Classical Mechanics: Fundamental Concepts and Principles

Vocabulary practice flashcards defining core concepts, equations, and principles in Classical Mechanics.

Principle of relativity

The fundamental principle that observers moving at a constant velocity relative to each other obey the same laws of mechanics. There’s no way to tell which object is truly at rest if both move at a constant relative velocity.

Inertial frame

A non-accelerating, non-rotating reference frame where Newton’s first law holds and objects in motion remain at rest or in uniform motion unless acted upon by an external force.

Newton's first law

A body acted on by no forces moves with uniform velocity in a straight line, which physically establishes the existence of inertial frames of reference.

Newton's second law

The equation of motion specifying that the time rate of change of momentum of a body is equal to the total force acting on it, $\frac{d\bf{p}}{dt} = m\bf{a} = \bf{F}$.

Newton's third law

The dynamical law asserting that mutual forces of action and reaction between two bodies are equal in magnitude and opposite in direction, satisfying $\bf{F}_{ji} = -\bf{F}_{ij}$.

Equivalence principle

The physical law derived from experimental observation stating that inertial mass in Newton's second law is proportional to gravitational mass.

Conservative force

Conservative Force: A force field depending only on position whose curl vanishes ($\nabla \times \bf{F} = \bf{0}$), enabling it to be represented as the negative gradient of a potential energy function, $\bf{F}(\bf{r}) = -\nabla V(\bf{r})$, and ensuring total mechanical energy is conserved.

Work done

The line integral of force over an infinitesimal displacement, defined in three dimensions by $dW = \bf{F} \times d\bf{r}$, which measures the amount of energy converted into kinetic energy.

Simple harmonic oscillator

A system executing motion near a stable equilibrium point under a linear restoring force $F = -kx$, described by the differential equation $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$.

Superposition principle

The property of linear differential equations stating that any linear combination of linearly independent solutions is also a solution to the equation.

Relaxation time

The characteristic time interval $\frac{1}{\gamma} = \frac{2m}{\lambda}$ during which the amplitude of an underdamped oscillator diminishes by a factor of $e^{-1}$.

Quality factor ($Q$)

A dimensionless parameter of an oscillator defined by $Q = \frac{m\times\frac{k}{m}^{1/2}}{\lambda} = \frac{\omega_0}{2\gamma}$, characterizing the sharpness of resonance and the persistence of stored mechanical energy.

Critical damping

The boundary damping condition occurring when $\gamma = \omega_0$, providing the most rapid return of an oscillator to equilibrium without any oscillatory overshoot.

Transient

The portion of the general solution of a driven damped oscillator that corresponds to the complementary function and decays exponentially with time, leaving only steady-state forced oscillations.

Resonance

The physical phenomenon occurring when the angular frequency of a periodic driving force approaches the natural frequency of an undamped oscillator ($\omega_1 \approx \omega_0$), generating maximum response amplitude.

Half-width of resonance

The parameter $\gamma$ specifying the frequency deviation from resonance ($\omega_1 = \omega_0 \pm \gamma$) at which the amplitude of forced oscillation drops to $\frac{1}{\sqrt{2}}$ of its resonance peak.

Impulse

The definite time integral of a force delivered over a short duration, $I = \int_{t}^{t+\Delta t} F\,dt$, which equals the resultant change in linear momentum $\Delta p$.

Green's function

The response function $G(t - t')$ representing the displacement at time $t$ of an oscillator at rest induced by an impulsive blow of unit impulse applied at time $t'$.

Elastic collision

A collision between bodies in which total mechanical kinetic energy is conserved before and after impact, reversing the relative velocity between the colliding bodies.

Coefficient of restitution ($e$)

The ratio of relative separation speed to relative approach speed between two colliding bodies, defined by $v_2 - v_1 = e(u_1 - u_2)$.

Moment of a force

The vector product $\bf{G} = \bf{r} \times \bf{F}$ of the position vector and applied force, which equals the time rate of change of angular momentum.

Angular momentum

The moment of linear momentum about a reference origin, defined for a body by $\bf{J} = \bf{r} \times \bf{p} = m(\bf{r} \times \frac{d\bf{r}}{dt})$.

Axial vector

A vector quantity, such as angular momentum or torque, whose assigned direction is determined by a right-hand screw convention rather than spatial inversion invariance.

Central force

A force that is always directed radially toward or away from a fixed point, having zero moment ($\bf{r} \times \bf{F} = \bf{0}$) and therefore conserving angular momentum.

Kepler's second law

The law stating that the radius vector of a particle moving under a central force sweeps out equal areas in equal times, satisfying $\frac{dA}{dt} = \frac{J}{2m} = \text{constant}$.

Euler-Lagrange equation

The second-order differential equation $\frac{\partial f}{\partial y} - \frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right) = 0$ that a function $y(x)$ must satisfy for the integral $I = \int_{x_0}^{x_1} f(y, y')\,dx$ to achieve a stationary value.

Lagrangian function ($L$)

The characteristic dynamical function defined as the difference between the kinetic energy and potential energy of a conservative system, $L = T - V$.

Hamilton's principle

The foundational principle stating that the actual trajectory followed by a conservative dynamical system renders the action integral $I = \int_{t_0}^{t_1} L\,dt$ stationary under arbitrary variations that vanish at the boundaries.

Generalized momentum

The conjugate momentum variable corresponding to a generalized coordinate $q_i$, defined in analytical mechanics by $p_i = \frac{\partial L}{\partial \dot{q}_i}$.

Effective potential energy

The composite radial potential function $U(r) = \frac{J^2}{2mr^2} + V(r)$ that combines the actual potential energy with the repulsive centrifugal potential term.

Impact parameter ($b$)

The perpendicular distance from the center of force to the asymptotic straight line along which an incident particle approaches from infinity.

Semi-latus rectum ($l$)

The geometric scale factor of a conic orbit under an inverse square law, defined by $l = \frac{J^2}{m|k|}$, which gives the radial distance at right angles to perihelion.

Kepler's third law

The astronomical law stating that the square of the orbital period of a body orbiting a primary mass $M$ is directly proportional to the cube of its semi-major axis, satisfying $\left(\frac{\tau}{2\pi}\right)^2 = \frac{a^3}{GM}$.

Differential cross-section

The ratio of the number of particles scattered into a unit solid angle per unit time to the incident particle flux per unit area, denoted by $\frac{d\sigma}{d\Omega}$.

Mean free path

The average straight-line distance $\lambda = \frac{1}{n\sigma}$ traversed by a particle through a scattering medium with particle number density $n$ and total collision cross-section $\sigma$ before colliding.

Rutherford scattering cross-section

The differential cross-section for Coulomb scattering of a charged particle by a central charge, given by $\frac{d\sigma}{d\Omega} = \frac{a^2}{4\left(\sin\left(\frac{1}{2}\theta\right)\right)^4}$.

Angular velocity vector

The axial vector $\bf{\omega} = \omega\bf{n}$ oriented along the axis of rotation by the right-hand screw rule, relating the position vector to velocity by $\bf{v} = \bf{\omega} \times \bf{r}$.