In the last part, the main results of the Euler-Poincare and Lie-Poisson reduction techniques are presented. The application of the above results and ideas to rigid body (RB) dynamics on SO(3) is described in Chapter 5.
A note on the notations
Chapter 2
Preliminaries
Variational Lagrangian mechanics
Using the variational principle, the fact that the evolution is symplectic is a consequence of the equation d2 = 0, applied to the bounded action on the solution space of the variational principle. Using the variational principle, Noether's theorem results from the infinite invariance of the bounded action on the solution space of .
Euler-Poincare and Lie-Poisson reduction
Namely, for a Lie group G, the canonical Poisson bracket in T*G is related to the Lie-Poisson bracket in g*. This formula will be used in Chapter 3 to construct the discrete Lie-Poisson (DLP) algorithm.
Chapter 3
Discrete Reduction on Lie Groups
Veselov discretization of mechanics
The discrete Lagrangian formalism in Veselov [ 50 , 51 ] can be applied to the case of the configuration manifold that is a Lie group G . In the remainder of this section, we review the derivation of the basic differential-geometric objects of discrete mechanics directly from the variational point of view.
The discrete Euler-Poincare and Lie-Poisson algo- rithms
As we will prove in the next theorem, reconstruction of the DEP algorithm (3.10) on G reproduces the DEL algorithm on G x G. The DEP algorithm thus approximates the flow of the Euler-Poincare equations if properly interpreted by means of reconstruction.
Discrete Lie-Poisson algorithm
Discretization using natural charts
Using the standard formula for the derivative of the exponential (see, for example, Dragt and Finn [12] or Channel and Scovel [10]) given by. The following lemma establishes that the discrete Lagrangian IL inherits the G-invariance property from the original Lagrangian L, so the discrete counterpart of the Euler-Poincare reduction is well-defined. The discrete Lagrangian IL : G x G - t lR is right (left) invariant under the diagonal action of G on G x G, whenever L : TG - t lR is right (left) invariant.
Using the explicit form of the graph 'l/Jgg together with the correct invariance of the Lagrangian L, we obtain that from (3.20) and (3.22). Using the discretization defined by {3.20}, the reduced discrete Lagrangian £, determined by the projection map {3.9} and given by £(91g21) JL{91,92), can be expressed in terms of the continuously reduced Lagrangian l by. The function X in (3.26) arises from transferring the derivative of the log function, viewed as a map of the Lie group, to its algebra.
Chapter 4
Poisson Structure and Invariant Manifolds on Lie Groups
- Dynamics on groupoids and algebroids
- Generating Lie-Poisson dynamics
- Some advantages of structure-preserving integra- tors
- Chapter 5
- Natural charts discretization
- Moser-Veselov discretization
- Poisson structures of the rigid body
This relation allows us to put a Poisson structure on the Lie group G and establish a correspondence between the dynamically invariant polyhedra of the corresponding continuous and discrete systems. Let L be the right invariant Lagrangian on TG and let IL be the Lagrangian of the corresponding discrete system on V c G X G. The proof is based on the commutativity of the diagram (4.6) and the G invariance of unreduced symplectic forms.
Using the left invariance of the metric, we can express the Lagrangian of a discrete rigid body as Recall that the reduced form of the Moser-Vesel Lagrangian on the group 80(3) is given by Its Symplectic sheets consist of invariant manifolds of reduced discrete dynamics corresponding to the Lagrangian (5.1).
Part II
Multisymplectic Geometry of Continuum Mechanics
Chapter 6
Compressible Continuum Mechanics
Configuration and Phase Spaces
The first sheaf of jets J1 Y is an affine sheaf over Y whose fiber over y E Yx consists of those linear mappings, : TxX -+ TyY, which satisfy T1fxy 0, = I dTxx. This section is labeled j1¢ and is called the first extension of the jet ¢. Note that we have introduced two different Riemannian structures into the configuration bundle. The internal metric on the spatial part B of the base manifold X is denoted by G, and the fiber or field metric on M by g.
There are two main cases we consider:. i) fluid dynamics on a fixed background with fixed boundaries, when Band M is the same and the fiber metric 9 coincides with the base metric G; a special case of this is fluid dynamics on a region in Euclidean space;. ii) elasticity on a fixed background, when the metric spaces (B,G) and (M,g) differ substantially. A very important remark here is that although metric 9 and G coincide in fluid dynamics, that is, on every fiber Yx , 9 is a copy of G, there is no cancellation because the metric tensors are evaluated at different points. Thus, only for fluid dynamics in Euclidean spaces, one can trivially raise and lower indices and drop all metric determinants and derivatives into the expressions below.
Ideal Fluid
Lagrangian Dynamics
Naturally, it must contain terms corresponding to the kinetic energy and the potential energy of the medium. The second term reflects the potential energy and depends on the spatial derivatives of the fields (restricted to the extensions of the first plane), i.e., on the deformation gradient F. These are generalizations of the notation used in the rest of part II of the thesis.
We summarize the results in the following theorem of [34] which illustrates the application of the variational principle to multisymplectic field theory. Since, in that case, entropy is advected, this dependence is subsumed in the material representation by the dependence of the stored energy function on the strain gradient. This relationship follows immediately from the form of the stored energy function; it recovers the Piola transformation law, which in conventional elasticity relates the first Piola-Kirchhoff stress tensor and the Cauchy stress tensor.
Chapter 7
Constrained Multisymplectic Field Theories
Lagrange Multipliers
The Lagrange multiplier theorem obviously makes use of the dual of the space of constraints. In the case of the incompressibility constraint, the vector space is one-dimensional and the constraint bundle is the real space of real-valued functions on the base space X. A dual of the constraint space is then defined with respect to an inner product structure on the vector bundle.
This is made explicit in the following statement of the Lagrange multiplication theorem, in which we assume that fields and Lagrange multipliers are sufficiently regular (see [31]). If [ is a trivial bundle over M, then in coordinates of the trivialization we have 0= (cp,.\), where.\: M -+ [1M is a Lagrange multiplication function. In the next section we will use this theorem to relate the constrained Hamilton principle to the extremum of the extended action integral containing the constraint combined with a Lagrange multiplier.
Multisymplectic Field Theories
In this picture, the Lagrange multiplier corresponds to a new field, which expands the dimension of the fiber space, and the augmented Lagrangian contains an additional part corresponding to the pairing of this field with the constraint. The Euler-Lagrange equations of motion then follow in a standard way from unconstrained Hamilton's principle. By construction, W is a map from the space M of parts of Y to the space COO(V) of parts of V, therefore it can be considered a smooth section W : M -+ e of the constraint bundle e.
The Lagrange multiplier theorem given in the previous subsection can be applied to conclude that this is equivalent to the existence of ¢ E e with 7rM,d¢) = ¢ which is an extremum of S. The stationarity of S with respect to variations in ¢ can be used to derive the constrained Euler-Lagrange equations, which have the form The Euler-Lagrange equations of motion are derived from Hamilton's unbounded principle in a standard way and coincide with (7.2).
Chapter 8
Incompressible Continuum Mechanics
- Configuration and Phase Spaces
- Lagrangian Dynamics
- Incompressible Ideal Hydrodynamics
- Incompressible Elasticity
- Chapter 9
The primary manifold of the constraint Q: is a sub-bundle of the dual jet set and corresponds to the incompressibility constraint. As we have already mentioned, the main distinguishing feature of incompressible continuum mechanics models is the presence of constraint (8.1). For the above Lagrangian choice, the Legendre transform thought of as a map of the fiber storage bundle lFLcp: Jl E -+ Jl E*.
For fluid dynamics, the stored energy term in the Lagrangian is a constant function precisely because of the incommensurability constraint. Recall the definition of the pressure function for barotropic fluids given by (6.16) as a partial derivative of the stored energy function W with respect to the Jacobian J . Compare this with the definition (8.2) of pressure as a Lagrange multiplier corresponding to the incompressibility constraint (8.1) (modulate a term Jdet[G]).
Symmetries, Momentum Maps and Noether Theorem
Relabeling Symmetry of Ideal Homogeneous Hy- drodynamics
Their corresponding Lagrangians differ only by the constraint term and both are equivariant with respect to the action of the group of volume-preserving diffeomorphisms. The action of the diffeomorphism group 'DtL(B) on the (spatial part of the) base manifold B C X captures exactly the meaning of particle relabeling. Both of these terms are functions of the Jacobian, which is equivariant with respect to the operation of volume-preserving diffeomorphisms given by (9.3).
Using (9.7), we can calculate the Noether current corresponding to the relabeling symmetry of the Lagrangian (6.3) to be. The differential of this quantity restricted to the solutions of the Euler-Lagrange equation is identically zero according to Theorem 9.1.1. The conditions in theorem 9.1.1 are fulfilled; therefore, the external difference for this Noether current dc (j1~)) is equal to zero for all sections ¢ which are solutions of the Euler-Lagrange equations.
Time Translation Invariance
We emphasize that the above equation is not equivalent to the Euler-Lagrange equations, i.e. the constraint cannot be recovered from the Noether flow. Indeed, the symmetry group is the same for both homogeneous compressible barotropic fluids and homogeneous incompressible fluids. Note also that the Noether fluxes (9.8) and (9.9) are different due to the difference in the corresponding Lagrangians.
Chapter 10
Conclusions and Future Directions
An explicit form of the Euler-Lagrange equations and conservation laws for rod and shell models is not included here, but can be easily derived by following the steps outlined above. A very important aspect of any multisymplectic field theory is the existence of the multisymplectic form formula (6.11) which is the covariant analogue of the fact that the flow of conservative systems consists of symplectic maps. However, preliminary results indicate that applications of the multisymplectic shape formula can not only be linked to some well-known principles in elasticity (such as the Betti reciprocity principle), but can also produce some new interesting relations that depend on the space-time direction of the first variations V, W in (6.11).
An accurate and consistent discretization of the model then results in so-called multisymplectic integrators, which preserve the discrete analogues of the multisymplectic form and the conservation laws. A structure that preserves discretization is one of the key aspects of the multisymplectic project and is currently under investigation. An implication of this statement for incompressible fluid dynamics is a discrete version of the vorticity conservation.
Bibliography
Marsden, Momentum Maps and the Hamiltonian Structure of Classical Relativistic Field Theories, I and II, (1997), preprints. West, Variational integrators and Newmark's algorithm for conservative and dissipative mechanical systems (submitted for publication). Lewis, The geometry of the Gibbs-Appell equations and Gauss's principle of least constraint, Mathematical Physics Reports.
Murray, Variational principles for constrained systems: theory and experiment, The International Journal of Nonlinear Mechanics.
