Archive for Rational Mechanics and Analysis

, Volume 115, Issue 1, pp 15–59

# Stability of relative equilibria. Part I: The reduced energy-momentum method

• J. C. Simo
• D. Lewis
• J. E. Marsden
Article

### Glossary: Simple mechanical systems with symmetry

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 TqQ, andδp ε Tq*Q.

〈·, ·>

Non-degenerate duality pairing betweenTqQ andTq*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,.

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;$$(Ad_{g^{ - 1} }^* \mu ) \cdot \eta = \mu \cdot Ad_{g^{ - 1} } \eta$$.

Infinitesimal adjoint action of on;$$ad_v \eta = [\nu ,\eta ] = \left. {\frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0} Ad_{(exp(\varepsilon \nu ))} \eta$$.

ηQ(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,ηQ(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:TQT*Q

Legendre transformation; 〈FL(vq),wq〉=〈vq, wqg.

Locked inertia tensor defined as.

Σ:P →J−1(0)

Shifting map:Σ(q,p) ≔ (q,p-pJ(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 givenqe 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,ηQ(qe)〉g=0 for all η∈.

Space of ‘rigid’ configuration variations.

identξ

(Minus) linearized ‘angular’ momentum in the directionξ for fixed locked velocity, i.e., identξ(δq):=-[Dj(qe·δqξ.

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:MV from a manifoldM to a vector spaceV, Dφ(q):TqM → V is given by$$D\phi (q) \cdot \delta q = \left. {\frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0} \phi (q_\varepsilon )$$ for any curveqε tangent toδq atq.

## Preview

Unable to display preview. Download preview PDF.

### References

1. R. Abraham &J. E. Marsden [1978],Foundations of Mechanics, Second Edition, Addison-Wesley, Reading.Google Scholar
2. 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.Google Scholar
3. 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.Google Scholar
4. 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.Google Scholar
5. 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).Google Scholar
6. V. I. Arnold [1978],Mathematical Methods of Classical Mechanics, Springer, Berlin.Google Scholar
7. V. I. Arnold [1988],Encyclopedia of Dynamical Systems III, Springer, Berlin.Google Scholar
8. S. Chandrasekhar [1977],Ellipsoidal Figures of Equilibrium, Dover Publications, Inc., New York.Google Scholar
9. H. Cohen &R. Muncaster [1988],The Theory of Pseudo-Rigid Bodies, Springer-Verlag, New York.Google Scholar
10. J. L. Ericksen &C. Truesdell [1958], Exact theory of stress and strain in rods and shells,Arch. Rational Mech. Anal. 1, 295–323.Google Scholar
11. J. M. Finn &G. Z. Sun [1987], Nonlinear stability and the energy-Casimir method,Comments on Plasma Physics and Control Fusion 11, 7–25.Google Scholar
12. D. D. Holm, J. E. Marsden, T. Ratiu &A. Weinstein [1985], Nonlinear stability of fluid and plasma equilibria,Physics Reports 123, 1–116.
13. J. Jellinek &D. H. Li [1989], Separation of the energy of overall rotation in anyN-body system,Physics Review Letters 62, 241–244.Google Scholar
14. P. S. Krishnaprasad &J. E. Marsden [1987], Hamiltonian structure and stability for rigid bodies with flexible attachments,Arch. Rational Mech. Anal. 98, 71–93.Google Scholar
15. D. Lewis [1989], Nonlinear stability of a rotating liquid drop,Arch. Rational Mech. Anal. 106, 287–333.Google Scholar
16. D. Lewis &J. C. Simo [1990], Nonlinear stability of pseudo-rigid bodies,Proc. Roy. Soc. London A 427, 281–319.Google Scholar
17. 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.Google Scholar
18. J. E. Marsden &A. Weinstein [1974], Reduction of symplectic manifolds with symmetry,Rep. Math. Phys. 5, 121–130.
19. R. Montgomery, J. E. Marsden &T. Ratiu [1984], Gauged Lie-Poisson Structures,Contemp. Math., American Math. Soc.29, 101–114.Google Scholar
20. 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.Google Scholar
21. G. Patrick [1990], Ph. D. thesis, Department of Mathematics, University of California at Berkeley.Google Scholar
22. 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.Google Scholar
23. J. W. S. Rayleigh [1920], On the dynamics of revolving fluids,Scientific Papers 6, 447–453, Cambridge, England.Google Scholar
24. E. J. Routh [1877],A Treatise on the Stability of a Given State of Motion, MacMillan, London.Google Scholar
25. 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.Google Scholar
26. 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).Google Scholar
27. 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.Google Scholar
28. J. J. Slawianowski [1988], Affinely rigid body and Hamiltonian systems onGL(n, ℝ), Reports on Mathematical Physics 26, 73–119.
29. S. Smale [1970a], Topology and Mechanics. I,Inventiones Math. 10, 305–331.
30. S. Smale [1970b], Topology and Mechanics. II,Inventiones Math. 11, 45–64.
31. S. Szeri &P. Holmes [1988], Nonlinear stability of axisymmetric swirling flows,Phil. Trans. Roy. Soc. London A326, 327–354.Google Scholar
32. R. Toupin [1964], Theory of elasticity with couple-stresses,Arch. Rational Mech. Anal. 17, 85–112.
33. E. B. Wilson, J. C. Decius &P. C. Cross [1955],Molecular Vibrations, McGraw-Hill (reprinted by Dover).Google Scholar

## Authors and Affiliations

• J. C. Simo
• 1
• 2
• 3
• D. Lewis
• 1
• 2
• 3
• J. E. Marsden
• 1
• 2
• 3
1. 1.Division of Applied MechanicsStanford UniversityStanford
2. 2.Department of MathematicsUniversity of CaliforniaSanta Cruz
3. 3.Department of MathematicsUniversity of CaliforniaBerkeley