Differential equations to describe elasticity are derived without the use of stress or strain. The points within the body are the independent parameters instead of strain and surface forces replace stress tensors. These differential equations are a continuous analytical model that can then be solved using any of the standard techniques of differential equations. Although the equations do not require the definition stress or strain, these quantities can be calculated as dependent parameters. This approach to elasticity is simple, which avoids the need for multiple definitions of stress and strain, and provides a simple experimental procedure to find scalar representations of material properties in terms of the energy of deformation. The derived differential equations describe both infinitesimal and finite deformations.

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Cerebellum is part of the brain that occupies only 10% of the brain volume, but it contains about 80% of total number of brain neurons. New cerebellar function model is developed that sets cerebellar circuits in context of multibody dynamics model computations, as important step in controlling balance and movement coordination, functions performed by two oldest parts of the cerebellum. Model gives new functional interpretation for granule cells-Golgi cell circuit, including distinct function for upper and lower Golgi cell dendritc trees, and resolves issue of sharing Granule cells between Purkinje cells. Sets new function for basket cells, and for stellate cells according to position in molecular layer. New model enables easily and direct integration of sensory information from vestibular system and cutaneous mechanoreceptors, for balance, movement and interaction with environments. Model gives explanation of Purkinje cells convergence on deep-cerebellar nuclei.

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The main purpose of this paper is to derive the fractional Euler-Lagrange equations which depend on the Riemann-Liouville derivatives of order for Lie algebroids. The fractional Hamiltonian formalism was also discussed. Two examples in particular the fractional geodesics for Lie algebroids and the Wong's fractional equations which arise in the dynamics of a colored particle in Yang-Mills field and on the falling cat theorem were also derived. Key Words: Fractional Action-like Variational Approach; Fractional Lagrangian and Hamiltonian Formalisms; Lie Algebroids

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We study the Euler-Lagrange cohomology and explore the symplectic or multisymplectic geometry and their preserving properties in classical mechanism and classical field theory in Lagrangian and Hamiltonian formalism in each case respectively. By virtue of the Euler-Lagrange cohomology that is nontrivial in the configuration space, we show that the symplectic or multisymplectic geometry and related preserving property can be established not only in the solution space but also in the function space if and only if the relevant closed Euler-Lagrange cohomological condition is satisfied in each case. We also apply the cohomological approach directly to Hamiltonian-like ODEs and Hamiltonian-like PDEs no matter whether there exist known Lagrangian and/or Hamiltonian associated with them.

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The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.

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We derive a new symmetric hyperbolic formulation of the Einstein-Euler equations in Lagrange coordinates that are adapted to the Frauendiener-Walton formulation of the Euler equations. As an application, we use this system to show that the densitized lapse and zero shift coordinate systems for the vacuum Einstein equations are equivalent to Lagrange coordinates for a fictitious fluid with a specific equation of state.

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We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings.

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We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings.

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It is commonly believed as a fundamental principle that energy-momentum conservation of a physical system is the result of space-time symmetry. However, for classical particle-field systems, e.g., charged particles interacting through self-consistent electromagnetic or electrostatic fields, such a connection has only been cautiously suggested. It has not been formally established. The difficulty is due to the fact that the dynamics of particles and the electromagnetic fields reside on different manifolds. We show how to overcome this difficulty and establish the connection by generalizing the Euler-Lagrange equation, the central component of a field theory, to a so-called weak form. The weak Euler-Lagrange equation induces a new type of flux, called the weak Euler-Lagrange current, which enters conservation laws. Using field theory together with the weak Euler-Lagrange equation developed here, energy-momentum conservation laws that are difficult to find otherwise can be systematically derived from the underlying space-time symmetry.