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Journal of Geometric Mechanics

2014 , Volume 6 , Issue 4

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The Hamilton-Jacobi equation, integrability, and nonholonomic systems
Larry M. Bates, Francesco Fassò and Nicola Sansonetto
2014, 6(4): 441-449 doi: 10.3934/jgm.2014.6.441 +[Abstract](358) +[PDF](348.9KB)
Abstract:
By examining the linkage between conservation laws and symmetry, we explain why it appears there should not be an analogue of a complete integral for the Hamilton-Jacobi equation for integrable nonholonomic systems.
Higher-order variational problems on lie groups and optimal control applications
Leonardo Colombo and David Martín de Diego
2014, 6(4): 451-478 doi: 10.3934/jgm.2014.6.451 +[Abstract](385) +[PDF](788.8KB)
Abstract:
In this paper, we describe a geometric setting for higher-order La- grangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.
Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems
Sebastián Ferrer and Francisco Crespo
2014, 6(4): 479-502 doi: 10.3934/jgm.2014.6.479 +[Abstract](352) +[PDF](718.7KB)
Abstract:
Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in $\mathbb{T}^*\mathbb{R}^4$ which is a homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. The key feature of the model is its nested Hamiltonian-Poisson structure, which appears as two extended Euler systems in the reduced equations. This is fully exploited in the process of integration, where we find two 1-DOF subsystems and a quadrature involving both of them. The solution is quasi-periodic, expressed by means of Jacobi elliptic functions and integrals, based on two periods. For a suitable choice of the parameters, some remarkable classical models such as the Kepler, geodesic flow, isotropic oscillator and free rigid body systems appear as particular cases.
Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
Janusz Grabowski, Katarzyna Grabowska and Paweł Urbański
2014, 6(4): 503-526 doi: 10.3934/jgm.2014.6.503 +[Abstract](317) +[PDF](560.5KB)
Abstract:
The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is $T M$, is based on the existence of canonical symplectic isomorphisms of double vector bundles $T^* TM$, $T^*T^* M$, and $TT^* M$, where the symplectic structure on $TT^* M$ is the tangent lift of the canonical symplectic structure $T^* M$. We show that there exists an analogous picture in the dynamics of objects for which the configuration space is $\wedge^n T M$, if we make use of certain structures of graded bundles of degree $n$, i.e. objects generalizing vector bundles (for which $n=1$). For instance, the role of $TT^*M$ is played in our approach by the manifold $\wedge^nT M\wedge^nT^*M$, which is canonically a graded bundle of degree $n$ over $\wedge^nT M$. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.
Nonlinear constraints in nonholonomic mechanics
Paul Popescu and Cristian Ida
2014, 6(4): 527-547 doi: 10.3934/jgm.2014.6.527 +[Abstract](363) +[PDF](430.3KB)
Abstract:
In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3,4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.
A dynamical condition for differentiability of Mather's average action
Alexandre Rocha and Mário Jorge Dias Carneiro
2014, 6(4): 549-566 doi: 10.3934/jgm.2014.6.549 +[Abstract](236) +[PDF](651.6KB)
Abstract:
We prove the differentiability of Mather's average action on all rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian asymptotically isolated graph, invariant by Tonelli Hamiltonians. We also show the relationship between differentiability of $\beta $ and local integrability of the Hamiltonian flow.

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