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Conceptual basis of classical mechanics
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
By enunciating Newton’s three laws of motion we addressed various aspects of classical mechanics that are inherent in these laws such as the concept of inertial frame, Galilean invariance and the homogeneous and isotropic character of the Newtonian Universe. We discussed the process of seeking the solutions of a physical system highlighting different coordinate frames that could be suitably employed to describe it. We looked, in particular, at the particular class of force called the conservative force which can be represented by a potential energy function. Use of the latter facilitates a great deal of simplification and its role was illustrated by turning to several problems that are of great importance in classical mechanics: the simple harmonic motion, the damped oscillator along with its various manifestations and the pendulum problem. Subsequently, the formulation of conservation principles was considered for the specific cases of linear momentum, angular momentum and energy. A general form of the time period formula was derived in terms of Jacobi elliptic function. We also introduced a treatment of perturbative analysis and discussed Lindstedt–Poincaré’s technique of avoiding the secular terms in approximating a periodic solution of a differential equation focusing on the specific case of the quartic oscillator. Finally, by the introduction of kinetic and potential energies, we showed how a Lagrangian function can be defined for a conservative system in a Cartesian frame. A detailed treatment of the Lagrangian dynamics will be taken up in Chapter 3.
Physics-Guided Deep Learning for Spatiotemporal Forecasting
Published in Anuj Karpatne, Ramakrishnan Kannan, Vipin Kumar, Knowledge-Guided Machine Learning, 2023
Rui Wang, Robin Walters, Rose Yu
Uniform motion is part of Galilean invariance and is relevant to all non-relativistic physics modeling. For a vector field X:ℝ2→ℝ2 and vector c∈ℝ2, uniform motion transformation is adding a constant vector field to the vector field X(v), Tcum(X)(v)=X(v)+c,c∈ℝ2. By the following corollary, enforcing uniform motion equivariance as above by requiring all layers of the CNN to be equivariant severely limits the model.
Experimental facilities
Published in Stefano Discetti, Andrea Ianiro, Experimental Aerodynamics, 2017
Wind tunnels are tools used in aerodynamic research to investigate the flow around solid objects. They are structures where a flow is produced, usually by means of a fan, under controlled conditions. The test model is placed in the tunnel test section. Their working principle relies upon the concept of Galilean invariance, according to which the laws of motion are the same in all inertial frames. This means that the same flow field is produced whether the model is in motion with respect to the fluid (as happens in actual flight) or the fluid is in motion with respect to the model (as occurs in wind tunnel tests).
On the equations of open channel flow
Published in Journal of Hydraulic Research, 2023
William Guerin Gray, Cass Timothy Miller
Newtonian continuum mechanical models should be invariant with respect to the inertial Euclidean reference frame used to define the system. This property is termed Galilean invariance. Systems of model equations that satisfy this property are referred to as objective (Hutter & Jöhnk, 2004). The principle of Galilean invariance implies certain constraints on model forms. For example, consider the microscale equation of total energy conservation (Bird et al., 2002; Bowen, 1989; Gray & Miller, 2014; Slattery, 1999) that may be written in the form where E is the internal energy, p is pressure, is the viscous stress tensor, is a heat flux vector, is the gravitational acceleration vector, and h is a heat source. Equation (8) can be written in material derivative form as where the material derivative is defined as Galilean invariance requires that a conservation equation remain valid for any constant change in the velocity of the system (Jou et al., 2001). Thus modifying all velocities in Eq. (9) by a constant velocity yields which may be written as The first line of Eq. (12) is identical to Eq. (9) – the original conservation of energy equation, which is equal to zero. Since can be any constant vector, it can be set to a non-zero vector that is orthogonal to the term in brackets in the second line, which implies that the term in brackets in the third line must be zero, which is the conservation of mass equation. This implies that the second line of Eq. (12) must also be zero for any choice of , which can only be so if the portion in brackets on the second line is the zero vector; this portion of the equation is the conservation of momentum equation. This calculation demonstrates the Galilean invariance of the conservation of energy equation and that the conservation of momentum and mass equations are consequences of this property. A similar calculation for Galilean invariance can be applied directly to the conservation of momentum equation to demonstrate that the conservation of mass equation is a consequence of Galilean invariance of the momentum equation. The Galilean invariance of the conservation of mass equation is self-evident.