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Newton's laws of motion can be recast into more powerful and general mathematical formalisms. These enable Newton's laws to be applied in the most appropriate coordinate system for the problem at hand. The Euler–Lagrange equations are introduced and used to show how the invariance of the equations under translations and rotations results in the laws of conservation of momentum and angular momentum. Time invariance results in the law of conservation of energy. An important application of the Euler–Lagrange equations is the characterisation the normal modes of oscillation of mechanical systems. Hamilton’s equations, Poisson brackets, the Hamilton–Jacobi equations and action-angle variables are introduced in order to solve increasingly complex problems, including the mathematics necessary for the description of quantum phenomena.
We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.
This chapter extends the discussion of waves beyond the longitudinal oscillations with which we began. Here, we look at the wave equation as it arises in electricity and magnetism, in Euler's equation and its shallow water approximation, in ``realistic" (extensible) strings, and in the quantum mechanical setting, culminating in a quantum mechanical treatment of the book's defining problem, the harmonic oscillator.
This paper focus on the mechanical and martensitic transformation behaviors of axially functionally graded shape memory alloy (AFG SMA) beams. It is taken into consideration that material properties, such as austenitic elastic modulus, martensitic elastic modulus, critical transformation stresses and maximum transformation strain vary continuously along the longitudinal direction. According to the simplified linear SMA constitutive equations and Bernoulli-Euler beam theory, the formulations of stress, strain, martensitic volume fraction and governing equations of the deflection, height and length of transformed layers are derived. Employing the Galerkin’s weighted residual method, the governing differential equation of the deflection is solved. As an example, the bending behaviors of an AFG SMA cantilever beam subjected to an end concentrated load are numerically analyzed using the developed model. Results show that the mechanical and martensitic transformation behaviors of the AFG SMA beam are complex after the martensitic transformation of SMA occurs. The influences of FG parameter on the mechanical behaviors and geometrical shape of transformed regions are obvious, and should be considered in the design and analysis of AFG SMA beams in the related regions.
The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.
This study proposed the application of a novel immersed boundary method (IBM) for the treatment of irregular geometries using Cartesian computational grids for high speed compressible gas flows modelled using the unsteady Euler equations. Furthermore, the method is accelerated through the use of multiple Graphics Processing Units – specifically using Nvidia’s CUDA together with MPI - due to the computationally intensive nature associated with the numerical solution to multi-dimensional continuity equations. Due to the high degree of locality required for efficient multiple GPU computation, the Split Harten-Lax-van-Leer (SHLL) scheme is employed for vector splitting of fluxes across cell interfaces. NVIDIA visual profiler shows that our proposed method having a computational speed of 98.6 GFLOPS and 61% efficiency based on the Roofline analysis that provides the theoretical computing speed of reaching 160 GLOPS with an average 2.225 operations/byte. To demonstrate the validity of the method, results from several benchmark problems covering both subsonic and supersonic flow regimes are presented. Performance testing using 96 GPU devices demonstrates a speed up of 89 times that of a single GPU (i.e. 92% efficiency) for a benchmark problem employing 48 million cells. Discussions regarding communication overhead and parallel efficiency for varying problem sizes are also presented.
We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler–Korteweg system in dimension one. Such solutions behave asymptotically in time like several travelling waves far away from each other. A kink is a travelling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.
Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form.
This paper integrates seemingly disjoint studies on consumer behavior in micro and macroanalyses via an intertemporal two-stage budgeting procedure with durable goods and liquidity constraints. The model specifies an indirect utility function as a function of nondurable consumption, commodity (nondurables) prices, and durables stock, and derives the demand functions for nondurable goods. A demand function for durable goods is derived in an adjustment cost framework. The consumption growth equation accounts for relative price effects with precautionary saving, durables stock, and liquidity constraints. The stochastic discount factor is approximated by a time-varying linear function of nondurable consumption growth, commodity price growth, durables stock growth, and disposable income growth. The demand functions for six nondurable goods and services are jointly estimated with the Euler equations for bonds, stocks, and durable goods with allowance for liquidity constraints, using US data. Estimation provides new findings for intertemporal consumption and a multifactor consumption-based capital asset pricing model.
Let
$p$
be an odd prime number and
$E$
an elliptic curve defined over a number field
$F$
with good reduction at every prime of
$F$
above
$p$
. We compute the Euler characteristics of the signed Selmer groups of
$E$
over the cyclotomic
$\mathbb{Z}_{p}$
-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above
$p$
and mixed signs in the definition of the signed Selmer groups.
This paper addresses three control implementation issues for trajectory tracking of robotic manipulators: unmodeled dynamics, unknown input saturation and peaking effects during the transient phase. A model-free first-order robust-adaptive control method is used to deal with the unmodeled dynamics. Robust optimality and stability of the controller are proved using the 𝓗∞ technique and the game-algebraic Riccati equation. An intuitive approach is devised to incorporate the unknown input saturation by modifying the speed of the desired trajectory. The trajectory scaling is performed by using only the state errors. Furthermore, two different techniques are utilized to suppress peaking during the transient response of the trajectory tracking. The first method adds an extra term in the input while the second method uses variable gain to improve the transient response. A systematic procedure for finding the controller parameters is formulated using features, such as rise time and settling time. A three-degree-of-freedom robot manipulator is used for the validation of the proposed controller in simulations and experiments.
Let X ⊂ ℂn be an equidimensional complex algebraic set and let f: X → ℂ be a polynomial function. For each c ∈ ℂ, we define the global Brasselet number of f at c, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of f at c. Then we establish several formulas relating these numbers to the topology of X and the critical points of f.
The second chapter gives a derivation of out-of-equilibrium fluid dynamics from first principles, based on the framework of effective field theory. Main results such as the Euler equations, Navier-Stokes equations, BRSSS equations are recovered and the divergence of the hydrodynamic gradient expansion is discussed. Novel features such as the Borel summability of the divergent series, hydrodynamic attractors, the role of nonhydrodynamic modes and far-from-equilibrium hydrodynamics are described. The chapter closes by a modern derivation of fluid dynamics in the presence of thermal fluctuations.
We define the notion of field, based on the example of electromagnetism. We write the relativistically covariant form of the Maxwell's equations in terms of a gauge field and field strength for it. We define the Euler–Lagrange equations for a field, and based on it, we derive the relativistic Maxwell's equations from a relativistically invariant Maxwell action.
Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of type Cn and Dn, completing the computation for all classical types.
where
$[x]$
denotes the integral part of real
$x$
. The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].
As internal energy is a function of entropy, volume and number of moles, its differential is given by the Gibbs relation, and temperature, pressure and chemical potentials are defined as conjugate variables. Extensivity implies the Euler relation. The Gibbs-Duhem relation will find applications later, in the analysis of phase transitions. Legendre transformations are introduced, leading to the definition of the thermodynamic potentials: free energy, enthalpy and Gibbs free energy. When a system is coupled to a thermal reservoir or heat bath, its equlibrium is characterised by a minimum of the free energy; when it is a work reservoir, the enthalpy is minimum, and when it is a work and heat reservoir, the Gibbs free energy is minimum. Maxwell relations establish relationships between quantities that would not immediately be associated. The cyclic chain rule links together the derivatives of one property function with respect to two others. It is conveniently applied to analyse the Joule expansion and Joule-Thomson effect.
This contribution covers the topic of my talk at the 2016-17 Warwick-EPSRC Symposium: 'PDEs and their applications'. As such it contains some already classical material and some new observations. The main purpose is to compare several avatars of the Kato criterion for the convergence of a Navier-Stokes solution, to a regular solution of the Euler equations, with numerical or physical issues like the presence (or absence) of anomalous energy dissipation, the Kolmogorov 1/3 law or the Onsager C^{0,1/3} conjecture. Comparison with results obtained after September 2016 and an extended list of references have also been added.