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An Introductory Review of Classical Mechanics
Published in Ramaswamy Jagannathan, Sameen Ahmed Khan, Quantum Mechanics of Charged Particle Beam Optics, 2019
Ramaswamy Jagannathan, Sameen Ahmed Khan
to first order in δx¯(t), any arbitrary small deviation in the path between the fixed initial and final points, x¯(ti) and χ x¯(tf), respectively. Usually, along the actual path, the action takes the least value, and hence Hamilton’s principle is often called the principle of least action. Nature seems to have chosen such variational principles in formulating its basic laws.
Variational Calculus
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Hamilton’s principle is also sometimes referred to as the principle of least action. However, this is generally a misnomer as the stationary function may also be a maximum or a saddle point. To separate the two approaches to mechanics, although they are often equivalent, we will refer to them as Newtonian and Lagrangian mechanics, respectively. It is also of interest to note that there is a third formulation of classical mechanics, known as Hamiltonianmechanics. Although Hamilton’s principle appears in Lagrangian mechanics, the two should generally not be confused with each other.
Structural Mechanics Fundamentals
Published in Colin H. Hansen, Foundations of Vibroacoustics, 2018
Invoking the requirement that the variation, δθ, vanish at the two instants, t1 and t2, the second term in Equation (c) reduces to zero. Moreover, δθ is arbitrary in the time interval between t1 and t2, so that the only way for the integral in the first term to be zero is for the coefficient of δθ to vanish for any time, t. Hence, the following must be set: () ddt(θ˙cos2θ)+θ˙2sin θcos θ+km(1−cos θ)sin θ−gLcos θ=0which is the desired equation of motion. Letting θ˙ = θ¨ = 0 the same equation for the system equilibrium position as the one derived in Example 2.1 is obtained. In general, it is not necessary to use Hamilton’s principle directly for the solution of dynamic problems. Instead, Hamilton’s principle is used to derive Lagrange’s equations of motion, which can then be used to solve dynamic problems.
Analytical solution of cross- and angle-ply nano plates with strain gradient theory for linear vibrations and buckling
Published in Mechanics of Advanced Materials and Structures, 2021
F. Cornacchia, F. Fabbrocino, N. Fantuzzi, R. Luciano, R. Penna
The dynamic version of the principle of the virtual works (Hamilton’s Principle) is employed in order to carry out the equations of motion. It is important to point out that the transverse shear stress, needed for the equilibrium of the plate, has been involved in the boundary conditions and equilibrium of forces. with, δU is the virtual strain energy, δV is the virtual work done by the applied forces, and δK is the virtual kinetic energy
Static and thermal instability analysis of embedded functionally graded carbon nanotube-reinforced composite plates based on HSDT via GDQM and validated modeling by neural network
Published in Mechanics Based Design of Structures and Machines, 2022
Ali Forooghi, Nasim Fallahi, Akbar Alibeigloo, Hosein Forooghi, Saber Rezaey
Hamilton’s principle is a generalization of the principle of virtual displacements to the dynamics of systems. This principle may be considered a dynamic version of the principle of virtual displacements (Reddy 2003). Therefore, the governing equations of motion of the system can be derived employing generalized Hamilton’s principle (Eq. (28)).
On the energy absorption of the reinforced sandwich curved beam equipped with piezoelectric layers through ANN and MCS analyses
Published in Mechanics of Advanced Materials and Structures, 2023
Minimizing the total energy of the structure is called Hamilton’s principle which presented the equations of motion of a structure. In the following equation, neglecting thermal effects, variations of all the energies in the structure are presented.