector fields. Representation of vector fields in coordinates, transformation under a change of coordinates. Notion of integral curve of a vector field, existence and uniqueness, examples. Motivation: In Hamiltonian Mechanics, the time evolution of states is given by the integral curves of a vector field, the Hamiltonian vector field. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 3 B. T. Bröcker: Analysis, Band II, II.3, II:4

5 3 Differential forms 3 talks of approx. 1 hour Algebraic basics Notion of exterior (aka alternating) form on a vector space. The vector spaces k V. exterior product and exterior algebra inner product Definition of the cotangent space T mm of M at m M as the dual space of the tangent space. Cotangent bundle T M and bundle of exterior forms k T M. Differential forms Notion of differential form Examples: Differentials (tangent mappings) df of functions f on M, as well as f 0 df 1 or f 0 df 1 df k. Representation in coordinates, transformation under a change of coordinates exterior product and pull-back exterior derivative de-rham complex, closed and exact forms, Poincaré Lemma Vector analysis in the language of differential forms Riemannian metric and associated Hodge operator Explain the relation between the operators of classical vector analysis grad, rot and div and the exterior derivative of differential forms on R 3. Generalize this to arbitrary 3 dimensional Riemannian manifolds. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 3 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989, Ch. 7 K. Jänich: Vektoranalysis, Springer 1993

6 4 Symplectic Geometry 2 talks of approx. 1 hour symplectic vector spaces Notion of symplectic form on a real vector space existence of a canonical basis symplectic mappings, symplectic group symplectic manifolds Definition and examples Theorem of Darboux (Existence of canonical coordinates) without proof canonical transformations Cotangent bundles Let Q be a manifold. Canonical 1-form θ on M = T Q Coordinates q i on M induce bundle coordinates (q i,p i ) on T M by (x,ξ) (q i (x),p i (x,ξ)), where ξ = p i (x,ξ)dq i. In bundle coordinates, θ = p i dq i. Define ω := dθ. Show that in bundle coordinates, ω = dp i dq i. Conclude that ω is symplectic. Motivation: If a system has configuration space Q, then the associated phase space is given by (T Q,ω). IntheexampleQ = R n,onehast Q = R n R n = R 2n andω = n i=1 dxi dp i. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 7 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989 F. Scheck: Mechanik, Springer 1994 R. Berndt: Einführung in die symplektische Geometrie, Vieweg 1998, , 2.1, 2.2 R. Abraham, J.E. Marsden: Foundations of Mechanics, Addision-Wesley 1978, 3.2, 3.3

7 5 Hamiltonian Mechanics 2 talks of approx. 1 hour mathematical basics Let M be a symplectic manifold. Notion of Hamiltonian vector field X f generated by a function f on M. Representation in canonical coordinates. Poisson bracket {f,g} of functions f,g on M. Under the bracket, the functions form a Lie Algebra and f X f is a Lie algebra homomorphism. The flow of a Hamiltonian vector field is a canonical transformation. physical interpretation Notion of Hamiltonian systems (M, ω, H) Example: M = T Q, where Q = configuration space and H = energy. The points of M are the states of the system: their time evolution is given by the integral curves of the Hamiltonian vector field X H. By writing the equation for the integral curves in terms of canonical coordinates, one obtains the Hamiltonian equations. Interpretation of the functions f : M R as observables, whose value in the state m is given by f(m). Show that the time derivative of the value f ( m(t) ) along the curve m(t) in phase space is given by {f,h} ( m(t) ). Conserved quantities Notion of conserved quantity. Use: the level sets of conserved quantities are invariant under time evolution dynamics reduces to level sets. Show that H is a conserved quantity and that f is conserved iff {H,f} = 0. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 7 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989 F. Scheck: Mechanik, Springer 1994 R. Berndt: Einführung in die symplektische Geometrie, Vieweg 1998 R. Abraham, J.E. Marsden: Foundations of Mechanics, Addision-Wesley 1978,

8 6 Application: the force-free rigid body 2 talks of approx. 1 hour Show that after separation of the motion of the center of mass, the configuration space Q is given by the group manifold of the rotation group SO(3). Hence, the phase space is T SO(3). As coordinates, use the Euler angles ϕ, ϑ, ψ and the induced coordinates p ϕ, p ϑ and p ψ on T SO(3). Write down the Lagrange function, perform the Legendre transformation and determine the Hamiltonian H Calculate the Poisson brackets of the components L i of angular momentum and derive the Euler equations Show that H, L 2, L 3 form a system of independent conserved quantities in involution (i.e., the differentials dh, dl 2, dl 3 are linearly independent and their Poisson brackets vanish). Using this example, explain the notion of integrable system: Definition Theorem of Arnold: Dynamics reduces to tori. (if possible, sketch the proof) Visualize the motion on the tori, explain action and angle coordinates and discuss the influence of the ratios of the frequencies. Discuss why one requires the functions to be independent and in involution. F. Scheck: Mechanik, Springer 1994, 3.14, N. Straumann: Klassische Mechanik, Lecture Notes in Physics 289, 11.5 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989, 49A R. Berndt: Einführung in die symplektische Geometrie, Vieweg 1998 R. Abraham, J.E. Marsden: Foundations of Mechanics, Addision-Wesley 1978,

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