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Abbreviations
- Q :
-
Configuration space, with elements denoted byq ε Q.
- TQ :
-
State space. Points\((q,\dot q)\) are configurations and velocities.
- P = T*Q :
-
Phase space. Pointsz = (q, p) ε P are configurations and momenta.
- δz = (δq, 6p) :
-
Configuration-momentum variations; whereδq ε T qQ, andδp ε T *q Q.
- 〈·, ·> :
-
Non-degenerate duality pairing betweenT qQ andT *q Q.
- G :
-
Lie group acting freely onQ on the left. The action ofG onP is symplectic, obtained by cotangent lifts.
- :
-
Lie algebra ofG, with bracket denoted by [·, ·].
- * :
-
Dual of, with duality pairing denoted by a dot. Thusμ · η ε ℝ,.
- Adg :
-
Adjoint action ofG on;\(\left. {Ad_g \eta = \frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0} g(\exp (\varepsilon \eta ))g^{ - 1} \).
- \(Ad_{g^{ - 1} }^* \) :
-
Coadjoint action of G on;\((Ad_{g^{ - 1} }^* \mu ) \cdot \eta = \mu \cdot Ad_{g^{ - 1} } \eta \).
- adv :
-
Infinitesimal adjoint action of on;\(ad_v \eta = [\nu ,\eta ] = \left. {\frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0} Ad_{(exp(\varepsilon \nu ))} \eta \).
- ad *v :
-
Infinitesimal coadjoint action of on * (ad * v µ)·η=µ·ad v η.
- η Q(q):
-
Infinitesimal generator;\(\eta _Q (q) = \left. {\frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0} \exp (\varepsilon \eta ) \cdot q\).
- :
-
Momentum map;J(q, p) · η = 〈P,η Q(q)〉.
- V:Q → ℝ:
-
G-invariant potential energy.
- K:P → ℝ:
-
G-invariant invariant kinetic energy.
- H:P → ℝ:
-
Hamiltonian function:H(z)=V(q)+K(z).
- :
-
Energy-momentum functional:\(H_{\mu _e } (z,\xi ) = V(q) + K(z) - (J(z) - \mu _e ) \cdot \xi \).
- 〈·, ·〉g :
-
Positive-definite form onQ associated with the kinetic energy.
- FL:TQ →T *Q:
-
Legendre transformation; 〈FL(vq),w q〉=〈v q , wq〉g.
- :
-
Locked inertia tensor defined as.
- Σ:P →J −1(0):
-
Shifting map:Σ(q,p) ≔ (q,p - p J (q, P)), where.
- :
-
Augmented potential.
- :
-
Amended potential.
- \(h_{\mu _e } :\mathbb{J}^{ - 1} (\mathbb{O}) \to \mathbb{R}\) :
-
Reduced Hamiltonian:\(h_{\mu _e } (z) = V_{\mu _e } (q) + K(z)\).
- :
-
Isotropy subalgebra of under the coadjoint action.
- :
-
Orthogonal complement to with respect to at a givenq e ε Q.
- :
-
Space of admissible configuration variations modulo variations generated by. Thus,\(\delta q \in T_{q_e } Q\) is in if and only if 〈δq,η Q(qe)〉g=0 for all η∈.
- :
-
Space of ‘rigid’ configuration variations.
- ident ξ :
-
(Minus) linearized ‘angular’ momentum in the directionξ∈ for fixed locked velocity, i.e., ident ξ (δq):=-[Dj(q e ·δq]ηξ.
- :
-
Space of ‘internal’ configuration variations.
- :
-
Space of admissible configuration-momentum variations modulo variations generated by. The variation\(\delta z = (\delta q,\delta p) \in T_{z_e } P\) is an element of if and only if\(T_{z_e } J \cdot \delta z = 0\) andδq∈.
- D :
-
Vector tangent map; given a mapφ:M →V from a manifoldM to a vector spaceV, Dφ(q):T qM → V is given by\(D\phi (q) \cdot \delta q = \left. {\frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0} \phi (q_\varepsilon )\) for any curveq ε tangent toδq atq.
References
R. Abraham &J. E. Marsden [1978],Foundations of Mechanics, Second Edition, Addison-Wesley, Reading.
S. S. Antman [1976a], Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of non-linearly elastic rods and shells,Arch. Rational Mech. Anal. 61, 307–351.
S. S. Antman [1976b], Ordinary differential equations of nonlinear elasticity II: Existence and regularity theory for conservative boundary value problems,Arch. Rational Mech. Anal. 61, 353–393.
V. I. Arnold [1966], Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,Ann. Inst. Fourier, XVI, 319–361.
V. I. Arnold [1968], On an a priori estimate in the theory of hydrodynamical stability,Izv. Vyssh. Uchebn. Zaved. Mat. Nauk 54, 225–226 (Russian).
V. I. Arnold [1978],Mathematical Methods of Classical Mechanics, Springer, Berlin.
V. I. Arnold [1988],Encyclopedia of Dynamical Systems III, Springer, Berlin.
S. Chandrasekhar [1977],Ellipsoidal Figures of Equilibrium, Dover Publications, Inc., New York.
H. Cohen &R. Muncaster [1988],The Theory of Pseudo-Rigid Bodies, Springer-Verlag, New York.
J. L. Ericksen &C. Truesdell [1958], Exact theory of stress and strain in rods and shells,Arch. Rational Mech. Anal. 1, 295–323.
J. M. Finn &G. Z. Sun [1987], Nonlinear stability and the energy-Casimir method,Comments on Plasma Physics and Control Fusion 11, 7–25.
D. D. Holm, J. E. Marsden, T. Ratiu &A. Weinstein [1985], Nonlinear stability of fluid and plasma equilibria,Physics Reports 123, 1–116.
J. Jellinek &D. H. Li [1989], Separation of the energy of overall rotation in anyN-body system,Physics Review Letters 62, 241–244.
P. S. Krishnaprasad &J. E. Marsden [1987], Hamiltonian structure and stability for rigid bodies with flexible attachments,Arch. Rational Mech. Anal. 98, 71–93.
D. Lewis [1989], Nonlinear stability of a rotating liquid drop,Arch. Rational Mech. Anal. 106, 287–333.
D. Lewis &J. C. Simo [1990], Nonlinear stability of pseudo-rigid bodies,Proc. Roy. Soc. London A 427, 281–319.
J. E. Marsden, J. C. Simo, D. Lewis &T. A. Posbergh [1989], A block diagonalization theorem in the energy-momentum method,Contemp. Math. American Math. Soc.97, 297–313.
J. E. Marsden &A. Weinstein [1974], Reduction of symplectic manifolds with symmetry,Rep. Math. Phys. 5, 121–130.
R. Montgomery, J. E. Marsden &T. Ratiu [1984], Gauged Lie-Poisson Structures,Contemp. Math., American Math. Soc.29, 101–114.
M. E. McIntyre &T. G. Shepard [1987], An exact local conservation theorem for finite amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnold's stability theorem,J. Fluid Mech. 181, 527–565.
G. Patrick [1990], Ph. D. thesis, Department of Mathematics, University of California at Berkeley.
T. A. Posbergh, P. S. Krishnaprasad &J. E. Marsden [1987], Stability analysis of a rigid body with a flexible attachement using the Energy-Casimir method,Contemp. Math., American Math. Soc.68, 253–273.
J. W. S. Rayleigh [1920], On the dynamics of revolving fluids,Scientific Papers 6, 447–453, Cambridge, England.
E. J. Routh [1877],A Treatise on the Stability of a Given State of Motion, MacMillan, London.
J. C. Simo, J. E. Marsden &P. S. Krishnaprasad [1988], The Hamiltonian structure of elasticity. The material and convective representation of solids, rods and plates,Arch. Rational Mech. Anal. 104, 125–183.
J. C. Simo, T. A. Posbergh &J. E. Marsden [1989], Stability of coupled rigid body and geometrically exact rods: Block diagonalization and the Energy-Momentum method,Physics Reports (to appear).
J. C. Simo, D. Lewis &J. E. Marsden [1989], Stability of relative equilibria. Part I: The reduced energy-momentum method, Report No. 89-3, Division of Applied Mechanics, Stanford University.
J. J. Slawianowski [1988], Affinely rigid body and Hamiltonian systems onGL(n, ℝ), Reports on Mathematical Physics 26, 73–119.
S. Smale [1970a], Topology and Mechanics. I,Inventiones Math. 10, 305–331.
S. Smale [1970b], Topology and Mechanics. II,Inventiones Math. 11, 45–64.
S. Szeri &P. Holmes [1988], Nonlinear stability of axisymmetric swirling flows,Phil. Trans. Roy. Soc. London A326, 327–354.
R. Toupin [1964], Theory of elasticity with couple-stresses,Arch. Rational Mech. Anal. 17, 85–112.
E. B. Wilson, J. C. Decius &P. C. Cross [1955],Molecular Vibrations, McGraw-Hill (reprinted by Dover).
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Simo, J.C., Lewis, D. & Marsden, J.E. Stability of relative equilibria. Part I: The reduced energy-momentum method. Arch. Rational Mech. Anal. 115, 15–59 (1991). https://doi.org/10.1007/BF01881678
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DOI: https://doi.org/10.1007/BF01881678