Laplace’s Equation

A set of three physical laws forming the basis of classical mechanics. The first law (inertia) states an object remains at rest or in uniform motion unless acted upon by a net external force. The second law quantifies force as mass times acceleration, [latex]\vec{F} = m\vec{a}[/latex]. The third law states that for every action, there is an equal and opposite reaction.

A Newtonian fluid's shear stress is directly proportional to the rate of shear strain. This linear relationship is defined by Newton's law of viscosity: [latex]\tau = \mu \frac{du}{dy}[/latex], where [latex]\tau[/latex] is the shear stress, [latex]\mu[/latex] is the dynamic viscosity (a constant of proportionality), and [latex]\frac{du}{dy}[/latex] is the shear rate or velocity gradient.

Fluid mechanics is the branch of applied mechanics concerned with the statics (fluids at rest) and dynamics (fluids in motion) of liquids and gases. It applies fundamental principles of mass, momentum, and energy conservation to analyze and predict fluid behavior. Its applications are vast, ranging from aerodynamics and hydraulics to meteorology and oceanography.

A second-order linear elliptic partial differential equation that describes systems in a steady-state or equilibrium condition. It is written as [latex]nabla^2 u = 0[/latex] or [latex]Delta u = 0[/latex], where [latex]nabla^2[/latex] (or [latex]Delta[/latex]) is the Laplace operator. Solutions, called harmonic functions, are the smoothest possible functions and represent potentials in fields like electrostatics, gravitation, and fluid flow.

A reformulation of classical mechanics based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy ([latex]L = T - V[/latex]). The equations of motion are derived from the Euler-Lagrange equation, [latex]\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0[/latex], using generalized coordinates, which simplifies analysis of complex systems with constraints.

The Navier–Stokes equations are a set of non-linear partial differential equations describing the motion of viscous fluid substances. They are a statement of Newton's second law, balancing momentum changes with pressure gradients, viscous forces, and external forces. For an incompressible fluid, the equation is [latex]\rho (\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}[/latex].

A fundamental second-order linear parabolic partial differential equation describing heat distribution or other diffusion processes. Its canonical form is [latex]\frac{partial u}{partial t} = \alpha \nabla^2 u[/latex], where [latex]u(\vec{x},t)[/latex] is temperature, [latex]t[/latex] is time, and [latex]\alpha[/latex] is thermal diffusivity. Solutions model how an initial temperature distribution evolves, smoothing out irregularities over time and approaching a steady state.

Generalized Hooke's Law is the constitutive equation for linear elastic materials, stating that the stress tensor is linearly proportional to the strain tensor. The relationship is expressed as [latex]\sigma = C : \varepsilon[/latex], where [latex]\sigma[/latex] is the stress tensor, [latex]\varepsilon[/latex] is the strain tensor, and [latex]C[/latex] is the fourth-order stiffness tensor containing the material's elastic constants.

This law states that every particle attracts every other particle in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is [latex]F = G \frac{m_1 m_2}{r^2}[/latex], where [latex]G[/latex] is the gravitational constant. It unified terrestrial and celestial mechanics.

Bernoulli's principle states that for an inviscid flow, an increase in a fluid's speed occurs simultaneously with a decrease in pressure or a decrease in its potential energy. It is a statement of the conservation of energy for a moving fluid, commonly expressed as [latex]p + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}[/latex] along a streamline.

In continuum mechanics, the principle of mass conservation states that the mass of a closed system must remain constant over time. For a fluid, this is expressed by the continuity equation. In its Eulerian differential form, it is written as [latex]\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0[/latex], where [latex]\rho[/latex] is the density and [latex]\mathbf{u}[/latex] is the velocity field.

These are two ways to describe motion in continuum mechanics:

• the Lagrangian specification follows individual material particles, tracking their properties over time, like watching a specific car in traffic
• the Eulerian specification focuses on fixed points in space, observing the properties (velocity, density) of whatever particles pass through those points, like a traffic camera observing a fixed intersection.

Solid mechanics is a branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, or other external loads. It is fundamental to engineering for the design and analysis of structures. Key areas include elasticity (recoverable deformation), plasticity (permanent deformation), and fracture mechanics (crack initiation and propagation).

The Cauchy stress tensor, denoted [latex]\boldsymbol{\sigma}[/latex], is a second-order tensor that completely defines the state of stress at a point inside a material. It relates the traction vector (force per unit area) [latex]\mathbf{T}[/latex] on any surface passing through that point to the surface's normal vector [latex]\mathbf{n}[/latex] via the linear relationship [latex]\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}[/latex].

A reformulation of classical mechanics that uses generalized coordinates and their conjugate momenta. It is based on the Hamiltonian function, [latex]H(q, p, t)[/latex], representing the system's total energy. The dynamics are described by Hamilton's equations: [latex]\dot{q}_i = \frac{\partial H}{\partial p_i}[/latex] and [latex]\dot{p}_i = -\frac{\partial H}{\partial q_i}[/latex]. This framework is central to quantum mechanics and statistical mechanics.