Hamel’s Formalism for Classical Field Theories

Abstract

This paper introduces Hamel’s formalism for classical field theories with the goal of analyzing the dynamics of continuum mechanical systems with velocity constraints. The developed formalism is utilized to prove the existence and uniqueness of motions of an infinite-dimensional generalization of the Chaplygin sleigh, a canonical example of nonholonomic dynamics. The formalism is very flexible and, for mechanical field theories, includes the Eulerian and Lagrangian representations of continuum mechanics as special cases. It also provides a useful approach to analyzing symmetry reduction.

Appendices

Functional-Analytic Technicalities

Recall that the field-theoretic states are smooth sections of a covariant configuration bundle $\pi _{BE} :E\rightarrow B$ with total space E, base B, and fiber F, which were assumed finite-dimensional in Sect. 2.3. The set of these sections is a functional space that needs to be equipped with a topology. This topology is a reflection of the nature of a problem under consideration. Consequently, the aforementioned functional space is not always Banach, and one needs to clarify what the bounded sets are in order to be able to utilize functional differentiation. We thus provide functional-analytic justification using convenient analysis, which is backward-compatible with more standard Banach and Hilbert settings. A comprehensive reference on convenient analysis is Kriegl and Michor (1997).

While in all known physical theories the bundle E is finite-dimensional, the constructs in Sect. 2 can be carried out for an infinite-dimensional fiber F. Thus, in this appendix, B, F, and E are smooth manifolds modeled on convenient vector spaces and F and E are, in general, infinite-dimensional.

A convenient space is a locally convex space with “tweaked” topology. By definition, the $c^\infty$-topology on a locally convex space W is the final topology with respect to all smooth curves $\gamma :{\mathbb {R}}\rightarrow W$, i.e., it is the finest topology on W with respect to which the aforementioned curves are continuous. Recall that smoothness is a bornological concept and is independent of a choice of topology of a locally convex space. In general, the $c^\infty$-topology is finer than any locally convex topology with the same collection of bounded sets. A locally convex vector space is said to be $c^\infty$-complete or convenient if it is $c^\infty$-closed in any locally convex space. For a Fréchet space, the $c^\infty$-topology coincides with the given locally convex topology. In general, a convenient space is not a topological vector space.

Letting the fiber F be infinite-dimensional allows one to elucidate connections between the infinite-dimensional and field-theoretic formalisms and provides background for possible future applications in infinite-dimensional multiscale mechanics. Indeed, recall that in the (finite-dimensional) multiscale dynamics, one benefits from introducing multiple time variables, as discussed in, e.g., Hunter (2004), Kuehn (2015), and Kuzmak (1959).

Addressing the infinite-dimensional mechanical formalism, for the trivial bundle $E=B \times F$, with $B={\mathbb {R}}$ representing the time, the field-theoretic Hamel equations recover the strong form of the Hamel equations for systems with configuration space F introduced and studied earlier in Shi et al. (2017). For instance, the configuration space for the fluid flow in a compact domain $C\subset {\mathbb {R}}^3$ is the group of diffeomorphisms ${\mathrm{Diff}}\,C$. In certain instances, the use of the infinite-dimensional formalism is more straightforward and efficient as it eliminates some of the technical requirements of the field-theoretic approach. On the other hand, the field-theoretic approach provides a complementary description of the infinite-dimensional formalism, highlights the importance of the compatibility condition, and reveals certain features of the velocity operator $\Psi$ that otherwise are difficult to notice.

In this Appendix, we adopt the notation convention of Kriegl and Michor (1997) for manifolds of smooth maps. In particular,

• Let $\pi :E\rightarrow B$ be a vector bundle. The space of all smooth sections of that bundle is denoted $C^\infty (B\leftarrow E)$. The space of all smooth sections of that bundle with compact support is denoted $C_c^\infty (B\leftarrow E)$. In the latter case, the base is assumed finite-dimensional and second countable.

• Assume that B is finite-dimensional and E admits a smooth local addition. The set ${\mathfrak {C}}^\infty (B, E)$ of smooth mappings from B to E is known to be an infinite-dimensional manifold. The model space for this manifold is the convenient space $C^\infty _c (B \leftarrow f^* TE)$ of sections with compact support of pullback bundles over B along smooth maps $f : B \rightarrow E$.

• Let M and N be smooth finite-dimensional manifolds. Let $q:N\rightarrow M$ be smooth. The set ${\mathfrak {C}}^\infty (q)$ of all smooth sections of q is a splitting smooth submanifold of $\mathfrak C^\infty (M,N)$. Its tangent space is $T{\mathfrak {C}}^\infty (q) = {\mathfrak {C}}_c^\infty (M, {\text {ker}} Tq) \subset \mathfrak C_c^\infty (M,TN)$. If $q:E\rightarrow B$ is a finite-dimensional vector bundle, the convenient vector space $C_c^\infty (B\leftarrow E)$ is a splitting smooth submanifold of ${\mathfrak {C}}^\infty (B, E)$. This statement extends to the fiber bundles with infinite-dimensional standard fiber under appropriate technical conditions; see Kriegl and Michor (1997) for details.

Using the above notations, the set of sections ${\mathcal {S}}$ introduced in Sect. 2.3 is just $\mathfrak C^\infty (\pi _{BE})$. Similarly, ${\mathcal {C}} = {\mathfrak {C}}^\infty (\pi _{DU})$. Thus, the set of sections ${\mathcal {S}}$ is a splitting submanifold of ${\mathfrak {C}}^\infty (B, E)$ when E is finite dimensional, which has been assumed in Sect. 2.3.

Next, invoking the Cartesian closedness (see Kriegl and Michor (1997)), one verifies that both the velocity operator $\Psi _\phi$ constructed from the family of operators $\psi _y: W_\Gamma \rightarrow {\mathcal {V}}_y E$ in Sect. 2.3 and its inverse $\Psi _\phi ^{-1}$ are bounded linear operators smoothly dependent on $\phi$. We point out that the operators $\Psi _\phi$ that introduce the spatial and convective velocity in continuum mechanics as in Gay-Balmaz et al. (2012) belong to this class. This class of operators is capable of producing the field-theoretic Lagrange–Poincaré equations of Ellis et al. (2011), relating Hamel’s formalism and the classical exposition of field theories.

The Field-Theoretic Reduction

Here demonstrate the flexibility of Hamel’s formalism and carry out the Lagrangian symmetry reduction in the field-theoretic context.

1.1 The Field-Theoretic Lagrange–Poincaré Equations

An intrinsic geometric description of the Lagrange–Poincaré field-theoretic equations in the absence of nonholonomic constraints has been given in Ellis et al. (2011), following a comprehensive exposition of the Lagrange–Poincaré reduction by stages in Cendra et al. (2001) and of the covariant Euler–Poincaré reduction in Castrillión López et al. (2000). See also Ellis et al. (2011) and the references therein for more details.

To put a field-theoretic system with symmetry in the geometric context, consider a principal fiber bundle $\pi _{RE}: E \rightarrow R$ with the Lie group G acting on the fibers (copies of G) by left translations. This action is denoted $\Phi : G \times E \rightarrow E$, with its associated mappings $\Phi _g:E\rightarrow E$ and $\Phi ^q:G\rightarrow E$ defined by

$$\begin{aligned} \Phi _g (q)= \Phi ^q (g):=\Phi (g, q) \quad \text {for}\quad (g ,q) \in G \times E. \end{aligned}$$

The base manifold R is called the shape space.¹⁴

Let ${\mathcal {A}} \in \Omega ^1(E, {\mathfrak {g}})$ be a principal connection form, where ${\mathfrak {g}}$ denotes the Lie algebra of the group G (see Bloch (2015) and Kriegl and Michor (1997) for a review of principal connections). We write the local expression for $\mathcal A$ in a principal fiber bundle atlas

$$\begin{aligned} \{U_\alpha , \psi _\alpha : E_\alpha \rightarrow U_\alpha \times G\} \end{aligned}$$

(B.1)

as

$$\begin{aligned} (\psi _\alpha ^{-1})^* {\mathcal {A}} (v_r, T_e L_{g} \eta ) = {\text {Ad}}_g ({\mathcal {A}}_\alpha (v_r) + \eta ) \quad \text {for}\quad v_r \in T_r U_\alpha ,\quad \eta \in {\mathfrak {g}}, \end{aligned}$$

where $E_\alpha : = E | U_\alpha$, ${\mathcal {A}}_\alpha := s_\alpha ^* {\mathcal {A}} \in \Omega ^1(U _\alpha , {\mathfrak {g}})$ and $s_\alpha \in {\mathfrak {C}}^\infty (\pi _{U_\alpha E_\alpha })$ are the local sections given by $s_\alpha (r) = \psi _\alpha ^{-1} (r, e)$. Here, e denotes the identity element of G.

Let $\Xi :=\Theta \circ {\mathcal {A}} \in \Omega ^1 (E; VE)$ be a principal connection that is a projection onto VE, where the vertical bundle$VE := {\mathrm{Ker}}\, T \pi _{RE}$ is a locally splitting vector subbundle of TE and where $\Theta$ defined by

$$\begin{aligned} {\mathfrak {g}}\ni \eta \mapsto \Theta _\eta (q):=T_e \Phi ^q \cdot \eta \in {\mathcal {X}} (E) \quad \text{ for }\quad q \in E \end{aligned}$$

is the fundamental vector field mapping. Then the horizontal bundle$HE:={\mathrm{Ker}}\, \Xi$ is a locally splitting subbundle of TE that is complementary to VE, i.e., $TE=HE \oplus VE$, or, using the bundle atlas (B.1),

$$\begin{aligned} T (\psi _\alpha )^{-1}(v_r, T_e L_g \eta )= & {} T (\psi _\alpha )^{-1} (v_r, -T_e L_g {\mathcal {A}}_\alpha v_r ) \\&+ T (\psi _\alpha )^{-1} (0,T_e L_g (\eta + {\mathcal {A}}_\alpha v_r)) \in {\mathrm{Ker}}\,\Xi \oplus {\text {Im}}\,\Xi . \end{aligned}$$

Consider a section $\phi \in {\mathfrak {C}}^\infty (\pi _{U E| U})$, where U is an open subset of B with compact closure ${{\bar{U}}}$. Let the pair of sections $(\rho , g) \in {\mathfrak {C}}^\infty (\pi _{U U_\alpha }) \times {\mathfrak {C}}^\infty (\pi _{U G})$ represent the section $\phi$ in the bundle atlas, i.e., $\phi = (\psi _\alpha ) ^{-1} (\rho , g)$. Thus, the following commutative diagram holds:

Hereafter, we assume, without loss of generality, that $\rho (U)\subset U_\alpha$ for some $\alpha$.¹⁵

In this setting, the model space $W_\Gamma$ is isomorphic to $\Omega ^1 (U, W_R) {\oplus } \Omega ^1 (U,{\mathfrak {g}})$. To proceed with Hamel’s formalism, define the operators $\Psi _\phi : \Omega ^1 (U, W_R) {\oplus } \Omega ^1 (U,{\mathfrak {g}}) \rightarrow \Omega ^1 (U, H_\phi E) {\oplus } \Omega ^1 (U, V_\phi E)$ by

$$\begin{aligned} (\varvec{\zeta }, {{\bar{\phi }}}) \mapsto T_\phi (\psi _\alpha ) ^{-1} ( (\varvec{\zeta },\, -T_e L_g {\mathcal {A}}_\alpha \varvec{\zeta }), (0, T_e L_g {{\bar{\phi }}})), \end{aligned}$$

where $W_R$ is the model space of the manifold R and where $H_\phi E$ and $V_\phi E$ are understood in the pointwise sense, i.e., as $H_{\phi (x)} E$ and $V_{\phi (x)} E$ at an unidentified $x\in U$. The relations among all of the mappings involved in the definition of the operator $\Psi$ are summarized in the following commutative diagram:

where $\Psi ^V (r, g, v, \eta ):= T \psi _\alpha ^{-1}(r, 0, g,T L_g \eta )$, $\Psi ^{H} (r,g,v, \eta ):= T \psi _\alpha ^{-1} (r, v, g, -TL_g {\mathcal {A}}_\alpha v)$, and the maps $\iota$ and $\kappa$ are determined uniquely by the diagram.

It is straightforward to verify that

$$\begin{aligned} \Psi _\phi ^H : \Omega ^1 (U, W_R) \rightarrow \Omega ^1 (U, H_\phi E) \quad \text {and}\quad \Psi _\phi ^V : \Omega ^1 (U,{\mathfrak {g}}) \rightarrow \Omega ^1 (U, V_\phi E) \end{aligned}$$

are bounded invertible linear operators smoothly dependent on $\phi \in {\mathfrak {C}} ^\infty ( \pi _{U E| U} )$. Each $\varvec{d}\phi =(\varvec{d} \rho ,\varvec{d} g)\in \Omega ^1 (U, W_R) \times \Omega ^1 (U,T_g G)$ is then uniquely written as

$$\begin{aligned} (\varvec{d} \rho , \varvec{d} g)= ({\varvec{\zeta }}, -T_e L_{g}{\mathcal {A}}_\alpha {\varvec{\zeta }}+ T_e L_{g} {{\bar{\phi }}}). \end{aligned}$$

(B.2)

Substituting (B.2) into the local expression of the connection ${\mathcal {A}}$, we obtain

$$\begin{aligned} \big (\psi _\alpha ^{-1}\big )^* {\mathcal {A}} (\varvec{d} \phi )= {\text {Ad}} _g {{\bar{\phi }}} \end{aligned}$$

and hence

$$\begin{aligned} \phi ^*{\mathcal {A}}={\text {Ad}} _g {{\bar{\phi }}}, \end{aligned}$$

which implies that $\phi ^*{\mathcal {A}} \in \Omega ^1 (U , {\mathfrak {g}})$ is a (local) principal connection. This connection will be used to derive the reconstruction condition later.

Definition B.1

We say that the Lagrangian is $\varvec{G}$-invariant if L is invariant under the induced action $\Phi ^1 : G \times J^1 E \rightarrow J^1 E$ defined by the formula $\Phi ^1_g (\gamma ):= T \Phi _g \circ \gamma .$

After replacing $j^1 \phi$ with its coordinate representation $(\rho , g, \varvec{d}\rho , \varvec{d} g)$ and using the invariance of the Lagrangian, we obtain the reduced Lagrangian as a smooth function of $(\rho , \varvec{\zeta }, {{\bar{\phi }}})$ on $C^\infty (U, U_\alpha ) \times \Omega ^1 (U, W_R) \times \Omega ^1 (U, {\mathfrak {g}})$:

$$\begin{aligned} l(\rho , \varvec{\zeta }, {{\bar{\phi }}}) := L(j^1 \phi ). \end{aligned}$$

Recall that the curvature of the connection ${\mathcal {A}}$ is the Lie-algebra-valued two-form defined by

$$\begin{aligned} {\mathcal {B}} (X, Y) = d {{\mathcal {A}}} ({\text {hor}} X, {\text {hor}} Y), \end{aligned}$$

where X and Y are two vector fields. The Cartan structure equation reads

$$\begin{aligned} {\mathcal {B}} (X, Y) = d {{\mathcal {A}}} (X, Y) - [{{\mathcal {A}}} (X), {{\mathcal {A}}} (Y)]. \end{aligned}$$

(B.3)

Theorem B.2

The field-theoretic equations for a G-invariant Lagrangian $L :J^1 E \rightarrow {\mathbb {R}}$ read

$$\begin{aligned} {\text {div}}\frac{\delta l}{\delta \varvec{d} \rho } - \frac{\delta l}{\delta \rho }&= - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}}, {{\mathbf {i}}}_{\varvec{d} \rho } \mathcal {B}_{\alpha } - \big \langle {\text {ad}}_{\mathcal {A}_{\alpha }}, {{\bar{\phi }}} \big \rangle \right\rangle , \end{aligned}$$

(B.4)

$$\begin{aligned} {\text {div}} \frac{\delta l}{\delta {{\bar{\phi }}} }&= {\text {ad}}^*_{{{\bar{\phi }}}} {\frac{\delta l}{\delta {{\bar{\phi }}} }} - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}} }, {\text {ad}}_{\rho ^* \mathcal {A}_{\alpha }} \right\rangle , \end{aligned}$$

(B.5)

$$\begin{aligned} \varvec{d} g&= T_e L_g ({{\bar{\phi }}} - \rho ^* \mathcal {A}_{\alpha }). \end{aligned}$$

(B.6)

Proof

A straightforward calculation verifies that

$$\begin{aligned}&[(u_1, - T_e L_g {\mathcal {A}}_\alpha u_1), (u_2, -T_e L_g \mathcal A_\alpha u_2)] \\&\quad =(0, - u_1[T_e L_g {\mathcal {A}}_\alpha u_2 ] + u_2 [T_e L_g {\mathcal {A}}_\alpha u_1 ] +[T_e L_{g}{\mathcal {A}}_\alpha u_1, T_e L_{g}{\mathcal {A}}_\alpha u_2]) \\&\quad =(0, -T_e L_g ( d {\mathcal {A}}_ \alpha (u_1, u_2) - [\mathcal A_\alpha u_1, {\mathcal {A}}_\alpha u_2] )) =(0, -T_e L_g {\mathcal {B}}_ \alpha (u_1, u_2)), \end{aligned}$$

where $u_1, u_2 \in W_R$. Thus,

$$\begin{aligned}{}[ (u_1 , 0), (u_2, 0)] _\phi =\Psi _\phi ^{-1} [ \Psi u_1, \Psi u_2 ](\phi )=(0, -\mathcal {B_\alpha } ( u_1 , u_2)). \end{aligned}$$

(B.7)

Similarly, we have

$$\begin{aligned}{}[(u_1, 0), (0, w_1 ) ]_\phi = (0, -[{\mathcal {A}}_\alpha u_1, w_1]), \quad [(0, w_1), (0, w_2)]_\phi = (0, {\text {ad}}_{w_1}{w_2}),\nonumber \\ \end{aligned}$$

(B.8)

where $w_i \in {\mathfrak {g}}$, $i=1,2.$ Thus, equations (B.4) and (B.5) follow from (2.18), (B.7), (B.8), and the formula $\varvec{d} \rho =\varvec{\zeta }$. Finally, (B.6) follows from formula (B.2). $\square$

Remarks. 

• (i) The representation (B.4) and (B.5) of the Lagrange–Poincaré field-theoretic equations originated in Ellis et al. (2011), where the theory has been presented in a global form.

• (ii) Given a solution $(\rho , \varvec{d} \rho , {{\bar{\phi }}})$ of the reduced Lagrange–Poincaré field-theoretic equations (B.4) and (B.5), one needs the reconstruction condition to obtain the group dynamics from the reconstruction equation (B.6), which was pointed out in Castrillión López et al. (2000) and presented for the Lagrange–Poincaré field-theoretic equations in Ellis et al. (2011). For completeness, below we give a brief proof utilizing Hamel’s formalism.

The Reconstruction Condition. If $\phi$ is the solution of the original Euler–Lagrange equations, then for every $q= \phi (x)$, $x\in U$, and $v_q \in T_q {\text {Im}}(\phi )$ we have

$$\begin{aligned} {\mathcal {A}} (v_q) ={\mathcal {A}} (T_x \phi v_x)={\mathcal {A}} (T_q (\phi \circ \pi _{BE} ) v_q) = (\phi \circ \pi _{BE})^*{\mathcal {A}} (v_q), \end{aligned}$$

i.e., $v_q \in {\mathrm{Ker}}\, \omega ^\phi$, where $\omega ^\phi := {\mathcal {A}} - (\phi \circ \pi _{BE})^* {\mathcal {A}}$ is a new principal connection on $E | U_\alpha$. It is easy to verify that ${\mathrm{Ker}}\, \omega ^\phi \subset T{\text {Im}}\phi$, and hence ${\mathrm{Ker}}\, \omega ^\phi = T {\text {Im}} \phi$. This shows that ${\mathrm{Ker}}\, \omega ^\phi$ is involutive and the principal connection $\omega ^\phi$ is flat, i.e., for any $X_1, X_2 \in {\mathcal {X}} (B)$ we have

$$\begin{aligned} 0&= {\mathcal {A}} [T \phi X_1, T \phi X_2] - (\phi \circ \pi _{BE} )^* {\mathcal {A}} [T \phi X_1, T \phi X_2] = (\psi _\alpha ^{-1})^* \mathcal A [\varvec{d} \phi X_1, \varvec{d} \phi X_2] \\&\quad - \phi ^* {\mathcal {A}} [X_1, X_2 ] \\&= {\text {Ad}}_g (- {\mathcal {B}}_\alpha (\varvec{d} \rho X_1, \varvec{d} \rho X_2) + \varvec{d} {{\bar{\phi }}} (X_1, X_2) - [{\mathcal {A}}_\alpha \varvec{d} \rho X_1, {{\bar{\phi }}} X_2] \\&\quad + [{\mathcal {A}}_\alpha \varvec{d} \rho X_2, {{\bar{\phi }}} X_1] + [{{\bar{\phi }}} X_1, {{\bar{\phi }}} X_2]), \end{aligned}$$

which means that the following reconstruction condition holds:

$$\begin{aligned} \rho ^* {\mathcal {B}}_\alpha = d {{\bar{\phi }}} - \rho ^* {\mathcal {A}}_\alpha \wedge {{\bar{\phi }}} + \tfrac{1}{2}{{\bar{\phi }}} \wedge {{\bar{\phi }}}, \end{aligned}$$

(B.9)

where the wedge product is defined by

$$\begin{aligned} \Upsilon _1 \wedge \Upsilon _2 (X_1, X_2) := [\Upsilon _1(X_1), \Upsilon _2(X_2)] - [\Upsilon _1(X_2), \Upsilon _2(X_1)] \quad \text{ for }\quad \Upsilon _1, \Upsilon _2 \in \Omega ^1 (U, \mathfrak {g}). \end{aligned}$$

Conversely, given a solution of the Lagrange–Poincaré field-theoretic equations, the reconstruction condition (B.9) implies that the induced principal connection $\omega ^\phi$ is flat. Assuming further that G is a regular Lie group, the horizontal subbundle ${\mathrm{Ker}}\, \omega ^\phi$ is integrable and thus defines a foliation (see Kriegl and Michor (1997) for details), which determines the solution of the original Euler–Lagrange equations provided that we have chosen U as a simply connected open set.

Remarks.

• (i) One can derive the reconstruction condition (B.9) by substituting ${\varvec{\xi }} =(\varvec{\zeta }, {{\bar{\phi }}})$ into the compatibility condition (2.22) and using the bracket formulae (B.7) and (B.8). However, it is necessary to introduce the horizontal subbundle ${\mathrm{Ker}}\, \omega ^\phi$ for reconstructing the solution of the original Euler–Lagrange equations.

• (ii) Define the covariant exterior derivative of ${{\bar{\phi }}}$ by the formula

  $$\begin{aligned} d^{{\mathcal {A}}} {{\bar{\phi }}}:=d {{\bar{\phi }}} - \rho ^* {\mathcal {A}}_\alpha \wedge {{\bar{\phi }}}. \end{aligned}$$

  Then, (B.2) becomes

  $$\begin{aligned} d ^{{\mathcal {A}}} {{\bar{\phi }}} + \tfrac{1}{2} {{\bar{\phi }}} \wedge {{\bar{\phi }}} = \rho ^ *{\mathcal {B}}, \end{aligned}$$

  which coincides with the reconstruction condition given in Ellis et al. (2011).

1.2 Reduced Dynamics of Nonholonomic Systems with Symmetry

Here, we construct a suitable connection form and obtain the reduced nonholonomic dynamics in the field-theoretic setting.

Assume that a locally splitting subbundle ${\mathcal {D}}$ of TE is invariant with respect to the induced action of the symmetry group G on TE. Then ${\mathcal {S}}: = {\mathcal {D}} \cap VE$ is invariant with respect to the induced action as well. We assume that ${\text {dim}} {\mathcal {S}}_q > 0$.

Thus, ${\mathcal {S}}$ is a locally splitting finite-dimensional subbundle of ${\mathcal {D}}$, and one can select locally G-invariant subbundles ${\mathcal {V}}$ and ${\mathcal {D}}^{\mathcal {U}}$ such that $VE = {\mathcal {S}} \oplus {\mathcal {V}}$ and ${\mathcal {D}} = {\mathcal {S}} \oplus {\mathcal {D}}^{{\mathcal {U}}}$. It follows from the left-invariance of the distributions ${\mathcal {S}}$ and ${\mathcal {V}}$ that there exist subspaces ${\mathfrak {g}} ^{{\mathcal {S}}}$ and ${\mathfrak {g}} ^{{\mathcal {U}}}$ of the Lie algebra ${\mathfrak {g}}$ such that in a local trivialization ${\mathcal {S}} _q = {\Psi }_q {\mathfrak {g}} ^{{\mathcal {S}}}$ and ${\mathcal {V}}_q ={\Psi }_q {\mathfrak {g}} ^{{\mathcal {U}}}$. Given $\xi \in {\mathfrak {g}}$, we write its components along these subspaces as $\xi ^{{\mathcal {S}}}$ and $\xi ^{{\mathcal {V}}}$. For simplicity, we make the following assumption:

The Dimension Assumption.The constraints and the vertical directions span the entire tangent space to the spaceE:

$$\begin{aligned} {\mathcal {D}} + VE = TE. \end{aligned}$$

Thus, in a local trivialization of TE induced by a principal fiber bundle atlas $\{U_\alpha \subset R, \psi _\alpha : E |U_\alpha \rightarrow U_\alpha \times G\}$, we can describe $\mathcal D^{{\mathcal {U}}}$ by means of a principal connection, i.e., we describe the subspaces ${\mathcal {D}}_q^{{\mathcal {U}}}$ as the horizontal spaces

$$\begin{aligned} \{(r_t, -{\mathcal {A}}^{\mathcal {U}} (r)r_t) \mid (r, r_t) \in U_\alpha \times W_R\}, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {A}}^{\mathcal {U}} : U_\alpha \rightarrow L\big (W_R , \mathfrak g^{\mathcal {U}}\big ) \end{aligned}$$

is the local representation of the connection.

Hence, in the same local trivialization of TE the constraints ${\mathcal {D}}$ read

$$\begin{aligned} \xi ^{{\mathcal {U}}} + {\mathcal {A}}^{{\mathcal {U}}} r_t = 0,\quad \text{ where }\quad \xi =T L_g^{-1} g_t. \end{aligned}$$

(B.10)

Then a section $\phi \in {\mathfrak {C}}^\infty (U E| U)$ consistent with constraints satisfies the condition

$$\begin{aligned} {{\bar{\phi }}}_t^{{\mathcal {U}}} = \xi ^{{\mathcal {U}}}+ {\mathcal {A}}^{{\mathcal {U}}} \rho _t = 0. \end{aligned}$$

Let ${\mathcal {A}}_s$ be a connection defined in a local trivialization by the formula

$$\begin{aligned} {\mathcal {A}}_s = {\text {Ad}}_g (\xi + {\mathcal {A}} r_t), \end{aligned}$$

where $\xi \in {\mathfrak {g}}$ and where ${\mathcal {A}}$ is a $\mathfrak g$-valued form on R. That is, the ${\mathcal {U}}$-component of the form ${\mathcal {A}}$ is defined by the constraints while the ${\mathcal {S}}$-component of ${\mathcal {A}}$ will be determined later. Let $\pi ^{\mathcal {S}} : U_\alpha \rightarrow L ({\mathfrak {g}}, \mathfrak g^{\mathcal {S}})$ be a smooth mapping such that $\pi ^{\mathcal {S}} (r)$ is a projection on ${\mathfrak {g}}^{\mathcal {S}}$.

Define the ${\mathfrak {g}} \otimes {\mathfrak {g}}^*$-valued one-form ${\mathcal {E}}$ by

$$\begin{aligned} \langle {\mathcal {E}} v,\eta \rangle := (d \pi ^{\mathcal {S}} \cdot v ) \eta - [{\mathcal {A}} v, \eta ] \quad \text {for}\quad v \in W_R,\quad \eta \in {\mathfrak {g}}. \end{aligned}$$

(B.11)

Using the constrained Hamel equations (2.34) and the G-invariance of the constraint distribution ${\mathcal {D}}$, one obtains the reduced equations from (B.4) and (B.5) by projecting equation (B.5) onto the fibers of the bundle ${\mathfrak {g}}^*_{{\mathcal {S}}}$ and imposing constraints, i.e., by replacing ${{\bar{\phi }}}_t$ with ${{\bar{\phi }}}_t^{{\mathcal {S}}}:= \pi ^\mathcal S {{\bar{\phi }}}_t$. Note that in this case we need the formula

$$\begin{aligned}{}[(u, 0), (0, \pi ^ {{\mathcal {S}}} w ) ] _\phi = u [ \pi ^ {{\mathcal {S}}} w]-[{\mathcal {A}} u, \pi ^ {{\mathcal {S}}} w] \end{aligned}$$

in order to evaluate the bracket that appeared in the proof of Theorem B.2. Summarizing, we have:

Theorem B.3

The reduced field-theoretic constrained dynamics is given by the equations

$$\begin{aligned} {\text {div}} \frac{\delta l}{\delta \varvec{d} \rho } - \frac{\delta l}{\delta \rho }&= - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}_t}, {{\mathbf {i}}}_{\rho _t} {\mathcal {B}} + \big \langle {\mathcal {E}}, {{\bar{\phi }}}_t^{{\mathcal {S}}} \big \rangle \right\rangle - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}_x}, {{\mathbf {i}}}_{\varvec{d}_x\!\rho }{\mathcal {B}} - \big \langle {\text {ad}}_{{\mathcal {A}}}, {{\bar{\phi }}}_x \big \rangle \right\rangle , \end{aligned}$$

(B.12)

$$\begin{aligned} \left[ {\text {div}} \frac{\delta l}{\delta {{\bar{\phi }}}} \right] _{{\mathcal {S}}}&= \left[ {\text {ad}}^*_{{{\bar{\phi }}}_t^{{\mathcal {S}}} } \frac{\delta l}{\delta {{\bar{\phi }}}_t} + {\text {ad}}^*_{{{\bar{\phi }}}_x } \frac{\delta l}{\delta {{\bar{\phi }}}_x} + \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}}, {{\mathbf {i}}}_{\varvec{d} \rho } {\mathcal {E}} \right\rangle \right] _{{\mathcal {S}}} . \end{aligned}$$

(B.13)

In the above, the Lagrangian l is a function of $(\rho , \varvec{d} \rho , {{\bar{\phi }}})$ while the curvature ${\mathcal {B}}$ of the form ${\mathcal {A}}$ and the quantity ${\mathcal {E}}$ are defined by (B.3) and (B.11), respectively. Note that the partial derivatives of l in (B.12) and (B.13) are computed before setting ${{\bar{\phi }}}_t = {{\bar{\phi }}}_t^{{\mathcal {S}}}$.

Once $\rho$ and ${{\bar{\phi }}}$ are found from equations (B.12) and (B.13), the solutions of the corresponding Euler–Lagrange equation on the simply connected set U are recovered by solving the reconstruction equations

$$\begin{aligned} \varvec{d} g = T_e L_g ({{\bar{\phi }}} - \rho ^* {\mathcal {A}}) \end{aligned}$$

in combination with the reconstruction condition

$$\begin{aligned} \rho ^* {\mathcal {B}}_\alpha = \tfrac{1}{2}{{\bar{\phi }}} \wedge {{\bar{\phi }}} + d {{\bar{\phi }}} - \rho ^ *{\mathcal {A}}_\alpha \wedge {{\bar{\phi }}}. \end{aligned}$$

Equations (B.12) and (B.13) can be rewritten as

$$\begin{aligned} {\text {div}} \frac{\delta l_c}{\delta \varvec{d} \rho } - \frac{\delta l_c}{\delta \rho }&= - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}_t}, {{\mathbf {i}}}_{\rho _t} {\mathcal {B}} + \big \langle {\mathcal {E}}, {{\bar{\phi }}}_t^{{\mathcal {S}}} \big \rangle \right\rangle - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}_x}, {{\mathbf {i}}}_{{\varvec{d}}_x \rho } {\mathcal {B}} - \big \langle {\text {ad}}_{{\mathcal {A}}}, {{\bar{\phi }}}_x \big \rangle \right\rangle , \end{aligned}$$

(B.14)

$$\begin{aligned} \frac{\partial }{\partial t}\frac{\delta l_c}{\delta {{\bar{\phi }}}_t^{{\mathcal {S}}}} + \left[ {\text {div}}_x \frac{\delta l_c}{\delta {{\bar{\phi }}}_x} \right] _{{\mathcal {S}}}&= \left[ {\text {ad}}^*_{{{\bar{\phi }}}_t^{{\mathcal {S}}}} \frac{\delta l}{\delta {{\bar{\phi }}}_t} + {\text {ad}}^*_{{{\bar{\phi }}}_x} \frac{\delta l}{\delta {{\bar{\phi }}}_x} + \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}}, {\mathbf {i}}_{\varvec{d} \rho } {\mathcal {E}} \right\rangle \right] _{{\mathcal {S}}}, \end{aligned}$$

(B.15)

where $l _c (\rho , \varvec{d} \rho , {{\bar{\phi }}}_t^{{\mathcal {S}}}, {{\bar{\phi }}}_X) := l(\rho , \varvec{d} \rho , {{\bar{\phi }}}_t^{{\mathcal {S}}}, {{\bar{\phi }}}_X)$ is the constrained reduced Lagrangian. These equations follow directly from (B.12) and (B.13) as

$$\begin{aligned} \frac{\delta l _c }{\delta \varvec{d} \rho } = \frac{\delta l}{\delta \varvec{d} \rho } \Big | _{{{\bar{\phi }}}_t = {{\bar{\phi }}}_t ^{{\mathcal {S}}} }, \quad {\text {div}} \frac{\delta l _c }{\delta \varvec{d} \rho } ={\text {div}} \frac{\delta l}{\delta \varvec{d} \rho } \Big |_{{{\bar{\phi }}}_t = {{\bar{\phi }}}_t ^{{\mathcal {S}}}}, \quad \frac{\delta l_c}{\delta \rho } = \frac{\delta l}{\delta \rho } \Big |_{{{\bar{\phi }}}_t = {{\bar{\phi }}}_t ^{{\mathcal {S}}}}, \quad \frac{\delta l_c}{\delta {{\bar{\phi }}}_t^{{\mathcal {S}}}} = \frac{\delta l}{\delta {{\bar{\phi }}}_t^{{\mathcal {S}}}} \Big |_{{{\bar{\phi }}}_t = {{\bar{\phi }}}_t^{{\mathcal {S}}}}. \end{aligned}$$

Note that, in general,

$$\begin{aligned} \frac{\delta l _c }{\delta {{\bar{\phi }}}_t ^{{\mathcal {S}}}} \ne \frac{\delta l}{\delta {{\bar{\phi }}}_t } \Big | _{{{\bar{\phi }}}_t = {{\bar{\phi }}}_t^{{\mathcal {S}}} }. \end{aligned}$$

1.3 The Moving Body Frame and Nonholonomic Connection

In this section, we will obtain the reduced dynamics for a class of field-theoretic nonholonomic systems with symmetry by specifying the ${\mathcal {S}}$-component of the connection form ${\mathcal {A}}$ when the Lagrangian equals the kinetic minus potential energy with the kinetic energy induced by a Riemannian metric $\langle \!\langle \cdot , \cdot \rangle \!\rangle$ on the space E, i.e.,

$$\begin{aligned} K(\phi ) := \int _{U_X} \langle \!\langle \phi _t ,\phi _t \rangle \!\rangle \,\mu _X \quad \text{ for }\quad \phi \in \mathfrak C^\infty (U E|U), \end{aligned}$$

where $U_X = \pi _{XB} U$ and $\mu _X$ is the volume form on X satisfying $\mu = dt \wedge \mu _X$. We follow the arguments given in Bloch et al. (2009).

Definition B.4

In the principal case and for a Lagrangian of the form kinetic minus potential energy, the nonholonomic connection$\mathcal A^{\mathrm {nhc}}$ is, by definition, the connection on the principal bundle $E \rightarrow R$ whose horizontal space at $q \in E$ is given by the orthogonal complement to the space ${\mathcal {S}}_q = {\mathcal {D}}_q \cap V_q E$ within the space ${\mathcal {D}}_q$.

In a local trivialization of E, the inertia tensor

$$\begin{aligned} {\mathcal {I}} : {\mathfrak {g}} = {\mathfrak {g}}^{{\mathcal {S}}} \oplus \mathfrak g^{{\mathcal {V}}} \rightarrow \big ({\mathfrak {g}}^{{\mathcal {S}}}\big )^* \oplus \big ({\mathfrak {g}}^{{\mathcal {V}}}\big )^* \end{aligned}$$

is given by

$$\begin{aligned} \langle {\mathcal {I}} (r) \xi , \eta \rangle = \langle \!\langle \Theta _ \xi , \Theta _\eta \rangle \!\rangle , \quad \xi , \eta \in {\mathfrak {g}}. \end{aligned}$$

The decomposition ${\mathfrak {g}} = {\mathfrak {g}}^{{\mathcal {S}}} \oplus {\mathfrak {g}}^{{\mathcal {V}}}$ induces the block tensor representation

$$\begin{aligned} \begin{bmatrix} {\mathcal {I}}_{{\mathcal {S}}} &{}{\mathcal {I}}_{{{\mathcal {S}}}{{\mathcal {V}}}} \\ {\mathcal {I}}_{{{\mathcal {V}}}{{\mathcal {S}}}} &{} {\mathcal {I}}_{{\mathcal {V}}} \end{bmatrix} \end{aligned}$$

of the tensor ${\mathcal {I}}$. In particular, ${\mathcal {I}}_{{\mathcal {S}}}$ is called constrained locked inertia tensor. Recall that $\Theta : {\mathfrak {g}} \rightarrow {\mathcal {X}} (E)$ is the fundamental vector field mapping. Since ${\mathfrak {g}}^{{\mathcal {S}}}$ is finite-dimensional, the inverse of the constrained locked inertia tensor ${\mathcal {I}}^{-1}_{{\mathcal {S}}}$ exists. Under the assumption that the Lagrangian and the distribution ${\mathcal {D}}$ are invariant, the nonholonomic connection in a local trivialization is defined by the formula

$$\begin{aligned} {\mathcal {A}}^{\mathrm {nhc}} = {\text {Ad}}_g (\xi + {\mathcal {A}} r_t), \end{aligned}$$

where $\xi \in {\mathfrak {g}}$ and where ${\mathcal {A}}$ is a $\mathfrak g$-valued one-form on R. Define the body angular velocity$\Omega \in {\mathfrak {g}}$ by the formula

$$\begin{aligned} \Omega = \xi + {\mathcal {A}} r_t, \end{aligned}$$

then the constraints (B.10) read

$$\begin{aligned} \Omega ^{{\mathcal {V}}} = 0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} {\mathcal {A}}^{{\mathcal {V}}} r_t = -\xi ^{{\mathcal {V}}}. \end{aligned}$$

Let the kinetic energy metric in a local trivialization be written as

$$\begin{aligned} \langle \! \langle q_t, q_t \rangle \! \rangle = \langle {G (r)} r_t, r_t \rangle + 2 \langle {{\mathcal {K}} (r)} r_t, \xi \rangle + \langle {{\mathcal {I}} (r)} \xi , \xi \rangle . \end{aligned}$$

Define the ${\mathcal {S}}$-component of ${\mathcal {A}}$ by

$$\begin{aligned} {\mathcal {A}}^{{\mathcal {S}}} = {\mathcal {I}}_{{\mathcal {S}}}^{-1} (r) \mathcal K_{{\mathcal {S}}} (r) - {\mathcal {I}}_{{\mathcal {S}}}^{-1} (r) \mathcal I_{{{\mathcal {S}}}{{\mathcal {V}}}} (r) {\mathcal {A}}^{{\mathcal {V}}}, \end{aligned}$$

where ${\mathcal {K}}_{{\mathcal {S}}}$ is the inclusion of ${\mathcal {K}}$ from ${\mathfrak {g}}^*$ to $\big ({\mathfrak {g}}^{{\mathcal {S}}}\big )^*$. Then, ${\mathcal {A}}$ satisfies definition B.4, and the constrained kinetic energy metric written as a function of $(r_t, \Omega ^{{\mathcal {S}}})$ is in the following block-diagonal form:

$$\begin{aligned} \big \langle G _c (r) r_t, r_t \big \rangle + \big \langle {\mathcal {I}} _{{\mathcal {S}}} (r) \Omega ^{{\mathcal {S}}}, \Omega ^{{\mathcal {S}}} \big \rangle . \end{aligned}$$

Introduce the nonholonomic momentum relative to the body frame$p = \delta l_c / \delta {{\bar{\phi }}}_t^{{\mathcal {S}}}$, then

$$\begin{aligned} \frac{\delta l}{\delta {{\bar{\phi }}}_t} = p + {\mathcal {I}}_{{{\mathcal {V}}}{{\mathcal {S}}}} {\mathcal {I}}_{{\mathcal {S}}}^{-1} p + \mathbf {i}_{r_t} \Lambda , \end{aligned}$$

(B.16)

where $\Lambda \in \Omega ^1\big (U_\alpha , \big ({\mathfrak {g}}^{\mathcal V}\big )^*\big )$ is given by

$$\begin{aligned} \Lambda = \big ( {\mathcal {K}} _{{\mathcal {V}}} - {\mathcal {I}}_{{{\mathcal {V}}}{{\mathcal {S}}}} {\mathcal {A}}^{{\mathcal {S}}} - {\mathcal {I}}_{{\mathcal {V}}} \mathcal A^{{\mathcal {V}}} \big ) dr. \end{aligned}$$

Substituting (B.16) into (B.14) and (B.15), we obtain the reduced dynamics:

$$\begin{aligned} {\text {div}} \frac{\delta l_c}{\delta \varvec{d} \rho } - \frac{\delta l_c}{\delta \rho }&= - \bigg \langle p, \frac{\partial {\mathcal {I}}_{{\mathcal {S}}}^{-1} }{\partial r}\, p \bigg \rangle \nonumber \\&\quad \, - \Big \langle p + {\mathcal {I}} _{{{\mathcal {V}}}{{\mathcal {S}}}} {\mathcal {I}} _{{\mathcal {S}}}^{-1} p + {\mathbf {i}}_{r_t} \Lambda , {\mathbf {i}}_{\rho _t} {\mathcal {B}} + \big \langle {\mathcal {E}}, {{\bar{\phi }}}_t^{{\mathcal {S}}} \big \rangle \Big \rangle \nonumber \\&\quad - \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}_x}, {\mathbf {i}}_{\varvec{d}_x\!\rho } {\mathcal {B}} - \big \langle {\text {ad}}_{{\mathcal {A}}}, {{\bar{\phi }}}_x \big \rangle \right\rangle , \end{aligned}$$

(B.17)

$$\begin{aligned} p_t + \left[ {\text {div}}_x \frac{\delta l_c}{\delta {{\bar{\phi }}}_x} \right] _{{\mathcal {S}}}&= \left[ {\text {ad}}^*_{\mathcal I_{{\mathcal {S}}}^{-1} p} p + {\text {ad}}^*_{{\mathcal {I}}_{\mathcal S}^{-1} p} {\mathcal {I}}_{{{\mathcal {V}}}{{\mathcal {S}}}} {\mathcal {I}}_{{\mathcal {S}}}^{-1} p \right. \nonumber \\&\quad \, + \left. \big \langle p, {\mathbf {i}}_{\rho _t} \tau \big \rangle + \big \langle \mathbf {i}_{\rho _t} \Lambda , {\mathbf {i}}_{\rho _t} {\mathcal {E}} \big \rangle + {\text {ad}}^*_{{{\bar{\phi }}}_x} \frac{\delta l}{\delta {{\bar{\phi }}}_x} + \left\langle \frac{\delta l}{\delta {{\bar{\phi }}}_x}, {\mathbf {i}}_{\varvec{d}_x \rho } {\mathcal {E}} \right\rangle \right] _{{\mathcal {S}}}. \end{aligned}$$

(B.18)

Equation (B.18) is called the momentum equation in body representation. The reduced Lagrangian in these equations is represented as a function of $(\rho , \varvec{d} \rho , p, {{\bar{\phi }}}_x)$ and $\tau \in \Omega ^1\big (U_\alpha ,{\mathfrak {g}}^{{\mathcal {S}}} \otimes \big ({\mathfrak {g}}^{{\mathcal {S}}}\big )^*\big )$ is defined by the formula

$$\begin{aligned} \big \langle p, {\mathbf {i}}_{\rho _t} \tau \big \rangle = \left[ {\text {ad}}^*_{{\mathcal {I}}_{{\mathcal {S}}}^{-1} p} {\mathbf {i}}_{\rho _t} \Lambda + \left\langle p + {\mathcal {I}}_{{{\mathcal {V}}}{{\mathcal {S}}}} {\mathcal {I}}_{{\mathcal {S}}}^{-1} p, {\mathbf {i}}_{\rho _t} {\mathcal {E}} \right\rangle \right] _{{\mathcal {S}}}. \end{aligned}$$

1.4 The Equivalence of the Lagrange–Poincaré Equations and the Lagrange–Poincaré Field-Theoretic Equations

Using the approach of Cendra and Marsden (1987) for treating the Lin constraints, it is possible to derive the Lagrange–Poincaré field-theoretic equations directly from the Lagrange–Poincaré equations by utilizing a trivial connection. Our strategy is to introduce additional independent variables and impose the compatibility conditions using the Lagrange multiplier $\lambda$.

More specifically, under the assumptions of Sect. B.1, introduce the extended reduced Lagrangian on $T (C^\infty (U, U_\alpha ) \times C^\infty (U, L(W_X, {\mathfrak {g}}) )\times C^\infty (U, W_B \otimes {\mathfrak {g}}^*) )\times C^\infty (U, {\mathfrak {g}})$ by the formula

$$\begin{aligned} {{\bar{l}}} (\rho ,\gamma ,\lambda , \rho _t, \gamma _t, \lambda _t, \zeta )&:= l(\rho ,\rho _t, \varvec{d}_x \rho , \gamma , \zeta ) + \langle \lambda , \gamma _t - \varvec{d}_x \zeta + [\zeta , \gamma ] \rangle \\&\;= l (\rho ,\rho _t, \varvec{d}_x \rho , \gamma , \zeta ) + \langle \lambda , \gamma _t \rangle + \langle C(\gamma , \lambda ), \zeta \rangle , \end{aligned}$$

where $C(\gamma , \lambda ) : C^\infty (U, L(W_X, {\mathfrak {g}})) \times C^\infty (U, W_B \otimes {\mathfrak {g}}^*) \rightarrow C^\infty (U, {\mathbb {R}} \otimes {\mathfrak {g}}^*)$ is defined by

$$\begin{aligned} \langle C (\gamma , \lambda ), \eta \rangle := \langle \lambda , -\varvec{d}_x \eta + [\eta , \gamma ]\rangle \end{aligned}$$

for $\eta \in C ^\infty (U , {\mathfrak {g}})$.

Substituting the extended Lagrangian ${{\bar{l}}}$ into the Euler–Lagrange–Poincaré equations of Shi et al. (2017), we obtain the horizontal part of the equations written in components as

$$\begin{aligned} 0&= \frac{\delta {{{\bar{l}}}}}{\delta \rho } - \frac{d}{dt} \frac{\delta {{{\bar{l}}}}}{\delta \rho _t} \nonumber \\&= \frac{\delta l}{\delta \rho } - {\text {div}}_x \left( \frac{\delta l}{\delta \varvec{d}_x \rho } \right) - \frac{d}{dt} \frac{\delta l}{\delta \rho _t} \nonumber \\&= \frac{\delta l}{\delta \rho } - {\text {div}} \left( \frac{\delta l}{\delta \varvec{d} \rho } \right) , \end{aligned}$$

(B.19)

$$\begin{aligned} 0&= \frac{\delta {{{\bar{l}}}}}{\delta \gamma } - \frac{d}{dt} \frac{\delta {{{\bar{l}}}}}{\delta \gamma _t} = \frac{\delta l}{\delta \gamma } - \lambda _t + {\text {ad}}^*_\zeta \lambda , \end{aligned}$$

(B.20)

$$\begin{aligned} 0&= \frac{\delta {{{\bar{l}}}}}{\delta \lambda } - \frac{d}{dt} \frac{\delta {{{\bar{l}}}}}{\delta \lambda _t} = \gamma _t -\varvec{d}_x \zeta + [\zeta , \gamma ], \end{aligned}$$

(B.21)

respectively, and the corresponding vertical component of the Lagrange–Poincaré equations reads

$$\begin{aligned} 0 = {\text {ad}}^ *_{\zeta }\frac{\delta {{{\bar{l}}}}}{\delta \zeta } - \frac{d}{dt} \frac{\delta {{{\bar{l}}}}}{\delta \zeta } = {\text {ad}}^ *_{\zeta }\left( \frac{\delta l}{\delta \zeta } + C (\gamma , \lambda )\right) - \frac{d}{dt} \left( \frac{\delta l}{\delta \zeta } + C (\gamma , \lambda )\right) . \end{aligned}$$

(B.22)

Using (B.20) and (B.21), the definition of $C(\gamma , \lambda )$, the Leibnitz formula, and the Jacobi identity, we obtain, for every $\eta \in C^\infty (U_X, L({\mathbb {R}}, {\mathfrak {g}}))$:

$$\begin{aligned} \big \langle {\text {ad}}^*_\zeta&C(\gamma , \lambda ), \eta \big \rangle - \left\langle \frac{d}{dt} C (\gamma , \lambda ), \eta \right\rangle \nonumber \\&= \big \langle \lambda , - \varvec{d}_x [\zeta , \eta ] + [[\zeta , \eta ], \gamma ] \big \rangle - \big \langle \lambda _t, -\varvec{d}_x \eta + [\eta , \gamma ] \big \rangle - \big \langle \lambda , [\eta , \gamma _t] \big \rangle \nonumber \\&= \big \langle \lambda , - \varvec{d}_x [\zeta , \eta ] + [[\zeta , \eta ], \gamma ] \big \rangle - \left\langle \frac{\delta l}{\delta \gamma } + {\text {ad}}^*_\zeta \lambda , - \varvec{d}_x \eta + [\eta , \gamma ] \right\rangle \nonumber \\&\quad - \big \langle \lambda , [\eta ,\varvec{d}_x \zeta - [\zeta , \gamma ]] \big \rangle \nonumber \\&= \big \langle \lambda , -\varvec{d}_x [\zeta , \eta ] + [\zeta , \varvec{d}_x \eta ] + [\varvec{d}_x \zeta , \eta ] + [[\zeta , \eta ], \gamma ] + [[\eta , \gamma ], \zeta ] + [[\gamma , \zeta ], \eta ] \big \rangle \nonumber \\&\quad + \left\langle - {\text {div}}_x \frac{\delta l}{\delta \gamma } + {\text {ad}}^*_\gamma \frac{\delta l}{\delta \gamma }, \eta \right\rangle \nonumber \\&= \left\langle - {\text {div}}_x \frac{\delta l}{\delta \gamma } + {\text {ad}}^*_\gamma \frac{\delta l}{\delta \gamma }, \eta \right\rangle , \end{aligned}$$

(B.23)

where ${\text {div}}_x \frac{\delta l}{\delta \gamma }$ is defined by

$$\begin{aligned} \left\langle {\text {div}}_x \frac{\delta l}{\delta \gamma }, \eta \right\rangle = {\text {div}} \left\langle \frac{\delta l}{\delta \gamma }, \eta \right\rangle - \left\langle \frac{\delta l}{\delta \gamma },\varvec{d}_x \eta \right\rangle . \end{aligned}$$

Combining (B.22) and (B.23) yields the vertical component,

$$\begin{aligned} {\text {ad}}^*_\zeta \frac{\delta l}{\delta \zeta } + {\text {ad}}^*_{\gamma } \frac{\delta l}{\delta \gamma } - \frac{d}{dt} \frac{\delta l}{\delta \zeta } - {\text {div}}_x \frac{\delta l}{\delta \gamma } = 0, \end{aligned}$$

of the Lagrange–Poincaré field-theoretic equations. After selecting the trivial connection, the horizontal component of the Lagrange–Poincaré field-theoretic equations becomes (B.19).

Remark. If the configuration bundle is a principal bundle and the structure group coincides with the symmetry group, the equivalence of the dynamical and covariant reductions have been shown in Castrillión López and Marsden (2008) in both the Lagrangian and Hamiltonian settings, and in Gay-Balmaz and Ratiu (2010) by using the gauge group as the configuration space.

Geometrically Exact Rods

The Configuration Space. We begin by selecting a reference configuration for the rod of length $\ell$ and a uniform cross section given by a compact subset $\varvec{A}$ of the Euclidean plane ${\mathbb {R}}^2$. To give a parametrization of the reference configuration hereafter denoted $\varvec{B}$, introduce an orthonormal frame $\varvec{e}_1$, $\varvec{e}_2$, $\varvec{e}_3$ so that $\varvec{e}_2$ is parallel to the initial rod axis and $\varvec{e}_1$ and $\varvec{e}_3$ span the cross section plane of the undeformed rod. Accordingly, the reference configuration $\varvec{B}$ is given by

$$\begin{aligned} \varvec{B} = \big \{x \in {\mathbb {R}}^3 \mid x = x_1 \varvec{e}_1 + s \varvec{e}_2 + x_3 \varvec{e}_3,\quad \text {where}\quad (s, (x_1, x_3)) \in [0, \ell ] \times \varvec{A} \big \}. \end{aligned}$$

The basic kinematic assumption are that the cross sections of the rod are connected by a curve through their centroids, that they undergo rigid motions and that they are not necessarily orthogonal to the centroid curve. A deformed configuration of the rod in the ambient Euclidean space ${\mathbb {R}}^3$ is then specified by giving a centroid curve $\varvec{\varphi } :[0,\ell ] \rightarrow {\mathbb {R}}^3$ and a ${\text {SO}}(3)$-valued curve

$$\begin{aligned} \Lambda :[0, \ell ] \rightarrow {\text {SO}}(3). \end{aligned}$$

The latter describes the orientation of the cross sections so that each vector $\varvec{t}_2 (s) := \Lambda (s) \varvec{e}_2$, $0\le s\le \ell$, is normal to the cross section through $\varvec{\varphi }(s)$ in the deformed configuration. Consequently, the set

$$\begin{aligned} {\mathcal {C}} = \big \{\varvec{\phi } = (\varvec{\varphi }, \Lambda ) \mid \varvec{\varphi }: [0,\ell ] \rightarrow {\mathbb {R}}^3,\ \Lambda : [0,\ell ] \rightarrow {\text {SO}}(3)\big \} \end{aligned}$$

is the configuration space for the rod. The position vector of a material point $x = (x_1, s, x_3)$ then reads

$$\begin{aligned} \varvec{\varphi }(s)+\Lambda \varvec{r}, \quad \text {where}\quad \varvec{r} = (x_1, 0, x_3). \end{aligned}$$

A motion is a curve of configurations. Let $\varvec{\phi }^t$ be the configuration of the rod at the moment of time t.

Consider a rigid motion of the ambient space given by the Euclidean transformation $\varvec{x} \rightarrow O \varvec{x} + \varvec{c}$, where O is a rotation matrix and $\varvec{c}$ is a constant vector. The left action of the special Euclidean group ${\text {SE}}(3)$ on ${\mathcal {C}}$ is defined by the formula $\varvec{\phi } \rightarrow (O \varvec{\varphi } + \varvec{c}, O \Lambda )$. Given a motion $\varvec{\phi }^t$, define $\varvec{\phi }^{t+} := (O(t) \varvec{\varphi }_t + {\mathbf {c}}(t), O(t) \Lambda (t))$ and use the superscript $+$ to denote the objects associated with $\varvec{\phi }^{t+}$.

The Strain Measure. The convected shear-axial strain and convected bending-torsional strain are defined as

$$\begin{aligned} {\varvec{\Gamma }}= \Lambda ^{-1} (\varvec{\varphi }_s - \varvec{t}_2) \end{aligned}$$

and the vector representation ${\varvec{\Upsilon }}$ of the skew-symmetric operator $\Lambda ^{-1} \Lambda _s$, respectively (see Auricchio et al. (2008) for their precise physical meaning). As before, the subscript s denotes partial derivation with respect to the variable s.

The deformation gradient$\varvec{F} = T\varvec{\phi }$ reads

$$\begin{aligned} \varvec{F} = \Lambda (I + (\Lambda ^{-1} (\varvec{\varphi }_s - \varvec{t}_2) + \Lambda ^{-1} \Lambda _s \varvec{r}) \otimes \varvec{e}_2) = \Lambda (I + \varvec{a}^r \otimes \varvec{e}_2), \end{aligned}$$

where

$$\begin{aligned} \varvec{a}^r = \varvec{\Gamma } + {\varvec{\Upsilon }} \times \varvec{r} \end{aligned}$$

(C.1)

measures the strain in the material frame. The strain vanishes for a rod that undergoes a rigid motion without deformation. Since the fields $\varvec{\Gamma }$, ${{\hat{\varvec{\Upsilon }}}}$, and $\varvec{a}^r$ are rigid-motion-invariant, i.e., since

$$\begin{aligned} {\varvec{\Gamma }}^+ = {\varvec{\Gamma }},\quad {\varvec{\Upsilon }}^+ = {\varvec{\Upsilon }},\quad \text {and}\quad {\varvec{a}^r}^+ = \varvec{a}^r, \end{aligned}$$

the Green–Lagrange strain tensor

$$\begin{aligned} \varvec{E} = \tfrac{1}{2} (\varvec{F}^T \varvec{F} - I) = \tfrac{1}{2} ((\varvec{a}^r \otimes \varvec{e}_2) + (\varvec{a}^r \otimes \varvec{e}_2)^T + (\varvec{a}^r \cdot \varvec{a}^r)\, \varvec{e}_2 \otimes \varvec{e}_2) \end{aligned}$$

(C.2)

is also invariant under the left action of the special Euclidean group, i.e., $\varvec{E}^+ = \varvec{E}$.

The Constitutive Equation. Under the finite deformation and small strains hypotheses, a plausible linear constitutive relation consistent with the frame indifference and locality for a purely elastic isotropic material reads

$$\begin{aligned} \varvec{S} = \lambda ({\text {tr}} \varvec{E})\, \varvec{I} + 2 \mu \varvec{E}, \end{aligned}$$

(C.3)

where $\varvec{S}$ is the second Piola–Kirchhoff stress tensor and $\lambda ,\mu$ are the Lamé constants (see, e.g., Marsden and Hughes (1983)). Focusing on the small strain case and using the formulae ${\varvec{a}^r}^+ = \varvec{a}^r$, (C.2), and (C.3), we obtain the invariant approximations

$$\begin{aligned} \widetilde{\varvec{E}} = \tfrac{1}{2} \big [(\varvec{a}^r \otimes \varvec{e}_2) + (\varvec{a}^r \otimes \varvec{e}_2)^T\big ],\quad \widetilde{{\mathbf {S}}} = \lambda ({\text {tr}} \widetilde{\varvec{E}}) \varvec{I} + 2\mu \widetilde{\varvec{E}} \end{aligned}$$

(C.4)

of the tensors $\varvec{E}$ and $\varvec{S}$.

The Invariant Stored Energy Function. Recall that, in terms of the second Piola–Kirchhoff stress tensor $\varvec{S}$, the stress power is given by the formula

$$\begin{aligned} \int _{{{\mathbf {A}}}\times [0,\ell ]} {\text {tr}} \big (\varvec{S} \varvec{E}_t^T \big )\,dA\,ds, \end{aligned}$$

(C.5)

where the subscript t denotes the time derivative. Substituting (C.1) and (C.4) in (C.5), we arrive at an invariant approximation

$$\begin{aligned} \int _{{\mathbf {A}} \times [0,\ell ]} D\, ({\varvec{\Gamma }} + {\varvec{\Upsilon }} \times \varvec{r}) \cdot ({\varvec{\Gamma }}_t + {\varvec{\Upsilon }}_t \times \varvec{r})\,dA\,ds = \int _0^\ell D_f\, {\varvec{\Gamma }} \cdot {\varvec{\Gamma }}_t + D_m\, {\varvec{\Upsilon }} \cdot {\varvec{\Upsilon }}_t \,ds\nonumber \\ \end{aligned}$$

(C.6)

of the mechanical power (see Auricchio et al. (2008) for more details), where $D={\text {diag}} \{\mu , \lambda + 2 \mu , \mu \}$,

$$\begin{aligned} D_f = D \int _{\mathbf {A}} dA,\quad D_m = \begin{bmatrix} (\lambda + 2 \mu ) J_{33}&{} 0 &{} -(\lambda + 2 \mu ) J_{13} \\ 0 &{} \mu ( J_{11} + J_{33} ) &{} 0 \\ - (\lambda + 2 \mu ) J_{13} &{} 0 &{} (\lambda + 2 \mu ) J_{11} \end{bmatrix},\qquad \end{aligned}$$

(C.7)

and where

$$\begin{aligned} J_{\alpha \beta } = \int _{{\mathbf {A}}} x_\alpha x_\beta \,dA. \end{aligned}$$

Without loss of generality, we assume that $J_{13}=0$ which holds for the rod with axisymmetric cross sections when appropriate coordinates are chosen. Using (C.6) and (C.7), see Auricchio et al. (2008) for details, we finally obtain the properly invariant stored energy function

$$\begin{aligned} \tfrac{1}{2} \left[ {\varvec{\Gamma }}^T, {\varvec{\Upsilon }}^T \right] {\text {diag}} \big \{GA_1, EA, GA_3, EI_3, GJ, EI_1\big \} \begin{bmatrix} {\varvec{\Gamma }} \\ {\varvec{\Upsilon }} \end{bmatrix}. \end{aligned}$$

Here EA is the axial stiffness, $GA_1$ and $GA_3$ are the shear stiffnesses along axes $\varvec{t}_1 = \Lambda \varvec{e}_1$ and $\varvec{t}_3 =\Lambda \varvec{e}_3$, GJ is the torsional stiffness and $EI_1$ and $EI_3$ are the principal bending stiffnesses relative to $\varvec{t}_1$ and $\varvec{t}_3$ (see, for example, Simo and Vu-Quoc (1986a, b)).