Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 193–200

A REDUCTION PROPERTY FOR A CLASS OF MECHANICAL SYSTEMS WITH FRICTION

F. Cantrijn

Abstract. A reduction scheme is established for a special class of non-Hamiltonian systems, defined on a cotangent bundle, which are invariant under the Hamiltonian action of a compact Lie group and for which the corresponding momentum map yields a conformal invariant. The reduction is constructed in terms of a suitably normalized momentum map and the reduced spaces are canonical manifolds. The class of systems under consideration in particular encompasses the phase space description of certain types of mechanical systems with friction. Keywords. Symmetry, reduction, canonical manifolds. MS classification. 58F05, 70G99.

1. Introduction

In a recent paper by Landi et al. [?], a general reduction procedure for dynamical systems has been proposed. The key observation in their analysis is that, given a vector field X on a manifold M, a reduction of X essentially requires the following two ingredients: (i) a submanifold M′ of M which is invariant for X (i.e. such that X is everywhere tangent to M′), (ii) an equivalence relation on M′ which, in an appropriate sense, is compatible with the flow of X. In case this equivalence relation induces a regular foliation on M′, the restriction of X to M′ will then project onto the corresponding quotient manifold.

The most prominent example of such a reduction scheme in mechanics is undoubtedly provided by the reduction of Hamiltonian (and Lagrangian) systems with symmetry. In brief the situation can be described as follows. (For a detailed treatment, see e.g. [?, ?, ?]). Let (M, ω) be a symplectic manifold and X_{H} a global Hamiltonian vector field on M, with Hamiltonian H ∈ C^{∞}(M). Let there be given a symplectic action of a Lie group G on (M, ω) with an Ad^{∗}- equivariant momentum map J : M ⟶ G^{∗}, where G^{∗} is the linear dual space of the Lie algebra G of G, and Ad^{∗} denotes the coadjoint representation of G on G^{∗}. Suppose this action leaves H invariant and, hence, defines a symmetry of X_{H}. Then, J is a G^{∗}-valued first integral of X_{H}. This is in fact a geometrical transcription of Noether’s theorem. Let μ be a weakly regular value of J and let G_{μ} denote the isotropy subgroup of G at μ under the coadjoint action. According to the Marsden-Weinstein reduction theorem [?], if G_{μ} acts freely and properly on the level set J^{−1}(μ), there exists a uniquely determined symplectic manifold structure on the orbit space P_{μ} = J^{−1}(μ)/G_{μ}. Exploiting the symmetry property of the given system again, one can then prove that X_{H}|_{J^{−1}(μ)} projects onto P_{μ} and the projected vector field is also Hamiltonian. Referring to the general reduction scheme mentioned above, we thus have here that the invariant submanifolds of the given dynamical system X_{H} are the (regular) level sets of the momentum map, and the appropriate equivalence relation on such a level set is determined by the action of the symmetry group.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The purpose of the present paper now is to study the reduction of a particular class of non-Hamiltonian systems with symmetry defined on a cotangent bundle T^{∗}Q. The systems we have in mind may serve as a model for the phase space description of mechanical systems with certain types of non-conservative forces (such as, for instance, frictional forces proportional to the velocity). The symmetry is again defined in terms of a strongly Hamiltonian group action (i.e. a symplectic action with an Ad^{∗}-equivariant momentum map), which arises from the cotangent lift of an action on the base manifold Q. For the systems under consideration it turns out that the momentum map J, rather than being an invariant, yields a conformal invariant. (Noether’s theorem does not apply to non-Hamiltonian systems!) In case the symmetry goup is compact, it will be shown that one can still establish a reduction scheme, similar to the one obtained for Hamiltonian systems with symmetry. The invariant submanifolds of the given system are now provided by the level sets of the normalized momentum map, where the normalization of J is defined with respect to a G-invariant norm on G^{∗}. The equivalence relation on such an invariant submanifold, needed to perform a reduction, is again determined by the symmetry group action. According to a result described elsewhere (see [?]), the resulting quotient spaces turn out to be Poisson manifolds or, more precisely, canonical manifolds.

In the next section we will first investigate the behaviour of the systems under consideration in the presence of an arbitrary (i.e. not necessarily compact) symmetry group. Conformal invariance of the momentum map will then only result in a reduction of the system with respect to the zero level set of the momentum map. Section 3 then deals with the case of a compact symmetry group and describes the reduction procedure in terms of the normalized momentum map.

We close this introduction by fixing some notation. Let G be a Lie group, with Lie algebra G, and suppose there is given a smooth (left) action of G on a manifold M, ϕ : G×M ⟶ M. For any g ∈ G we put ϕ_{g} = ϕ(g, .). The isotropy subgroup of G at a point m ∈ M is denoted by G_{m}, i.e. G_{m} = {g ∈ G| ϕ(g, m) = m}. The fundamental vector field of the given action, corresponding to ξ ∈ G, is the vector field ξ_{M} on M defined by

ξ_{M}(m) = \frac{d}{dt}ϕ(exp(−tξ), m)|_{t=0}.

Note that in defining ξ_{M} we follow the conventions of [?]. In particular, the assignment ξ ⟼ ξ_{M} then yields a Lie algebra homomorphism from G into the Lie algebra X (M) of smooth vector fields on M. The Lie algebra of the isotropy subgroup G_{m} is the subalgebra G_{m} of G given by G_{m} = {ξ ∈ G| ξ_{M}(m) = 0}. Finally, the natural pairing between G and G^{∗} and, more generally, the pairing between any linear space and its dual, will be denoted by ⟨ , ⟩.

2. Mechanical systems with friction and symmetry

Let Q be a smooth, finite dimensional, connected manifold and T^{∗}Q its cotangent bundle, with projection π_{Q} : T^{∗}Q ⟶ Q. On T^{∗}Q we have the canonical Liouville 1-form θ_{Q} and the corresponding symplectic form ω_{Q} = dθ_{Q} which, in canonical coordinates (q^{i}, p_{i}), read θ_{Q} = p_{i}dq^{i} and ω_{Q} = dp_{i}∧dq^{i}. We now consider a dynamical system, described by a vector field X on T^{∗}Q which is determined by

i_{X}ω_{Q} = −dH − fθ_{Q},  (1)

for some functions H, f ∈ C^{∞}(T^{∗}Q). In canonical coordinates, the corresponding equations of motion read

q̇^{i} = \frac{∂H}{∂p_{i}} (q, p) ,  ṗ_{i} = −\frac{∂H}{∂q^{i}} (q, p) − f(q, p)p_{i}.  (2)

These equations can be thought of as representing the phase space description of a mechanical system with Hamiltonian H = T + V (kinetic + potential energy) and F_{i} = −fp_{i} the i^{th} ‘generalized’ component of a nonconservative force. In particular, this nonconservative force can be a frictional or resistive force, in which case f will be strictly positive. An example of such a system is provided by a model for the Kepler problem with drag, with vector equation of motion with respect to a Cartesian reference frame (in the appropriate units):

r̈ + f(r, ṙ)ṙ + k \frac{r}{r^{3}} = 0

where k is a real constant (see e.g. [?]). When passing to the phase space description we indeed obtain a system of the form (2). Although we will mainly think of (1) as representing a mechanical system with friction, for the subsequent analysis we do not make any specific assumption about the sign of the function f.

Suppose we are given a smooth (left) action ϕ : G×Q ⟶ Q of a connected Lie group G on Q. For simplicity we henceforth assume this to be a free action. Let Φ : G×T^{∗}Q ⟶ T^{∗}Q be the canonically lifted action of G on T^{∗}Q, i.e. Φ is completely determined by

⟨v, Φ_{g}(α)⟩ = ⟨T ϕ_{g^{−1}}(v), α⟩

for each α ∈ T^{∗}Q and v ∈ T_{ϕ_{g}(m)}Q, with m = π_{Q}(α). It is well-known that Φ is a strongly Hamiltonian action on (T^{∗}Q, ω_{Q}), i.e. a symplectic action with an Ad^{∗}- equivariant momentum map J : T^{∗}Q ⟶ G^{∗} (cf. [?, ?]). In fact, we have for each g ∈ G,

Φ_{g}^{∗}θ_{Q} = θ_{Q}  (3)

and

i_{ξ_{T^{∗}Q}}ω_{Q} = −dJ_{ξ},  (4)

where J_{ξ} = ⟨ξ_{T^{∗}Q}, θ_{Q}⟩ ∈ C^{∞}(T^{∗}Q) is a global Hamiltonian for ξ_{T^{∗}Q}. The momentum map J is then determined by

⟨ξ, J(α)⟩ = J_{ξ}(α)

for each α ∈ T^{∗}Q and each ξ ∈ G. From the general properties of a momentum map we recall, in particular, that the image (Im) of the tangent map of J at a point α ∈ T^{∗}Q is given by

Im(T_{α}J) = G_{α}^{0}  (5)

where G_{α}^{0} ⊂ G^{∗} is the annihilator of the isotropy subalgebra G_{α} (see e.g. [?, ?, ?]). Since ϕ was assumed to be a free action, the lifted action Φ is also free and, hence, G_{α} = {0}. From (5) it then follows that J is a submersion and thus, for each ν ∈ G^{∗}, J^{−1}(ν) is a closed regular submanifold of T^{∗}Q (or the empty set).

Returning to the dynamical system X defined by (1), we first derive the following property:

Proposition 1. If Φ leaves H invariant, i.e. H ∘ Φ_{g} = H for all g ∈ G, then

X(J) = −fJ.

Proof: Taking the contraction of both sides of (1) with ξ_{T^{∗}Q}, for ξ ∈ G, we find

−i_{X}(i_{ξ_{T^{∗}Q}}ω_{Q}) = −ξ_{T^{∗}Q}(H) − f⟨ξ_{T^{∗}Q}, θ_{Q}⟩.

Since H is invariant under Φ it follows that ξ_{T^{∗}Q}(H) = 0 and thus, with (4), we obtain that

X(J_{ξ}) = −fJ_{ξ}.

This holds for every ξ ∈ G, from which the result follows. □

Under the conditions of the proposition, we thus have that the momentum map of the given action Φ is a conformal invariant of X. Moreover, J being a submersion, 0 ∈ G^{∗} is a regular value of J and so we readily have:

Corollary 1. If Φ leaves H invariant, then J^{−1}(0) is an invariant submanifold of X. □

Note that for the previous properties to hold, it suffices that H is invariant under the given group action, i.e. G need not be a symmetry group for the dynamical system X. If we now also require invariance of f, then we can easily derive the following result. (Note that, in view of the equivariance of J and the fact that G_{0} = G, the action Φ restricts to an action of G on J^{−1}(0)).

Theorem 1. If both H and f are invariant under Φ, then G is a symmetry group of X. Moreover, if G acts properly on J^{−1}(0), X|_{J^{−1}(0)} projects onto the reduced Marsden-Weinstein symplectic manifold P_{0} = J^{−1}(0)/G.

Proof: Using (3) and the invariance of H and f, it immediately follows from (1) that

i_{Φ_{g}^{∗}X}ω_{Q} = −dH − fθ_{Q}
= i_{X}ω_{Q}.

Hence, Φ_{g}^{∗}X = X for each g ∈ G, which proves that G is a symmetry group of X. From the previous corollary we already know that X is everywhere tangent to J^{−1}(0). The action of G induced on J^{−1}(0) is free (since Φ is free) and proper (by assumption). The Marsden-Weinstein reduction theorem then tells us that the orbit space P_{0} = J^{−1}(0)/G is a smooth symplectic manifold. Since G is a symmetry group of X, the flow of X|_{J^{−1}(0)} commutes with the group action and, therefore, induces a flow on P_{0}. □

Let us denote the symplectic form on P_{0} by ω_{0}. In case the actio ϕ of G on Q, which we already assumed to be free, is also proper, one can prove that (P_{0}, ω_{0}) is symplectomorphic to (T^{∗}(Q/G), ω_{Q/G} (cf. [?]). One can then easily verify that the reduced vector field X_{0}, i.e. the projection of X|_{J^{−1}(0)} onto P_{0}, satisfies the equation

i_{X_{0}}ω_{Q/G} = −dH_{0} − f_{0}θ_{Q/G},

where H_{0} and f_{0} are the functions induced by H and f, respectively, on P_{0}≅T^{∗}(Q/G).

In the next section we continue the analysis of the dynamical system X under the additional hypothesis that the symmetry group G is compact.

3. Compact symmetry groups and reduction

Henceforth we assume that G is a compact connected Lie group, acting freely on the connected manifold Q. Let Φ again denote the corresponding lifted Hamiltonian action on T^{∗}Q with associated Ad^{∗}- equivariant momentum map J : T^{∗}Q ⟶ G^{∗}.

G being compact we know that its Lie algebra G admits a G- invariant positive definite metric (see e.g. [?]). Fixing such a metric, let ( , ) denote the corresponding induced metric on G^{∗}. In particular, we then have for any ν_{1}, ν_{2} ∈ G^{∗} and for each g ∈ G, that

(Ad_{g}^{∗}ν_{1}, Ad_{g}^{∗}ν_{2}) = (ν_{1}, ν_{2}).  (6)

This metric further determines a G-invariant norm on G^{∗} according to ∥ν∥ = (ν, ν)^{1/2}. We can now introduce the normalized momentum map associated with the given Hamiltonian action Φ, namely

Ĵ: T^{∗}Q\J^{−1}(0) ⟶ G^{∗}, α⟼Ĵ(α) = \frac{J(α)}{∥J(α)∥}.

Next, let S_{G^{∗}} denote the unit sphere in G^{∗} with respect to the given metric, i.e. S_{G^{∗}} ={μ ∈ G^{∗}| ∥μ∥ = 1}, and consider the natural projection

ρ : G^{∗}\\{0\} ⟶ S_{G^{∗}}, ν⟼ρ(ν) = \frac{ν}{∥ν∥}.

For simplicity of notation, we put T^{∗}Q\J^{−1}(0) = T^{∗}Q_{0}. Clearly, T^{∗}Q_{0} is an open symplectic submanifold of T^{∗}Q, with symplectic form (the restriction of) ω_{Q}. One easily verifies that Ĵ can be regarded as a smooth map from T^{∗}Q_{0} into S_{G^{∗}} and we then have

Ĵ = ρ ∘ J|_{T^{∗}Q_{0}}.  (7)

Moreover, taking into account (6) and the Ad^{∗}-equivariance of J, it follows that

Ĵ(Φ_{g}(α)) = Ad_{g}^{∗}(Ĵ(α))

for each g ∈ G and each α ∈ T^{∗}Q_{0}, i.e. Ĵ is equivariant with respect to the actions of G induced on T^{∗}Q_{0} and S_{G^{∗}}, respectively (see also [?]).

With X again being the dynamical system defined by (1), suppose Φ leaves H invariant. From corollary 1 we then know that J^{−1}(0) is an invariant submanifold for X. We now consider the dynamical system obtained by taking the restriction of X to T^{∗}Q_{0}.

Proposition 2. If the action Φ of the compact connected Lie group G leaves H invariant, then Ĵ is constant along the integral curves of X|_{T^{∗}Q_{0}}.

Proof: Choose an orthonormal basis (e^{i}) of G^{∗} with respect to the given metric, i.e. such that (e^{i}, e^{j}) = δ^{ij}, and let (e_{i}) denote the dual basis of G. The momentum map J can then be written as J = J_{i}e^{i}, with J_{i} = J_{e_{i}} ∈ C^{∞}(T^{∗}Q). Furthermore, we then have ∥J∥ = (∑_{i}J_{i}^{2})^{1/2}. From proposition 1 one can infer that X(J_{i}) = −fJ_{i} and a straightforward computation then reveals that

X(\frac{J_{i}}{∥J∥}) = −f \frac{J_{i}}{∥J∥} − \frac{J_{i}}{∥J∥^{2}} X(∥J∥) = 0

and since Ĵ = (J_{i}/∥J∥)e^{i}, this indeed proves that Ĵ is a conserved quantity of X|_{T^{∗}Q_{0}}. □

As mentioned already in the previous section, the momentum map J is a submersion since the given Hamiltonian action Φ is assumed to be free. ¿From (7) it is then seen that Ĵ is also a submersion from T^{∗}Q_{0} into S_{G^{∗}}. For each μ ∈ Im(Ĵ), the corresponding fibre Ĵ^{−1}(μ) therefore determines a closed submanifold of T^{∗}Q_{0}. From the previous proposition we then immediately deduce:

Corollary 2. If Φ leaves H invariant, then for each μ ∈ S_{G^{∗}}, Ĵ^{−1}(μ) is an invariant submanifold of X|_{T^{∗}Q_{0}}. □

We now add the condition that f is also invariant under the given action, such that Φ defines a symmetry of X. In view of the Ad^{∗}- equivariance of Ĵ, it follows that G_{μ}, the isotropy subgroup of G at μ ∈ S_{G^{∗}}, induces an action on Ĵ^{−1}(μ). This action is free (since Φ is free) and proper (since G_{μ}, being a closed subgroup of a compact group, is also compact). Consequently, the orbit space Q_{μ} = Ĵ^{−1}(μ)/G_{μ} admits a smooth manifold structure such that the natural projection π̂_{μ} : Ĵ^{−1}(μ) ⟶ Q_{μ} becomes a submersion. According to the previous corollary, X is everywhere tangent to Ĵ^{−1}(μ) and, since Φ is a symmetry of X, the flow of X|_{Ĵ^{−1}(μ)} commutes with the action of G_{μ}. Hence, X|_{Ĵ^{−1}(μ)} projects onto Q_{μ} and we denote the projected vector field by X̂_{μ}. In particular, we thus have

X̂_{μ} ∘ π̂_{μ} = T π̂_{μ} ∘ X|_{Ĵ^{−1}(μ)} .

In [?] we have shown that under some appropriate regularity conditions, which are in particular verified in the present situation, the reduced space Q_{μ} admits the structure of a canonical manifold. Recall that, following Lichnerowicz [?], a canonical manifold is a triple (P, { , }, τ), consisting of a Poisson manifold (P, { , }) of constant rank (i.e. the corresponding Poisson tensor field has constant rank) and a regular map τ : P ⟶ IR, such that the connected components of the fibres of τ are the symplectic leaves of the Poisson structure.

One can now indeed prove that the reduced space Q_{μ}, constructed above, admits a Poisson stucture { , }_{μ} of constant rank, which is completely determined by (the restriction to Ĵ^{−1}(μ) of) the Poisson structure on T^{∗}Q. Moreover, there is a fibration τ_{μ} : Q_{μ} ⟶ IR_{0}^{+}, with IR_{0}^{+} the set of stictly positive real numbers, which is determined by

τ_{μ} ∘ π̂_{μ} = ∥J∥|_{Ĵ^{−1}(μ)}.

The symplectic leaves of the Poisson structure on G_{μ} are the connected components of the reduced Marsden-Weinstein spaces P_{ν} = J^{−1}(ν)/G_{ν}, for ν ∈ J(Ĵ^{−1}(μ)) ⊂ {kμ| k ∈ IR_{0}^{+}}, and the latter precisely coincide with the fibres of τ_{μ}. One can furthermore easily verify that Q_{μ} is odd dimensional with dim Q_{μ} = (dim M − dim G − dim G_{μ}) + 1. (See [?] for more details). Summarizing the above discussion we thus have:

Theorem 2. If both H and f are invariant under the (lifted) free Hamiltonian action of the compact connected Lie group G, then for each μ ∈ Im(Ĵ) ⊂ S_{G^{∗}}, the restriction of X to the invariant submanifold Ĵ^{−1}(μ) projects onto the quotient space Q_{μ} = Ĵ^{−1}(μ)/G_{μ}, which admits the structure of a canonical manifold. □

Let j_{μ} : Ĵ^{−1}(μ) ⟶ T^{∗}Q_{0} be the natural inclusion and let Ĥ_{μ} and f̂_{μ} denote the functions induced on Q_{μ} by H and f, respectively, i.e. Ĥ_{μ} ∘ π̂_{μ} = H ∘ j_{μ} and f̂_{μ} ∘ π̂_{μ} = f ∘j_{μ}. If Λ_{μ} denotes the Poisson tensor field on Q_{μ}, then one can check that the projected vector field X̂_{μ} is of the form (see also [?])

X̂_{μ} = −f̂_{μ}Δ̂_{μ} − Λ_{μ}(dĤ_{μ})  (8)

where, on the right-hand side, Λ_{μ} is regarded as a vector bundle homomorphism from T^{∗}Q_{μ} into T Q_{μ} and Δ̂_{μ} denotes the reduction of (the restriction to Ĵ^{−1}(μ) of) the canonical dilation vector field Δ = p_{i} ∂/∂p_{i} on T^{∗}Q. It is indeed easily verified that Δ is invariant under the lifted group action and, moreover, that Ĵ is a conserved quantity of Δ. From (1) it is seen that the given vector field X can still be written as X = −fΔ − Λ_{Q}(dH), with Λ_{Q} = ω_{Q}^{−1} the Poisson tensor field on T^{∗}Q, from which (8) now readily follows. We finally note that Δ̂_{μ}(τ_{μ}) = τ_{μ} (cf.[?]), from which it can be deduced that the component of X̂_{μ} transverse to the symplectic leaves of Q_{μ} is given by −f̂_{μ}τ_{μ} ∂/∂τ_{μ}. A more detailed analysis of this reduced system, as well as some examples, will be treated elsewhere.

Acknowledgements. The author is indebted to the Belgian National Fund for Scientific Research for a Research Grant.

References

[1] R. Abraham, and J.E. Marsden, Foundations of Mechanics, 2nd ed., (Benjamin/Cummings, Reading, 1978).

[2] F. Cantrijn, Normalized momentum maps and reduction, preprint, (Univ. of Gent, 1992).

[3] V. Guillemin, and S. Sternberg, Symplectic Techniques in Physics, (Cambridge Univ. Press, Cambridge, 1984).

[4] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, (Academic Press, New York, 1978).

[5] G. Landi, G. Marmo, G. Sparano, and G. Vilasi, A generalized reduction procedure for dynamical systems, Modern Phys. Letters 6A (1991) 3445–3453.

[6] P.G.L. Leach, The first integrals and orbit equation for the Kepler problem with drag, J. Phys. A: Math. Gen. 20 (1987) 1997–2002.

[7] P. Liberman, and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, (Reidel, Dordrecht, 1987).

[8] A. Lichnerowicz, Variétés symplectiques et variétés canoniques, in: Trends in applications of pure mathematics to mechanics, G. Fichera, ed. (Pitman, London, 1976) 249–261.

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Frans Cantrijn^{a} Instituut voor Theoretische Mechanica Universiteit Gent Krijgslaan 281 S9 B-9000 Gent, Belgium

aSenior Research Associate at the National Fund for Scientific Research, Belgium.

Received 14 October 1992