Fact-checked by Grok 2 weeks ago

Classical mechanics

Classical mechanics is the branch of physics that studies the motion of macroscopic bodies under the action of forces, using mathematical models based on the principles first formulated by Isaac Newton in his 1687 work Philosophiæ Naturalis Principia Mathematica.^[1]^[2] This framework assumes absolute space and time, treating objects as point particles or rigid bodies with continuous energies, and applies to low-energy, non-quantum systems in weak gravitational fields.^[2]^[3] The development of classical mechanics began with the scientific revolution, overthrowing Aristotelian views of motion.^[1] Key early contributions came from Nicolaus Copernicus, who proposed a heliocentric model in the 16th century, followed by Johannes Kepler's laws of planetary motion in the early 17th century and Galileo Galilei's experiments on free fall and inertia in the late 16th century.^[2]^[4]^[5] Newton synthesized these ideas in 1687, introducing his three laws of motion and the universal law of gravitation, which enabled precise predictions of celestial and terrestrial phenomena.^[1]^[4] Later advancements in the 18th and 19th centuries included Leonhard Euler's work on rigid body dynamics, Joseph-Louis Lagrange's variational formulation in 1764, and William Rowan Hamilton's phase space approach in 1835, providing more general and coordinate-independent descriptions.^[2]^[4]^[6] At its core, classical mechanics relies on fundamental concepts such as force, mass, momentum, energy, and torque.^[3] Newton's laws form the Newtonian formulation: the first law states that objects remain at rest or in uniform motion unless acted upon by a force; the second relates acceleration to net force via \mathbf{F} = m\mathbf{a}; and the third describes action-reaction pairs.^[2] The Lagrangian formulation, developed by Lagrange, uses the Lagrangian L = T - V (kinetic minus potential energy) and the Euler-Lagrange equations to derive equations of motion from the principle of least action, simplifying problems with constraints and generalized coordinates.^[6]^[4] The Hamiltonian formulation, introduced by Hamilton, employs the Hamiltonian H in phase space, yielding first-order canonical equations that highlight symmetries and conservation laws, such as those from Noether's theorem linking continuous symmetries to conserved quantities like energy and momentum.^[6]^[4] These reformulations reveal underlying geometric structures, including symplectic manifolds.^[4] Classical mechanics finds wide applications in engineering, astronomy, and everyday phenomena, predicting trajectories, oscillations, and rotations for systems ranging from pendulums to planetary orbits.^[1]^[3] It underpins fields like celestial mechanics, fluid dynamics, and structural analysis, with conservation laws providing powerful tools for solving complex problems without detailed force calculations.^[3]^[4] However, classical mechanics has limitations: it fails at speeds approaching the speed of light, where special relativity (introduced by Albert Einstein in 1905) is required, and at atomic scales, where quantum mechanics applies.^[2] Despite these, it remains a foundational theory, serving as the classical limit of more advanced physical descriptions.^[3]

Branches

Newtonian Mechanics

Newtonian mechanics is the foundational branch of classical mechanics that describes the motion of macroscopic objects under the influence of forces, primarily through Isaac Newton's three laws of motion as formulated in his seminal work Philosophiæ Naturalis Principia Mathematica published in 1687.^[7] This framework applies to everyday scales where velocities are much less than the speed of light and quantum effects are negligible, focusing on point particles or rigid bodies in inertial reference frames.^[8] It builds upon kinematics by incorporating the causes of motion—namely, forces—rather than merely describing trajectories.^[9] Newton's first law, known as the law of inertia, states that an object at rest remains at rest, and an object in uniform motion in a straight line continues in that motion, unless acted upon by an external force.^[10] This law defines inertial frames of reference, where no net force implies constant velocity; for example, a hockey puck sliding on frictionless ice maintains its speed and direction until an external push or friction intervenes.^[8] In non-inertial frames, such as a accelerating car, fictitious forces appear to violate this law, but true inertial frames reveal the underlying inertia.^[9] The second law quantifies the relationship between force and motion, stating that the net force on an object equals the rate of change of its linear momentum.^[11] Linear momentum \vec{p} is defined as the product of mass m and velocity \vec{v}, so \vec{p} = m \vec{v}.^[12] For constant mass, this derives to the familiar vector form \vec{F} = m \vec{a}, where \vec{F} is the net force and \vec{a} is the acceleration; the derivation follows from the definition of force as \vec{F} = \frac{d\vec{p}}{dt} = m \frac{d\vec{v}}{dt} = m \vec{a}, since mass is invariant.^[13] An example is a ball accelerating downward under gravity, where the gravitational force mg produces an acceleration g \approx 9.8 \, \mathrm{m/s^2}.^[14] Newton's third law asserts that for every action, there is an equal and opposite reaction; that is, if object A exerts a force on object B, then B exerts an equal force in the opposite direction on A.^[15] These forces act on different bodies and are simultaneous, as seen in a collision between two billiard balls, where the force one ball exerts pushes the other equally oppositely, altering their velocities symmetrically.^[16] From the third law, conservation of momentum emerges for isolated systems with no external forces: the total momentum remains constant.^[17] Consider two objects colliding; the force \vec{F}_{21} on object 2 due to 1 equals -\vec{F}_{12}, so the momentum change \Delta \vec{p}_2 = -\Delta \vec{p}_1, preserving \vec{p}_1 + \vec{p}_2.^[18] This principle underpins analyses of explosions or rocket propulsion in closed systems.^[16]

Analytical Mechanics

Analytical mechanics represents a reformulation of classical mechanics that shifts the emphasis from force-based descriptions to energy functionals, providing a more general and elegant framework for analyzing mechanical systems. Instead of directly applying forces as in the Newtonian approach, it employs principles derived from variational calculus to determine the equations of motion, often through the concept of a Lagrangian function defined as the difference between kinetic and potential energy. This methodology facilitates the treatment of complex systems by focusing on the overall energy structure rather than individual force interactions.^[19]^[6] In contrast to Newtonian mechanics, which relies on vector forces and typically Cartesian coordinates, analytical mechanics utilizes generalized coordinates—such as angles or other parameters suited to the system's symmetries—to describe the configuration space. This approach offers significant advantages in handling constraints, as it incorporates them directly into the formulation without the need to explicitly calculate constraint forces, thereby simplifying the analysis of systems like pendulums or rigid bodies with holonomic restrictions. The foundational ideas were developed by key figures including Leonhard Euler, who advanced variational principles in the 18th century; Joseph-Louis Lagrange, who formalized the Lagrangian method around 1764; and William Rowan Hamilton, who introduced the Hamiltonian framework in 1835.^[6]^[20]^[21] Central to analytical mechanics is the action principle, also known as Hamilton's principle, which posits that the actual path of a system extremizes the action integral over time. Complementing this is d'Alembert's principle, which extends the idea of virtual work from statics to dynamics by stating that the virtual work done by constraint forces is zero for any virtual displacement consistent with the constraints. These principles provide an equivalent foundation to Newton's laws while enabling the derivation of conservation laws, such as those for energy and momentum, in a more systematic manner for constrained systems.^[19]^[21]

Specialized Applications

Classical mechanics extends beyond point particles to specialized domains, adapting its core principles to describe complex systems such as deformable materials, rotating objects, and large ensembles of particles. These applications maintain the foundational laws of motion while incorporating additional mathematical structures to handle spatial distributions, internal constraints, and probabilistic behaviors. For instance, Newtonian mechanics serves as the basis for approximating continua as collections of interacting point particles, though continuum models introduce averaged quantities for practicality.^[22] In continuum mechanics, classical principles are applied to deformable solids and fluids, where matter is treated as a continuous medium rather than discrete particles. This framework introduces the stress tensor, a second-order tensor that quantifies internal forces per unit area at a point within the material, and the strain tensor, which measures local deformations such as stretching or shearing. The stress tensor \sigma_{ij} relates forces to surface orientations, enabling the description of equilibrium and motion via Cauchy's stress theorem, which states that the stress vector on any surface is given by \mathbf{t} = \boldsymbol{\sigma} \cdot \mathbf{n}, where \mathbf{n} is the unit normal. Similarly, the infinitesimal strain tensor \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) captures symmetric displacement gradients, with u_i denoting displacement components. These tensors facilitate the analysis of phenomena like elasticity in solids or viscosity in fluids, grounded in conservation laws of mass, momentum, and energy.^[22]^[23] Rigid body dynamics applies classical mechanics to objects that maintain fixed shape and size under motion, focusing on translational and rotational behaviors. Unlike point particles, rigid bodies possess moments of inertia, which quantify resistance to angular acceleration about specific axes; for a body rotating about its center of mass, the inertia tensor I_{ij} relates angular momentum \mathbf{L} to angular velocity \boldsymbol{\omega} via \mathbf{L} = \mathbf{I} \boldsymbol{\omega}. Euler's equations describe the rotational dynamics in the principal axis frame: I_1 \dot{\omega}_1 + (I_3 - I_2) \omega_2 \omega_3 = \tau_1, and cyclic permutations for the other components, where I_k are principal moments and \tau_k are torque components. A key relation is the torque equation \vec{\tau} = I \vec{\alpha}, which holds for rotation about a principal axis with moment of inertia I and angular acceleration \vec{\alpha}, illustrating how external torques induce changes in rotational motion. These formulations are essential for analyzing systems like spinning tops or spacecraft attitude control.^[24]^[25] Classical statistical mechanics emerges as a specialized application for systems with many particles, treating them in phase space—a $6N-dimensional space where each dimension represents position or momentum coordinates for N particles. This approach assumes the ergodic hypothesis, which posits that over long times, a system explores all accessible phase space regions uniformly, equating time averages of observables to ensemble averages. Hamiltonian methods underpin this by defining the total energy H(\mathbf{q}, \mathbf{p}) on phase space, with Liouville's theorem preserving phase space volume under time evolution. For large particle systems, this framework yields macroscopic properties like temperature and pressure from microscopic dynamics, as in the microcanonical ensemble where entropy S = k \ln \Omega relates to the phase space volume \Omega of constant energy surfaces.^[26] Representative examples highlight these applications. In planetary motion, Kepler's laws—elliptical orbits with the sun at one focus, equal areas swept in equal times, and T^2 \propto a^3 relating period T to semi-major axis a—emerge directly from Newtonian gravity applied to two-body problems, demonstrating inverse-square force laws in celestial mechanics. For simple harmonic motion in oscillators, such as a mass-spring system, the restoring force F = -kx leads to the equation \ddot{x} + \omega^2 x = 0 with angular frequency \omega = \sqrt{k/m}, yielding sinusoidal solutions x(t) = A \cos(\omega t + \phi) that model vibrations in pendulums or atomic bonds under small displacements.^[27]^[28]

Core Concepts

Kinematics

Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without regard to the forces or other agents causing the motion.^[29] It focuses on the geometric aspects of motion, such as position, velocity, and acceleration as functions of time.^[30] The position of an object specifies its location in space relative to a chosen origin and is represented by the position vector \vec{r}(t) in three-dimensional Euclidean space.^[31] In one dimension, position is a scalar function x(t), such as the coordinate along a straight line.^[32] Displacement is the change in position over a time interval, given by the vector \Delta \vec{r} = \vec{r}_f - \vec{r}_i, where \vec{r}_i and \vec{r}_f are the initial and final positions, respectively.^[33] In one dimension, displacement \Delta x = x_f - x_i can be positive or negative, indicating direction.^[34] The trajectory of an object is the path it traces in space, determined by the locus of its position vector over time, and can be a straight line in one dimension, a curve in a plane for two dimensions, or a space curve in three dimensions.^[32] Velocity quantifies the rate of change of position and has both magnitude and direction.^[35] Average velocity over a time interval \Delta t is defined as \vec{v}_\text{avg} = \frac{\Delta \vec{r}}{\Delta t}, representing the total displacement divided by the elapsed time.^[35] Instantaneous velocity at a specific time is the limit of the average velocity as \Delta t approaches zero, given by the derivative \vec{v}(t) = \frac{d\vec{r}}{dt}.^[35] Graphically, on a position-versus-time plot, the average velocity corresponds to the slope of a secant line between two points, while the instantaneous velocity is the slope of the tangent line to the curve at that time.^[35] In one dimension, velocity v(t) = \frac{dx}{dt} can be positive or negative, indicating direction along the line.^[36] Acceleration measures the rate of change of velocity.^[31] The instantaneous acceleration is \vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}.^[37] For curvilinear motion, acceleration decomposes into two perpendicular components: the tangential acceleration a_t = \frac{dv}{dt}, which is parallel to the velocity and changes the speed, and the centripetal (or normal) acceleration a_c = \frac{v^2}{\rho}, directed toward the center of curvature with radius \rho, which changes the direction of velocity.^[38] In one dimension, acceleration is simply a(t) = \frac{dv}{dt}, a scalar that can indicate speeding up or slowing down depending on its sign relative to velocity.^[37] Uniform motion occurs when velocity is constant, so acceleration is zero, resulting in a straight-line trajectory at constant speed; for example, a car traveling at a steady 60 km/h along a highway has \vec{v} = \text{constant} and \vec{a} = 0.^[39] Non-uniform motion involves changing velocity, hence non-zero acceleration; a classic example is projectile motion under gravity, where an object launched at an angle follows a parabolic trajectory in two dimensions, with constant horizontal velocity but vertically accelerated motion.^[39] In three dimensions, non-uniform motion might describe the helical path of a charged particle in a magnetic field, combining uniform circular motion in a plane with constant velocity perpendicular to that plane.^[40]

Dynamics and Forces

Dynamics, as a branch of classical mechanics, extends the descriptive framework of kinematics by incorporating the causes of motion, particularly the forces that alter an object's velocity or direction. While kinematics focuses solely on the geometric aspects of motion—such as position, velocity, and acceleration—dynamics examines how these quantities change under the influence of external influences, enabling predictions of an object's behavior in response to applied interactions.^[41] A key concept in dynamics is linear momentum, defined as the product of mass and velocity, \vec{p} = m \vec{v}. For constant mass, Newton's second law can be expressed as \vec{F}_\mathrm{net} = \frac{d\vec{p}}{dt} = m \vec{a}, linking force to the rate of change of momentum.^[42] Forces in classical mechanics are vector quantities that represent interactions capable of producing acceleration, categorized into fundamental and contact types. Gravitational forces arise between any two masses and follow Newton's law of universal gravitation, which states that the magnitude of the attractive force F between two point masses m_1 and m_2 separated by a distance r is given by

F = G \frac{m_1 m_2}{r^2},

where G = 6.67430 \times 10^{-11} \, \mathrm{m^3 \, kg^{-1} \, s^{-2}} is the gravitational constant; this inverse-square dependence means the force diminishes rapidly with distance.^[43] In vector form, the force on m_1 due to m_2 is

\vec{F}_{12} = -G \frac{m_1 m_2}{r^2} \hat{r},

where \hat{r} is the unit vector from m_1 to m_2, indicating the force's central and attractive nature. Electromagnetic forces, treated classically via Coulomb's law for static charges or the Lorentz force for moving charges, govern interactions between charged particles or magnets, often dominating over gravity at atomic scales.^[44] Contact forces include friction, which opposes relative motion between surfaces (kinetic) or prevents it (static), and tension, which acts along a taut string or rope to pull objects toward each other.^[44] For rotational motion, torque is the rotational equivalent of force, defined as \vec{\tau} = \vec{r} \times \vec{F}, where \vec{r} is the position vector from the axis of rotation to the point of force application. Torque causes angular acceleration according to \vec{\tau} = I \vec{\alpha}, with I the moment of inertia.^[45] To analyze forces on an isolated object, physicists employ free-body diagrams (FBDs), graphical representations that isolate the body and depict all external forces as vectors originating from its center, excluding internal or reaction forces from other bodies. These diagrams facilitate the identification of force components in chosen coordinate systems, aiding in the application of Newton's laws without considering the broader system.^[46] Equilibrium occurs when the vector sum of all forces on the body is zero, resulting in zero acceleration and either rest or constant-velocity motion; for translational equilibrium, this condition \sum \vec{F} = 0 must hold in all directions.^[47] The net force \vec{F}_\mathrm{net}, defined as the vector sum of all individual forces acting on an object, directly determines its acceleration according to Newton's second law: \vec{F}_\mathrm{net} = m \vec{a}, where m is the object's mass and \vec{a} is the resulting acceleration. This relation quantifies how unbalanced forces produce changes in kinematic quantities like acceleration, underscoring dynamics' role in causal explanations of motion.^[48]

Energy and Work

In classical mechanics, work is defined as the line integral of a force along the path traversed by a particle, quantifying the energy transferred by the force during the displacement. Mathematically, the work W done by a force \vec{F} from position \vec{r}_1 to \vec{r}_2 along a path C is given by

W_{12} = \int_{\vec{r}_1}^{\vec{r}_2 : C} \vec{F}(\vec{r}) \cdot d\vec{r}.

This scalar quantity depends on the specific path taken unless the force is conservative, in which case the work is path-independent.^[49] The work-energy theorem relates the net work done on a particle to the change in its kinetic energy, providing a foundation for understanding energy transformations. Kinetic energy K is the scalar measure of a particle's motion, defined as K = \frac{1}{2} m v^2, where m is the mass and v is the speed. To derive this, start from Newton's second law \vec{F}_{\rm tot} = m \vec{a} and take the dot product with velocity \vec{v}:

\vec{F}_{\rm tot} \cdot \vec{v} = m \vec{a} \cdot \vec{v} = m \vec{v} \cdot \frac{d\vec{v}}{dt} = \frac{1}{2} m \frac{d}{dt} (v^2),

using the product rule. Integrating over time from initial to final states, with d\vec{x} = \vec{v} \, dt, yields

\int_i^f \vec{F}_{\rm tot} \cdot d\vec{x} = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2,

or W_{\rm tot} = \Delta K. Thus, the net work equals the change in kinetic energy, applicable to any force and path.^[50]^[49] Potential energy arises for conservative forces, where the work done is path-independent and can be stored as a scalar function of position. A force \vec{F} is conservative if \oint \vec{F} \cdot d\vec{r} = 0 for any closed path, or equivalently if \vec{F} = -\nabla U(\vec{r}), with U the potential energy such that the work from \vec{r}_2 to \vec{r}_1 is U(\vec{r}_2) - U(\vec{r}_1).^[51] A classic example is the gravitational force near Earth's surface, approximated as constant \vec{F} = -mg \hat{y}. The corresponding potential energy is U_g = mgh, where h is the height above a reference level (typically set to U_g = 0), g \approx 9.80 \, \rm m/s^2, and this formula holds for small heights where the field is uniform. Lifting an object increases its potential energy by the work done against gravity.^[52] The total mechanical energy E = K + U is conserved for isolated systems interacting only through conservative forces, with no dissipative effects like friction. In such cases, \Delta E = 0, so changes in kinetic energy are balanced by opposite changes in potential energy, enabling reversible transformations (e.g., a pendulum converting between K and U_g). This principle simplifies analysis by focusing on energy rather than forces alone.^[53]^[51] Power is the time rate of doing work, measuring how quickly energy is transferred by a force. For a constant force, it is the dot product P = \vec{F} \cdot \vec{v}, with units of watts (joules per second). In general, P = \frac{dW}{dt}, extending the concept to variable forces and providing insight into energy flow in dynamic systems.^[41]

Formulations

Newtonian Formulation

The Newtonian formulation of classical mechanics describes the motion of point particles through a system of differential equations derived from Newton's second law of motion, which states that the net force \vec{F}_i on the i-th particle equals its mass m_i times its acceleration \vec{a}_i, or \vec{F}_i = m_i \vec{a}_i.^[54] For a system of N particles, this yields $3N coupled second-order ordinary differential equations (ODEs) in Cartesian coordinates, one for each component of acceleration, with forces \vec{F}_i comprising pairwise interactions between particles and external fields.^[55] Constraints, such as rigid body connections or holonomic restrictions, add algebraic equations that limit the degrees of freedom, often reducing the effective dimensionality of the problem while preserving the second-law structure.^[56] Solving these equations analytically is feasible for constant forces, where acceleration is uniform, allowing direct integration: for example, position \vec{r}(t) follows a quadratic form \vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a} t^2 from twice integrating \vec{a} = \constant.^[55] For variable forces, such as those depending on position or time, analytical solutions are rare beyond special cases, necessitating numerical methods like the Euler scheme or higher-order Runge-Kutta integrators to approximate trajectories by iterative stepping from initial conditions.^[57] These techniques propagate the state (position and velocity) forward in time, with accuracy improving via smaller time steps or adaptive algorithms, though they require computational resources for complex systems.^[58] A key application is the two-body problem, where two particles interact via a central force \vec{F} = f(r) \hat{r} directed along their separation vector.^[59] The system's motion separates into center-of-mass translation, which is uniform if no external forces act, and relative motion equivalent to a single particle of reduced mass \mu = \frac{m_1 m_2}{m_1 + m_2} orbiting a fixed center under the central force.^[60] This reduction simplifies the $6-dimensional problem to a $3-dimensional one for the relative coordinate, conserving angular momentum and enabling planar solutions.^[60] For gravitational central forces, f(r) = -\frac{[G](/page/G) m_1 m_2}{r^2}, the two-body problem yields closed elliptical orbits, but circular motion provides a specific solvable case where centripetal acceleration balances the force: \frac{v^2}{r} = \frac{[G](/page/G) [M](/page/M)}{r^2}, simplifying to v^2 = \frac{[G](/page/G) [M](/page/M)}{r}, with v the orbital speed, r the radius, G the gravitational constant, and M the central mass (for m_2 \ll M).^[61] In the International System of Units (SI), Newtonian mechanics employs kilograms (kg) for mass, meters (m) for length, seconds (s) for time, and derives force in newtons (N = kg·m/s²), acceleration in m/s², and velocity in m/s, ensuring dimensional consistency across equations.^[62] Initial positions and velocities draw from kinematic descriptions to set boundary conditions, while solutions can be verified against energy conservation principles for isolated systems.^[63]

Lagrangian Formulation

The Lagrangian formulation of classical mechanics provides a powerful framework for describing the motion of systems by employing the principle of least action, which reformulates dynamics in terms of energy differences rather than forces directly. Developed by Joseph-Louis Lagrange in his seminal work Mécanique Analytique, this approach uses generalized coordinates to handle complex systems efficiently.^[64] The core quantity is the Lagrangian L, defined as the difference between the kinetic energy T and the potential energy V of the system:

L = T - V.

This definition encapsulates the system's dynamics in a single scalar function, applicable to a wide range of mechanical systems from particles to rigid bodies.^[65] The equations of motion arise from Hamilton's principle, which states that the actual path of the system minimizes (or more precisely, makes stationary) the action integral S = \int_{t_1}^{t_2} L \, dt.^[66] This principle, introduced by William Rowan Hamilton, derives from variational calculus: the variation \delta S = 0 for the true trajectory leads to the Euler-Lagrange equations for each generalized coordinate q_i:

\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0.

Here, \dot{q}_i denotes the time derivative of q_i. The derivation involves expanding the action to first order in the variations \delta q_i and \delta \dot{q}_i, integrating by parts, and setting the coefficients of independent \delta q_i to zero, yielding these second-order differential equations that govern the system's evolution.^[67] For conservative systems without explicit time dependence in L, this formulation is equivalent to Newtonian mechanics in simple cases. A key advantage of the Lagrangian approach lies in its handling of holonomic constraints, which can be expressed as functions of coordinates and time (e.g., f(q_1, \dots, q_n, t) = 0). By selecting generalized coordinates that inherently satisfy these constraints—such as angles for rotational systems—the constraint forces are automatically eliminated from the equations, simplifying the analysis without introducing Lagrange multipliers unless necessary.^[19] This makes it particularly effective for systems with geometric restrictions, like beads on wires or pendulums, where Newtonian methods require explicit resolution of tensions or normals. As an illustrative example, consider a simple pendulum of mass m and length \ell, with the angle \theta from the vertical as the generalized coordinate. The kinetic energy is T = \frac{1}{2} m \ell^2 \dot{\theta}^2, and the potential energy is V = -m g \ell \cos \theta (taking the lowest point as zero). Thus, the Lagrangian is

L = \frac{1}{2} m \ell^2 \dot{\theta}^2 + m g \ell \cos \theta.

Applying the Euler-Lagrange equation gives

\frac{d}{dt} \left( m \ell^2 \dot{\theta} \right) - (- m g \ell \sin \theta) = 0,

which simplifies to the equation of motion \ddot{\theta} + \frac{g}{\ell} \sin \theta = 0. This derivation captures the nonlinear dynamics without invoking tension forces, highlighting the method's elegance for constrained motion.^[68]

Hamiltonian Formulation

The Hamiltonian formulation of classical mechanics represents a reformulation of the equations of motion in terms of generalized coordinates q_i and their conjugate momenta p_i, providing a symplectic structure that emphasizes the evolution in phase space. Developed by William Rowan Hamilton, this approach shifts the focus from velocities to momenta, facilitating the analysis of time evolution and conservation laws through the total energy function known as the Hamiltonian. Unlike the Lagrangian formulation, which uses coordinates and velocities, the Hamiltonian framework treats positions and momenta as independent variables, enabling a more symmetric treatment of dynamics.^[69] The Hamiltonian H for a system is typically expressed as the sum of kinetic energy T and potential energy V, but rewritten in terms of the generalized coordinates q_i and conjugate momenta p_i:

H(q, p, t) = T\left(q, \frac{\partial L}{\partial \dot{q}}\right) + V(q),

where L is the Lagrangian, and the momenta are defined as p_i = \frac{\partial L}{\partial \dot{q}_i}. This expression arises from a Legendre transformation of the Lagrangian L(q, \dot{q}, t), which converts the dependence on velocities \dot{q} to momenta p:

H(q, p, t) = \sum_i p_i \dot{q}_i(q, p, t) - L(q, \dot{q}(q, p, t), t).

The transformation ensures that H generates the dynamics via partial derivatives, preserving the physical content of the system while highlighting its energetic structure.^[69]^[70] The time evolution of the system is governed by Hamilton's canonical equations, a pair of first-order differential equations for each degree of freedom:

\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}.

These equations describe how coordinates and momenta change along trajectories in phase space, the $2n-dimensional space spanned by (q_1, \dots, q_n, p_1, \dots, p_n) for an n-degree-of-freedom system. In this space, the motion of a point representing the system's state follows deterministic paths determined by H, with the Hamiltonian often conserved if it does not explicitly depend on time (\frac{\partial H}{\partial t} = 0), implying \dot{H} = 0. The symplectic nature of the formulation ensures that transformations preserving the equations form a group under composition, underlying the geometric interpretation of dynamics.^[69]^[70] Phase space trajectories are incompressible due to Liouville's theorem, which states that the phase space volume occupied by an ensemble of systems remains constant under Hamiltonian evolution. Formally, for a density function \rho(q, p, t) in phase space, the theorem implies

\frac{d\rho}{dt} = \sum_i \left( \frac{\partial \rho}{\partial q_i} \dot{q}_i + \frac{\partial \rho}{\partial p_i} \dot{p}_i \right) = 0,

since the flow is divergence-free: \sum_i \left( \frac{\partial \dot{q}_i}{\partial q_i} + \frac{\partial \dot{p}_i}{\partial p_i} \right) = 0. This preservation of volume reflects the deterministic and reversible nature of classical mechanics, ensuring that information about initial conditions is conserved over time.^[71] A simple example is the one-dimensional harmonic oscillator, with Hamiltonian

H(q, p) = \frac{p^2}{2m} + \frac{1}{2} k q^2,

where m is mass and k is the spring constant. The canonical equations yield \dot{q} = \frac{p}{m} and \dot{p} = -k q, leading to elliptical trajectories in phase space centered at the origin, with the area enclosed by each orbit equal to $2\pi E / \omega (where E is total energy and \omega = \sqrt{k/m}), demonstrating Liouville's volume preservation as the ellipse is traversed without distortion.^[70]

Applicability and Limitations

Regions of Validity

Classical mechanics provides accurate descriptions of physical systems at macroscopic scales, encompassing everyday objects such as vehicles and machinery, as well as larger structures like planets and asteroids, where quantum effects are negligible. This framework is particularly effective for systems involving masses on the order of grams to solar masses, where the behavior aligns with Newtonian principles without requiring relativistic or quantum corrections. For instance, the motion of a baseball or a satellite in low Earth orbit can be precisely modeled using classical equations, as the particles or bodies involved are sufficiently large for wave-like quantum properties to be insignificant.^[72]^[73] The theory holds validity primarily at low speeds, where the velocities of objects satisfy v \ll c, with c being the speed of light in vacuum (c \approx 3 \times 10^8 m/s).^[72] In this regime, relativistic effects such as time dilation or mass increase are minimal, allowing classical predictions to match experimental observations to high precision. Classical mechanics finds extensive application in domains like engineering, where it underpins the design of structures and machines; ballistics, for predicting projectile trajectories under gravity and drag; and celestial mechanics, for calculating orbital paths of planets and spacecraft.^[41] A key criterion for classical behavior is that the de Broglie wavelength \lambda = h / p (where h is Planck's constant and p is momentum) must be much smaller than the characteristic size L of the system, i.e., \lambda \ll L, ensuring that quantum interference effects do not manifest.^[74]^[73] Even within its deterministic framework, classical mechanics encounters limitations in chaotic systems, where nonlinear dynamics lead to sensitive dependence on initial conditions, causing exponentially diverging trajectories from tiny perturbations.^[75] This sensitivity, quantified by positive Lyapunov exponents, implies that while the systems are fully deterministic, long-term predictability is constrained by the precision of initial measurements and computational resources, as errors amplify rapidly over time.^[75] For example, in weather models or planetary three-body problems, classical equations yield aperiodic behavior that appears random, highlighting practical boundaries in forecasting despite theoretical exactness.^[75]

Transitions to Modern Physics

Classical mechanics serves as an excellent approximation to both special relativity (at low speeds) and general relativity (in weak gravitational fields). In the low-velocity regime where particle velocities v are much smaller than the speed of light c (i.e., v \ll c), the Lorentz transformations reduce to the Galilean transformations of Newtonian mechanics. Similarly, in weak gravitational fields, Newtonian gravity approximates the curved spacetime of general relativity. This ensures that classical predictions for position, time, and gravitational interactions align with relativistic ones, as higher-order corrections become negligible.^[76] A key example of this approximation appears in the relativistic expressions for energy and momentum. The total relativistic energy is given by E = \gamma m c^2, where \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} and m is the rest mass; for v \ll c, \gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2}, so E \approx m c^2 + \frac{1}{2} m v^2. Here, m c^2 represents the rest energy, while \frac{1}{2} m v^2 recovers the classical kinetic energy. Similarly, the relativistic momentum p = \gamma m v approximates to p \approx m v \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right), with the leading term matching the Newtonian p = m v. These expansions highlight how classical mechanics emerges as the first-order approximation in the relativistic framework.^[77]^[78] However, classical mechanics fails in strong gravitational fields, such as near black holes or in the precise dynamics of compact objects like neutron stars, where general relativity predicts phenomena like gravitational waves and extreme time dilation not captured by Newtonian gravity. For instance, the anomalous precession of Mercury's orbit was explained by Einstein's general relativity in 1915, resolving a long-standing discrepancy. Practical implications of relativistic deviations are evident in technologies like the Global Positioning System (GPS), where satellites orbit at speeds around 14,000 km/h (v/c \approx 4 \times 10^{-6}) and at an altitude of about 20,000 km. Without accounting for both special relativistic time dilation (which slows satellite clocks by ~7 μs/day due to velocity) and general relativistic gravitational effects (which speed them up by ~45 μs/day due to weaker gravity), the net effect would cause satellite clocks to run faster by ~38 μs/day. This would accumulate to positional errors of about 10 km per day, rendering the system unusable for precise navigation. GPS compensates by designing onboard clocks to tick slightly slower and applying further corrections in receivers.^[79] In the quantum domain, classical mechanics arises as the limit of quantum mechanics when the reduced Planck's constant \hbar approaches zero, according to Bohr's correspondence principle. This principle states that for systems with large quantum numbers or macroscopic scales, quantum mechanical predictions converge to classical trajectories and energies, ensuring continuity between the theories. A notable failure of classical mechanics occurs in the photoelectric effect, where illuminating a metal surface with light ejects electrons only above a threshold frequency, independent of intensity; classical wave theory predicts emission proportional to intensity, but Einstein explained it via discrete photons with energy E = h \nu, where h is Planck's constant and \nu is frequency, resolving the discrepancy.^[80] To bridge the classical-quantum divide in intermediate regimes, semiclassical methods like the Wentzel-Kramers-Brillouin (WKB) approximation provide approximate solutions to the Schrödinger equation for slowly varying potentials. In WKB, the wave function is expressed as \psi(x) \approx \frac{1}{\sqrt{p(x)}} \exp\left(\frac{i}{\hbar} \int p(x) \, dx \right), where p(x) = \sqrt{2m(E - V(x))} is the classical momentum; as \hbar \to 0, this yields classical paths and turning points, offering insights into quantum tunneling and bound states near classical limits.^[81]

Historical Development

Origins and Newtonian Era

The origins of classical mechanics can be traced to ancient Greek philosophy, where Aristotle developed a teleological theory of motion in his Physics and related works. He classified motions as natural—such as the downward fall of heavy bodies toward the Earth's center or the circular paths of celestial spheres—or violent, requiring external forces to overcome natural tendencies. Aristotle's framework emphasized purpose-driven change, with objects seeking their "natural place" as the final cause of motion.^[82] Medieval scholars built upon and critiqued Aristotelian ideas, laying groundwork for empirical approaches. In the 14th century, Jean Buridan advanced the impetus theory, proposing that a mover imparts a permanent force (impetus) to a projectile, sustaining its motion until resisted by air or gravity, thus resolving inconsistencies in Aristotle's explanation of continued projectile flight.^[83] Concurrently, Nicole Oresme innovated by representing variations in qualities (like heat intensity) and motions (like velocity) through geometric figures in his Tractatus de configurationibus qualitatum et motuum, using a baseline for extension (time or space) and perpendiculars for intensity, enabling calculations of averages and totals that anticipated graphical methods in physics.^[84] The 17th century marked a pivotal shift toward experimental and mathematical foundations, led by Galileo Galilei. Through controlled experiments with balls rolling down inclined planes, Galileo established that acceleration due to gravity is uniform and independent of mass, with the distance fallen proportional to the square of time; this built on his earlier observations of free fall and pendulum motion. These principles were systematically presented in his 1638 Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze, emphasizing kinematics as a precursor to dynamics.^[85] Meanwhile, Johannes Kepler, using Tycho Brahe's observations, formulated three empirical laws of planetary motion between 1609 and 1619: orbits are ellipses with the Sun at one focus, a line from the Sun to a planet sweeps equal areas in equal times, and the square of the orbital period is proportional to the cube of the semi-major axis.^[2] Isaac Newton's Philosophiæ Naturalis Principia Mathematica (1687) culminated this evolution by integrating Galileo's insights with astronomical observations. Newton formulated three laws of motion and posited universal gravitation as an inverse-square force, deriving Johannes Kepler's empirical laws of planetary motion (elliptical orbits, equal areas in equal times, harmonic periods) as consequences of this attraction acting between all bodies.^[7] The popular anecdote of Newton conceiving gravity upon seeing an apple fall at Woolsthorpe Manor is apocryphal, first recorded decades later by associates like William Stukeley and lacking any basis in Newton's writings or contemporary accounts.^[86] However, Newton's development of the inverse-square law drew from earlier exchanges, including correspondence with Robert Hooke in 1679–1680, where Hooke suggested central forces varying inversely with distance squared to explain orbital paths.^[87]

Advances in the 18th and 19th Centuries

In the mid-18th century, Leonhard Euler advanced classical mechanics by developing foundational theories for rigid body motion and fluid dynamics. His 1765 work, Theoria Motus Corporum Solidorum, established the principles of rigid body dynamics, including the use of Euler angles to describe orientations and the derivation of equations governing rotational motion under torque.^[88] Euler's contributions to fluid dynamics in the 1750s, notably in Principia Motus Fluidorum (1757), introduced the Euler equations for inviscid flow, extending Newtonian principles to continuous media and laying the groundwork for hydrodynamics.^[89] Joseph-Louis Lagrange further refined analytical approaches in his 1788 treatise Mécanique Analytique, which formalized mechanics using variational principles and generalized coordinates, eliminating the need for geometric considerations in favor of purely algebraic methods.^[90] This work synthesized a century of progress since Newton, emphasizing the principle of least action to derive equations of motion for complex systems, influencing subsequent developments in theoretical physics. In the 1830s, William Rowan Hamilton introduced a transformative reformulation in his papers "On a General Method in Dynamics" (1834) and its sequel (1835), defining dynamics through a characteristic function in phase space, where position and momentum coordinates form a symplectic structure for conservative systems.^[91] This Hamiltonian approach provided a unified framework for integrating equations of motion, enhancing computational tractability for multi-body problems.^[92] A landmark application of Newtonian mechanics occurred in 1846 with the prediction and discovery of Neptune, where Urbain Le Verrier and John Couch Adams independently calculated its position by analyzing perturbations in Uranus's orbit, confirming gravitational theory's predictive power through inverse perturbation methods.^[93] Johann Galle observed the planet on September 23, 1846, at the Berlin Observatory, validating these computations.^[94] During the 19th century, classical mechanics integrated with emerging fields, as seen in James Clerk Maxwell's 1860s formulation of electromagnetism, where his equations described electromagnetic fields as propagating waves governed by mechanical analogies like stress and strain in elastic media, bridging optics and dynamics.^[95]^[96] Similarly, Ludwig Boltzmann's 1870s contributions to statistical mechanics, including the Boltzmann equation (1872) and the interpretation of entropy as S = k \ln W—where k is Boltzmann's constant and W the number of microstates—provided a probabilistic foundation for thermodynamic laws within classical frameworks.^[97] Lord Kelvin (William Thomson) proposed the vortex atom theory in the 1860s–1880s, modeling atoms as stable knotted vortices in a perfect fluid ether to explain chemical stability and periodicity without discrete particles, influencing early atomic models.^[98] Toward the century's end, Henri Poincaré's 1890s investigations into the three-body problem, detailed in Les Méthodes Nouvelles de la Mécanique Céleste (1892–1899), revealed sensitive dependence on initial conditions and homoclinic tangles, foreshadowing chaotic behavior in deterministic classical systems.^[99]

References

[1]
What is classical mechanics? - Richard Fitzpatrick
Classical mechanics is the study of the motion of bodies (including the special case in which bodies remain at rest) in accordance with the general principles.
[2]
[PDF] 8.01SC S22 Chapter 1: Introduction to Classical Mechanics
Classical mechanics is the mathematical science that studies the displacement of bodies under the action of forces. Gailieo Galilee initiated the modern era ...
[3]
Classical Mechanics | Physics | MIT OpenCourseWare
### Summary of Classical Mechanics from MIT OpenCourseWare (8.01SC, Fall 2016)
[4]
[PDF] Mechanics - OU Math Department
The entire subsequent development of classical mechanics is based on the understanding of the mathematical structures behind Newton's theory and its relation ...Missing: key | Show results with:key
[5]
[PDF] Chapter 2 Lagrange's and Hamilton's Equations - Rutgers Physics
In this chapter, we consider two reformulations of Newtonian mechanics, the. Lagrangian and the Hamiltonian formalism. The first is naturally associated with ...
[6]
Newton's Philosophiae Naturalis Principia Mathematica
Dec 20, 2007 · The second edition appeared in 1713, twenty six years after the first. It had five substantive changes of note. First, the structure of the ...Newton's Laws of Motion · Book 1 of the Principia · Book 3 of the Principia
[7]
Newton's Laws - HyperPhysics Concepts
Newton's First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
[8]
Newton's Three Laws of Motion
Newton's First Law of Motion: I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.
[9]
PHYS 200 - Lecture 3 - Newton's Laws of Motion - Open Yale Courses
The First Law on inertia states that every object will remain in a state of rest or uniform motion in a straight line unless acted upon by an external force.
[10]
Momentum, Work and Energy - Galileo and Einstein
rate of change of momentum = mass x rate of change of velocity. This means that Newton's Second Law can be rewritten: force = rate of change of momentum. Now ...
[11]
8.1 Linear Momentum and Force – College Physics chapters 1-17
NEWTON'S SECOND LAW OF MOTION IN TERMS OF MOMENTUM. The net external force equals the change in momentum of a system divided by the time over which it changes.<|control11|><|separator|>
[12]
Newton's Second Law of Motion - Richard Fitzpatrick
Newton's second law of motion essentially states that if a point object is subject to an external force, ${\bf f}$, then its equation of motion is given by
[13]
9 Newton's Laws of Dynamics - Feynman Lectures
Thus the law of gravity tells us that weight is proportional to mass; the force is in the vertical direction and is the mass times g. Again we find that the ...
[14]
Newton's Three Laws of Motion - Stanford CCRMA
Newton's three laws of motion may be stated as follows: Every object in a state of uniform motion will remain in that state of motion unless an external force ...
[15]
Conservation of momentum
By Newton's third law object 2 pushes on object 1 with a force F = 10 N for 2 s to the left. The momentum of object 1 changes by 20 Ns = 20 kgm/s to the left.
[16]
8.3 Conservation of Momentum – College Physics - UCF Pressbooks
We know from Newton's third law that \boldsymbol{F_2=-F_1}, and so. \boldsymbol{\Delta{p}_2=-F_1\Delta{t. Thus, the changes in momentum are equal and opposite, ...
[17]
10 Conservation of Momentum - Feynman Lectures - Caltech
Even though the law F=ma is false, and all the derivations of Newton were wrong for the conservation of momentum, in quantum mechanics, nevertheless, in the ...
[18]
[PDF] Chapter 1 A Review of Analytical Mechanics - MIT OpenCourseWare
Example: Hamilton's principle states that motion qi(t) extremizes the action, so in this case s = t, yi = qi, f = L, and J = S. Demanding δS = 0 then yields the ...
[19]
[PDF] Analytical Dynamics: Lagrange's Equation and its Application
Apr 9, 2017 · Implicit in the definition of Hamilton's principle is that the system will move along a dynamical path consistent with the system ...
[20]
[PDF] NOTES ON CLASSICAL MECHANICS Last updated: January 16, 2023
Definition 1.1 (Newton's principle of determinacy). A Newtonian system is given by N particles and a function F : R × (RNd r ∆) × R. Nd → RNd, called the ...
[21]
[PDF] Continuum Mechanics - MIT
May 11, 2012 · ... Strain ... Tensors. . . . . . . . . . . . . . . . . 66. 3.4 Rate of Change of Length, Orientation, and Volume. . . . . . . . . . . . . . 68.
[22]
[PDF] Introduction to Continuum Mechanics
This is a generalization of the classical stress-strain law of linear elasticity (compare to Equation (6.263) later in this chapter), and is known as the ...
[23]
[PDF] Euler's Equations - 3D Rigid Body Dynamics - MIT OpenCourseWare
We now turn to the task of deriving the general equations of motion for a three-dimensional rigid body. These equations are referred to as Euler's equations ...
[24]
[PDF] Lecture Notes for PHY 405 Classical Mechanics
Nov 21, 2004 · The following will derive Euler's equations for a rigid body, which describe how a rigid body's orientation is altered when a torque is applied.
[25]
[PDF] Chapter 4: Statistical Mechanics [version 1204.1.K] - Caltech PMA
It retains its phase-space volume, but gets strung out into a winding, contorted surface (Fig. 4.2b) which (by virtue of the ergodic hypothesis) ultimately ...
[26]
[PDF] 8.223 IAP 2017 Lecture 8 Kepler's Laws - MIT OpenCourseWare
Kepler's laws: 1. Orbits are elliptical with the Sun at one focus 2. The line from the planet to the Sun sweeps out equal area in equal time 3. The square of ...
[27]
[PDF] Chapter 23 Simple Harmonic Motion - MIT OpenCourseWare
Jul 23, 2022 · Now substitute the simple harmonic oscillator equation of motion, (Eq. ... 8.01 Classical Mechanics. Spring 2022. For information about citing ...
[28]
[PDF] Kinematics Study Guide
Understanding Kinematics: The Basics. Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects), and systems of ...
[29]
[PDF] Kinematics - MIT OpenCourseWare
Sep 12, 2007 · Kinematics deals with the motions of bodies. Kinematics has to do with geometry and physical constraints. • Kinetics deals with the evolution of ...Missing: mechanics | Show results with:mechanics
[30]
[PDF] Position, Displacement, Velocity, and Acceleration Vectors - BoxSand
But we must first go over 4 specific vectors that are immensely important in classical physics: Position, Displacement, Velocity, and Acceleration vectors.Missing: mechanics | Show results with:mechanics
[31]
[PDF] University Physics I: Classical Mechanics - ScholarWorks@UARK
Feb 8, 2019 · Kinematics is the part of mechanics that deals with the mathematical description of motion ... position vectors, and the displacement vector. (b) ...
[32]
1.3 Displacement Vector in 1D | Classical Mechanics | Physics
The displacement vector is defined as . In one dimension, the displacement vector has one component. For example, if the motion is along the x-axis, the ...Missing: trajectory | Show results with:trajectory
[33]
[PDF] ONE-DIMENSIONAL KINEMATICS - UNLV Physics
Jan 1, 2008 · The formula definition—but not the explicit definition which you'll get in intro calculus—is df dx. = lim. ∆x→0. ∆f. ∆x . (3). Note that as ...
[34]
[PDF] Chapter 4 One Dimensional Kinematics - MIT OpenCourseWare
The SI units for average velocity are meters per second ⎡m⋅s−1 ⎤⎦ . ... Figure 4.5 Plot of instantaneous velocity instantaneous velocity as a function of time.
[35]
[PDF] Chapter 4 One Dimensional Kinematics
May 4, 2013 · We first consider how the instantaneous velocity changes over an interval of time and then take the limit as the time interval approaches zero.
[36]
[PDF] Chapter 1 Particle Kinematics - Rutgers Physics
Classical mechanics, narrowly defined, is the investigation of the motion of systems of particles in Euclidean three-dimensional space, under the influence.
[37]
Path kinematics | ME 274: Basic Mechanics II - Purdue University
The tangental component of acceleration is the “rate of change of speed”: If the speed of P is INCREASING, then the angle theta is less than 90 degrees. If the ...
[38]
[PDF] Kinematics
Oct 19, 2022 · It is to the average speed what the instantaneous velocity is to the average velocity. In 1D, the instantaneous speed is the absolute value of ...
[39]
[PDF] Chapter 6 Circular Motion - MIT OpenCourseWare
Any radial inward acceleration is called centripetal acceleration. Recall that the direction of the velocity is always tangent to the circle. Therefore the ...
[40]
[PDF] Classical Mechanics - Richard Fitzpatrick
Classical mechanics is the study of the motion of bodies (including the special case in which bodies remain at rest) in accordance with the general ...
[41]
Newton's Universal Law of Gravitation | Physics - Lumen Learning
In equation form, this is F = G mM r 2 , where F is the magnitude of the gravitational force. G is the gravitational constant, given by G = 6.673 × 10−11 N·m2/ ...
[42]
Types of Forces - The Physics Classroom
Types of Forces ; Frictional Force. Gravitational Force ; Tension Force. Electrical Force ; Normal Force. Magnetic Force ; Air Resistance Force ; Applied Force ...
[43]
5.7 Drawing Free-Body Diagrams – General Physics Using Calculus I
A free-body diagram is a useful means of describing and analyzing all the forces that act on a body to determine equilibrium according to Newton's first law or ...
[44]
12.1 Conditions for Static Equilibrium - University Physics Volume 1
Sep 19, 2016 · We say that a rigid body is in equilibrium when both its linear and angular acceleration are zero relative to an inertial frame of reference ...
[45]
5.3 Newton's Second Law - University Physics Volume 1 | OpenStax
Sep 19, 2016 · According to Newton's second law, a net force is required to cause acceleration. Significance. These questions may seem trivial, but they are ...<|separator|>
[46]
[PDF] Classical Mechanics Lecture Notes: Work and Energy
Mar 14, 2024 · In general, the work does depend on the path, so the path has to be specified. The definition of work can be used both for the net force acting ...
[47]
The Work–Energy Theorem – Introductory Physics
The kinetic energy is defined as K=\frac{1}{2}mv^2. The total work as a particle moves from \vec x_{\rm i} to \vec x_{\rm f} is equal to the change in . In ...
[48]
[PDF] PHYS 419: Classical Mechanics Lecture Notes
In general, the work does depend on the path, so the path has to be specified. The definition of work can be used both for the net force acting on the body ...
[49]
7.3 Gravitational Potential Energy – College Physics
ΔPEg=mgh for any path between the two points. Gravity is one of a small class of forces where the work done by or against the force depends only on the starting ...
[50]
[PDF] 8.01SC S22 Chapter 14: Potential Energy and Conservation of Energy
Jan 14, 2023 · Then the gravitational force is an internal conservative force ... conservative forces, the total mechanical energy of the system is constant,.
[51]
[PDF] Formulations of Classical Mechanics 1. Introduction - PhilSci-Archive
The Hamiltonian and Lagrangian formulations are both more coordinate- independent than the Newtonian formulation. Each of them is given in terms of generalized ...
[52]
[PDF] Newtonian Mechanics
Sep 30, 2015 · Systems of N particles are described by in generally coupled system of ODE's consisting of second Newton's laws for each particle miri = Fi ...
[53]
[PDF] Chapter 1 Newtonian particle mechanics - Physics
Classical mechanics begins by analyzing the motion of particles. Classical particles are idealizations: they are pointlike, with no internal degrees of.
[54]
[PDF] Numerical Solutions of Classical Equations of Motion - Physics
While some of the numerical schemes that we will discuss here are particularly suitable for integrat- ing classical equations of motions, we will also described ...Missing: techniques | Show results with:techniques
[55]
[PDF] Phys 410 - Three Formulations of Classical Mechanics. - UMD Physics
Newtonian Forces are closest to our own experiences, and perhaps give us ... 4) The Euler Method for numerical solutions to. ODE, is easy to do. Since.Missing: techniques | Show results with:techniques
[56]
Two-body problem
An isolated dynamical system consisting of two freely moving point objects exerting forces on one another is conventionally termed a two-body problem.
[57]
[PDF] The Two-Body Problem - UCSB Physics
is the reduced mass of the system. Thus, our problem has effectively been reduced to a one-particle system - mathematically, it is no different than a single.
[58]
[PDF] Chapter 13. Newton's Theory of Gravity
Orbital Energetics. We know that for a satellite in a circular orbit, its speed is related to the size of its orbit by v2 = GM/r. The satellite's kinetic ...
[59]
[PDF] force and - newton's laws - KITP
In the SI system of units, mass is measured in kg and acceleration in m/s². To impart an acceleration of 1 m/s² to a mass of 1 kg requires a force of 1 kg·m ...
[60]
[PDF] Advanced General Physics I Lecture 1 Classical Mechanics
There are three dynamical quantities that are conserved for closed systems: momentum, angular momentum and energy. Because these quantities are conserved they ...<|control11|><|separator|>
[61]
Mécanique analytique : Lagrange, J. L. (Joseph Louis), 1736-1813
Jan 18, 2010 · Publication date: 1811 ; Topics: Mechanics, Analytic ; Publisher: Paris, Ve Courcier ; Collection: thomasfisher; universityofottawa; toronto; ...
[62]
13.4: The Lagrangian Equations of Motion - Physics LibreTexts
Aug 7, 2022 · The quantity L = T − V is known as the lagrangian for the system, and Lagrange's equation can then be written. (13.4.16) d d ⁢ t ⁢ ∂ L ...
[63]
[PDF] Hamilton's Principle and Lagrange's Equation - Duke Physics
Properties of the Lagrangian. So Hamilton's principle has given us Eq (1) for the Lagrangian. What do we know about. L beyond the variables it depends on? We ...
[64]
6.4: Lagrange equations from Hamilton's Principle - Physics LibreTexts
Jun 28, 2021 · That is, both Hamilton's Action Principle, and d'Alembert's Principle, can be used to derive Lagrangian mechanics leading to the most general ...
[65]
[PDF] The Lagrangian Method
Given a classical mechanics problem, we can solve it with F = ma, or we can solve it with the E-L equations, which are a consequence of the principle of ...<|control11|><|separator|>
[66]
[PDF] ON A GENERAL METHOD IN DYNAMICS By William Rowan Hamilton
The paper On a General Method in Dynamics has also been republished in The Mathe- matical Papers of Sir William Rowan Hamilton, Volume II: Dynamics, edited for ...
[67]
[PDF] SECOND ESSAY ON A GENERAL METHOD IN DYNAMICS By ...
Second Essay on a General Method in Dynamics. By William Rowan Hamil- ton, Member of several Scientific Societies in Great Britain and in Foreign. Countries ...
[68]
[PDF] Note sur la Théorie de la Variation des constantes arbitraires
Sur la Théorie de la Variation des constantes arbitraires : PAR J LIOUVILLE. Soient η un nombre entier positif, x une fonction de t dont nous désignerons par χ' ...Missing: Joseph | Show results with:Joseph
[69]
[PDF] Classical Mechanics - Rutgers Physics
Oct 5, 2010 · Classical mechanics, narrowly defined, is the investigation of the motion of systems of particles in Euclidean three-dimensional space, ...
[70]
https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/SecEssay.pdf
[71]
[PDF] Quantum wave packets in space and time and an improved criterion ...
Apr 26, 2009 · This useful criterion states that if the physical size of an object is large compared to its de Broglie wavelength, then the behavior of that ...<|separator|>
[72]
Chaos - Stanford Encyclopedia of Philosophy
Jul 16, 2008 · Stephen Kellert defines chaos theory as “the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems” ( ...
[73]
Special Relativity and Its Newtonian Limit from a Group Theoretical ...
It is about an approximation when the relevant velocities of particle motion have magnitudes small relative to the speed of light c, i.e., β i < < 1 . The ...
[74]
Relativistic Energy | Physics - Lumen Learning
Relativistic kinetic energy is KErel = (γ − 1)mc2, where γ = 1 1 − v 2 c 2 . At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
[75]
Relativistic Momentum | Physics - Lumen Learning
p = γmu, where m is the rest mass of the object, u is its velocity relative to an observer, and the relativistic factor γ = 1 1 − u 2 c 2 . At low velocities, ...
[76]
Real-World Relativity: The GPS Navigation System
Mar 11, 2017 · A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.Missing: classical | Show results with:classical
[77]
[1201.0150] What is the limit $\hbar \to 0$ of quantum theory ? - arXiv
Dec 30, 2011 · Abstract:An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit \hbar \to 0.
[78]
[PDF] The WKB Method† 1. Introduction - University of California, Berkeley
The WKB method is important both as a practical means of approximating solutions to the. Schrödinger equation and as a conceptual framework for ...
[79]
Aristotle's Natural Philosophy
May 26, 2006 · Aristotle argues at the opening of Physics bk. 8 that motion and change in the universe can have no beginning, because the occurrence of change ...
[80]
John Buridan - Stanford Encyclopedia of Philosophy
May 13, 2002 · Buridan's major contribution here was to develop and popularize the theory of impetus, or impressed force, to explain projectile motion. ... The ...
[81]
Nicole Oresme - Stanford Encyclopedia of Philosophy
Jul 23, 2009 · Oresme's discussion of the infinite in his Physics Commentary is another fascinating testimony to the originality of this outstanding medieval ...Missing: graphs | Show results with:graphs
[82]
Galileo Galilei - Stanford Encyclopedia of Philosophy
Jun 4, 2021 · Particularly in the cases of the pendulum, the inclined plane, free fall ... Galileo's second new science, in Days Three and Four of the Two New ...
[83]
Newton's apple | Notes and Records of the Royal Society of London
Newton's attention was directed to the possibility of there being a universal force of gravitation through meditation on the fall of an apple from a tree in ...<|control11|><|separator|>
[84]
Isaac Newton (Stanford Encyclopedia of Philosophy)
### Summary of Newton's Apple Anecdote and Correspondence with Hooke on Inverse Square Law
[85]
Leonhard Euler (1707 - 1783) - Biography - MacTutor
He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765). We owe to Euler the notation ...
[86]
[PDF] PDEs of fluids - How Euler Did It
We could summarize Euler's contribution to the subject by saying that he extended the principles described by Archimedes in On floating bodies from statics to ...
[87]
Mécanique Analytique - Cambridge University Press
Joseph-Louis Lagrange. Publisher: Cambridge University Press. Online publication date: February 2015. Print publication year: 2009. First published in: 1788.
[88]
XV. On a general method in dynamics - Journals
Hamilton William Rowan. 1834XV. On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is ...
[89]
VII. Second essay on a general method in dynamics - Journals
Second essay on a general method in dynamics. William Rowan Hamilton. Google Scholar · Find this author on PubMed · Search for more papers by this author.
[90]
Perturbations of a Mystery Planet | Science | AAAS
Neptune was first spotted on this night in 1846. This was the first time that Newton's theory of gravitation had been used to deduce the position of an ...
[91]
Gravitational Perturbations and the Prediction of New Planets
In 1846, the planet Neptune was discovered after its existence was predicted because of discrepancies between calculations and data for the planet Uranus. ...
[92]
[PDF] The Scientific Theories of Michael Faraday and James Clerk Maxwell
The origins of modern electromagnetism and electrodynamics can be traced back to the influential work of two nineteenth century thinkers: Michael Faraday and ...
[93]
Boltzmann's Work in Statistical Physics
Nov 17, 2004 · His earlier contributions clearly belong to the kinetic theory of gases (although his 1868 paper already applies probability to an entire gas ...Missing: 1870s | Show results with:1870s
[94]
[PDF] THE MODERN LEGACIES OF THOMSON'S ATOMIC VORTEX ...
In our tale of two centuries, we have traced the impact of Sir William. Thomson's (Lord Kelvin's) work on his atomic vortex theory in the nine- teenth century ...
[95]
Henri Poincaré - Stanford Encyclopedia of Philosophy
Sep 3, 2013 · Poincaré then considers classical mechanics, which again extends our knowledge while relying on the mathematics that came before it. Finally ...