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

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

Glossary: Simple mechanical systems with symmetry


Configuration space, with elements denoted byq ε Q.


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.


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


Infinitesimal coadjoint action of on* (adv*µη=µ·advη.


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.


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.


(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.


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.


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Copyright information

© Springer-Verlag 1991

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

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