Geometric mechanics

Last updated January 12, 2025

Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics and control theory.

Geometric mechanics applies principally to systems for which the configuration space is a Lie group, or a group of diffeomorphisms, or more generally where some aspect of the configuration space has this group structure. For example, the configuration space of a rigid body such as a satellite is the group of Euclidean motions (translations and rotations in space), while the configuration space for a liquid crystal is the group of diffeomorphisms coupled with an internal state (gauge symmetry or order parameter).

Momentum map and reduction

One of the principal ideas of geometric mechanics is reduction, which goes back to Jacobi's elimination of the node in the 3-body problem, but in its modern form is due to K. Meyer (1973) and independently J.E. Marsden and A. Weinstein (1974), both inspired by the work of Smale (1970). Symmetry of a Hamiltonian or Lagrangian system gives rise to conserved quantities, by Noether's theorem, and these conserved quantities are the components of the momentum map J. If P is the phase space and G the symmetry group, the momentum map is a map $\mathbf {J} :P\to {\mathfrak {g}}^{*}$ , and the reduced spaces are quotients of the level sets of J by the subgroup of G preserving the level set in question: for $\mu \in {\mathfrak {g}}^{*}$ one defines $P_{\mu }=\mathbf {J} ^{-1}(\mu )/G_{\mu }$ , and this reduced space is a symplectic manifold if $\mu$ is a regular value of J.

Variational principles

Geometric integrators

One of the important developments arising from the geometric approach to mechanics is the incorporation of the geometry into numerical methods. In particular symplectic and variational integrators are proving particularly accurate for long-term integration of Hamiltonian and Lagrangian systems.

History

The term "geometric mechanics" occasionally refers to 17th-century mechanics.^[1]

As a modern subject, geometric mechanics has its roots in four works written in the 1960s. These were by Vladimir Arnold (1966), Stephen Smale (1970) and Jean-Marie Souriau (1970), and the first edition of Abraham and Marsden's Foundation of Mechanics (1967). Arnold's fundamental work showed that Euler's equations for the free rigid body are the equations for geodesic flow on the rotation group SO(3) and carried this geometric insight over to the dynamics of ideal fluids, where the rotation group is replaced by the group of volume-preserving diffeomorphisms. Smale's paper on Topology and Mechanics investigates the conserved quantities arising from Noether's theorem when a Lie group of symmetries acts on a mechanical system, and defines what is now called the momentum map (which Smale calls angular momentum), and he raises questions about the topology of the energy-momentum level surfaces and the effect on the dynamics. In his book, Souriau also considers the conserved quantities arising from the action of a group of symmetries, but he concentrates more on the geometric structures involved (for example the equivariance properties of this momentum for a wide class of symmetries), and less on questions of dynamics.

These ideas, and particularly those of Smale were central in the second edition of Foundations of Mechanics (Abraham and Marsden, 1978).

Applications

Computer graphics
Control theory — see Bloch (2003)
Liquid Crystals — see Gay-Balmaz, Ratiu, Tronci (2013)
Magnetohydrodynamics
Molecular oscillations
Nonholonomic constraints — see Bloch (2003)
Nonlinear stability
Plasmas — see Holm, Marsden, Weinstein (1985)
Quantum mechanics
Quantum chemistry — see Foskett, Holm, Tronci (2019)
Superfluids
Thermodynamics — see Gay-Balmaz, Yoshimra (2018)
Trajectory planning for space exploration
Underwater vehicles
Variational integrators; see Marsden and West (2001)

Related Research Articles

<span class="mw-page-title-main">Hamiltonian mechanics</span> Formulation of classical mechanics using momenta

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities $used in Lagrangian mechanics with (generalized) momenta . Both theories provide interpretations of classical mechanics and describe the same physical phenomena.$

In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by S. Pancharatnam (1956), in classical optics and by H. C. Longuet-Higgins (1958) in molecular physics; it was generalized by Michael Berry in (1984). It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase. It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C₆H₃F₃⁺ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be nonzero holonomy.

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries.

In mathematics, the coadjoint representation $of a Lie group is the dual of the adjoint representation. If denotes the Lie algebra of, the corresponding action of on, the dual space to, is called the coadjoint action . A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on .$

In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic blow up. The cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds.

In classical mechanics, the Hannay angle is a mechanics analogue of the geometric phase. It was named after John Hannay of the University of Bristol, UK. Hannay first described the angle in 1985, extending the ideas of the recently formalized Berry phase to classical mechanics.

Jerrold Eldon Marsden was a Canadian mathematician. He was the Carl F. Braun Professor of Engineering and Control & Dynamical Systems at the California Institute of Technology. Marsden is listed as an ISI highly cited researcher.

In theoretical physics, the BRST formalism, or BRST quantization denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.

<span class="mw-page-title-main">Alan Weinstein</span> American mathematician (born 1943)

Alan David Weinstein is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry.

In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.

In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theory in the way that a field is represented as a system that varies both in space and in time. This generalization is different from the canonical Hamiltonian formalism in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.

In mathematics a Dirac structure is a geometric structure generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of the Dirac bracket constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein.

Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physics and chemistry, for example it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions, without doing the exact rigorous calculations. To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Among all the molecular symmetries, diatomic molecules show some distinct features and they are relatively easier to analyze.

Tudor Stefan Rațiu is a Romanian-American mathematician who has made contributions to geometric mechanics and dynamical systems theory.

David Gregory Ebin is an American mathematician, specializing in differential geometry.

References

↑ Sébastien Maronne, Marco Panza. "Euler, Reader of Newton: Mechanics and Algebraic Analysis". In: Raffaelle Pisano. Newton, History and Historical Epistemology of Science, 2014, pp. 12–21.

Abraham, Ralph; Marsden, Jerrold E. (1978), Foundations of Mechanics (2nd ed.), Addison-Wesley
Arnold, Vladimir (1966), "Sur la géométrie différentielle des groupes de Lie de dimension infine et ses applications a l'hydrodynamique des fluides parfaits" (PDF), Annales de l'Institut Fourier, 16: 319–361, doi: 10.5802/aif.233
Arnold, Vladimir (1978), Mathematical Methods for Classical Mechanics, Springer-Verlag
Bloch, Anthony (2003). Nonholonomic Mechanics and Control. Springer-Verlag.
Foskett, Michael S.; Holm, Darryl D.; Tronci, Cesare (2019). "Geometry of Nonadiabatic Quantum Hydrodynamics". Acta Applicandae Mathematicae. 162 (1): 63–103. arXiv: 1807.01031 . doi:10.1007/s10440-019-00257-1. S2CID 85531406.
Gay-Balmaz, Francois; Ratiu, Tudor; Tronci, Cesare (2013). "Equivalent Theories of Liquid Crystal Dynamics". Arch. Ration. Mech. Anal. 210 (3): 773–811. arXiv: 1102.2918 . Bibcode:2013ArRMA.210..773G. doi:10.1007/s00205-013-0673-1. S2CID 14968950.
Holm, Darryl D.; Marsden, Jerrold E.; Ratiu, Tudor S.; Weinstein, Alan (1985). "Nonlinear stability of fluid and plasma equilibria". Physics Reports. 123 (1–2): 1–116. Bibcode:1985PhR...123....1H. doi:10.1016/0370-1573(85)90028-6.
Libermann, Paulette; Marle, Charles-Michel (1987). Symplectic geometry and analytical mechanics . Mathematics and its Applications. Vol. 35. Dordrecht: D. Reidel. doi:10.1007/978-94-009-3807-6. ISBN 90-277-2438-5.
Marsden, Jerrold; Weinstein, Alan (1974), "Reduction of Symplectic Manifolds with Symmetry", Reports on Mathematical Physics, 5 (1): 121–130, Bibcode:1974RpMP....5..121M, doi:10.1016/0034-4877(74)90021-4
Marsden, Jerrold; Ratiu, Tudor S. (1999). Introduction to mechanics and symmetry. Texts in Applied Mathematics (2 ed.). New York: Springer-Verlag. ISBN 0-387-98643-X.
Meyer, Kenneth (1973). "Symmetries and integrals in mechanics". Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971). New York: Academic Press. pp. 259–272.
Ortega, Juan-Pablo; Ratiu, Tudor S. (2004). Momentum maps and Hamiltonian reduction. Progress in Mathematics. Vol. 222. Birkhauser Boston. ISBN 0-8176-4307-9.
Smale, Stephen (1970), "Topology and Mechanics I", Inventiones Mathematicae, 10 (4): 305–331, Bibcode:1970InMat..10..305S, doi:10.1007/bf01418778, S2CID 189831616
Souriau, Jean-Marie (1970), Structure des Systemes Dynamiques, Dunod

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[1] Sébastien Maronne, Marco Panza. "Euler, Reader of Newton: Mechanics and Algebraic Analysis". In: Raffaelle Pisano. Newton, History and Historical Epistemology of Science, 2014, pp. 12–21.

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