Geometric mechanics

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 $\backslash mathbf\{J\}:P\backslash to\backslash mathfrak\{g\}^*$, and the reduced spaces are quotients of the level sets of J by the subgroup of G preserving the level set in question: for $\backslash mu\backslash in\backslash mathfrak\{g\}^*$ one defines $P\_\backslash mu=\backslash mathbf\{J\}^\{-1\}(\backslash mu)/G\_\backslash mu$, and this reduced space is a symplectic manifold if $\backslash mu$ is a regular value of J.

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• Hamilton's principle
• Lagrange d'Alembert principle
• Maupertuis' principle of least action
• Euler–Poincaré
• Vakonomic

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.

The term "geometric mechanics" occasionally refers to 17th-century mechanics.Sébastien Maronne, Marco Panza. [https://hal.archives-ouvertes.fr/hal-00415933/document "Euler, Reader of Newton: Mechanics and Algebraic Analysis".] In: Raffaelle Pisano. Newton, History and Historical Epistemology of Science, 2014, pp. 12–21.

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).

• Computer graphics
• Control theory — see Bloch (2003)
• Liquid Crystals — see [https://link.springer.com/article/10.1007/s00205-013-0673-1 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 [https://link.springer.com/article/10.1007/s10440-019-00257-1 Foskett, Holm, Tronci (2019)]
• Superfluids
• Thermodynamics — [https://www.mdpi.com/1099-4300/21/1/8 see Gay-Balmaz, Yoshimra (2018)]
• Trajectory planning for space exploration
• Underwater vehicles
• Variational integrators; see [https://doi.org/10.1017/S096249290100006X Marsden and West (2001)]

{{Reflist}}

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• {{Citation | last=Souriau | first=Jean-Marie | author-link=Jean-Marie Souriau | title=Structure des Systemes Dynamiques | publisher=Dunod | year=1970}}

Category:Classical mechanics

Category:Hamiltonian mechanics

Category:Dynamical systems

Category:Symplectic geometry

Category:Lagrangian mechanics

Category:Variational principles