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Journal of Geometric Mechanics

2015 , Volume 7 , Issue 4

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Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory
L. Búa, T. Mestdag and M. Salgado
2015, 7(4): 395-429 doi: 10.3934/jgm.2015.7.395 +[Abstract](622) +[PDF](586.5KB)
Abstract:
We investigate the reduction process of a $k$-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincaré field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a $k$-connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' $k$-connection.
Geometric arbitrage theory and market dynamics
Simone Farinelli
2015, 7(4): 431-471 doi: 10.3934/jgm.2015.7.431 +[Abstract](672) +[PDF](672.4KB)
Abstract:
We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to:
    $\bullet$ Write arbitrage as curvature of a principal fibre bundle.
    $\bullet$ Parameterize arbitrage strategies by its holonomy.
    $\bullet$ Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.
    $\bullet$ Characterize Geometric Arbitrage Theory by five principles and show they are consistent with the classical theory of stochastic finance.
    $\bullet$ Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where:
       - Arbitrage is allowed but minimized.
       - Arbitrage is not allowed.
    $\bullet$ Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio to be satisfied.
A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena
E. Minguzzi
2015, 7(4): 473-482 doi: 10.3934/jgm.2015.7.473 +[Abstract](553) +[PDF](307.9KB)
Abstract:
A mechanical covariant equation is introduced which retains all the effectingness of the Lagrange equation while being able to describe, in a unified way, other phenomena including friction, non-holonomic constraints and energy radiation (Lorentz-Abraham-Dirac force equation). A quantization rule adapted to the dissipative degrees of freedom is proposed which does not pass through the variational formulation.
Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures
Giovanni Rastelli and Manuele Santoprete
2015, 7(4): 483-515 doi: 10.3934/jgm.2015.7.483 +[Abstract](578) +[PDF](478.1KB)
Abstract:
We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Using this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on $ \mathfrak{ so}^\ast (3) $ and $ \mathfrak{ so}^\ast (4)$) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.
Invariant metrics on Lie groups
Gerard Thompson
2015, 7(4): 517-526 doi: 10.3934/jgm.2015.7.517 +[Abstract](665) +[PDF](369.5KB)
Abstract:
Index formulas for the curvature tensors of an invariant metric on a Lie group are obtained. The results are applied to the problem of characterizing invariant metrics of zero and non-zero constant curvature. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics.

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