Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [2] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space. [3]
Classical mechanics utilises many equations —as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory. [4] This article gives a summary of the most important of these.
This article lists equations from Newtonian mechanics, see analytical mechanics for the more general formulation of classical mechanics (which includes Lagrangian and Hamiltonian mechanics).
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Linear, surface, volumetric mass density | λ or μ (especially in acoustics, see below) for Linear, σ for surface, ρ for volume. | kg m−n, n = 1, 2, 3 | M L−n | |
Moment of mass [5] | m (No common symbol) | Point mass: Discrete masses about an axis : Continuum of mass about an axis : | kg m | M L |
Center of mass | rcom (Symbols vary) | i-th moment of mass Discrete masses: Mass continuum: | m | L |
2-Body reduced mass | m12, μ Pair of masses = m1 and m2 | kg | M | |
Moment of inertia (MOI) | I | Discrete Masses: Mass continuum: | kg m2 | M L2 |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Velocity | v | m s−1 | L T−1 | |
Acceleration | a | m s−2 | L T−2 | |
Jerk | j | m s−3 | L T−3 | |
Jounce | s | m s−4 | L T−4 | |
Angular velocity | ω | rad s−1 | T−1 | |
Angular Acceleration | α | rad s−2 | T−2 | |
Angular jerk | ζ | rad s−3 | T−3 |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Momentum | p | kg m s−1 | M L T−1 | |
Force | F | N = kg m s−2 | M L T−2 | |
Impulse | J, Δp, I | kg m s−1 | M L T−1 | |
Angular momentum about a position point r0, | L, J, S | Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point. | kg m2 s−1 | M L2 T−1 |
Moment of a force about a position point r0, | τ, M | N m = kg m2 s−2 | M L2 T−2 | |
Angular impulse | ΔL (no common symbol) | kg m2 s−1 | M L2 T−1 |
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Mechanical work due to a Resultant Force | W | J = N m = kg m2 s−2 | M L2 T−2 | |
Work done ON mechanical system, Work done BY | WON, WBY | J = N m = kg m2 s−2 | M L2 T−2 | |
Potential energy | φ, Φ, U, V, Ep | J = N m = kg m2 s−2 | M L2 T−2 | |
Mechanical power | P | W = J s−1 | M L2 T−3 |
Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Generalized coordinates | q, Q | varies with choice | varies with choice | |
Generalized velocities | varies with choice | varies with choice | ||
Generalized momenta | p, P | varies with choice | varies with choice | |
Lagrangian | L | where and p = p(t) are vectors of the generalized coords and momenta, as functions of time | J | M L2 T−2 |
Hamiltonian | H | J | M L2 T−2 | |
Action, Hamilton's principal function | S, | J s | M L2 T−1 |
In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector
defines the axis of rotation, = unit vector in direction of r, = unit vector tangential to the angle.
Translation | Rotation | |
---|---|---|
Velocity | Average: Instantaneous: | Angular velocity Rotating rigid body: |
Acceleration | Average: Instantaneous: | Angular acceleration Rotating rigid body: |
Jerk | Average: Instantaneous: | Angular jerk Rotating rigid body: |
Translation | Rotation | |
---|---|---|
Momentum | Momentum is the "amount of translation" For a rotating rigid body: | Angular momentum Angular momentum is the "amount of rotation": and the cross-product is a pseudovector i.e. if r and p are reversed in direction (negative), L is not. In general I is an order-2 tensor, see above for its components. The dot · indicates tensor contraction. |
Force and Newton's 2nd law | Resultant force acts on a system at the center of mass, equal to the rate of change of momentum: For a number of particles, the equation of motion for one particle i is: [7] where pi = momentum of particle i, Fij = force on particle iby particle j, and FE = resultant external force (due to any agent not part of system). Particle i does not exert a force on itself. | Torque Torque τ is also called moment of a force, because it is the rotational analogue to force: [8] For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation: Likewise, for a number of particles, the equation of motion for one particle i is: [9] |
Yank | Yank is rate of change of force: For constant mass, it becomes; | Rotatum Rotatum Ρ is also called moment of a Yank, because it is the rotational analogue to yank: |
Impulse | Impulse is the change in momentum: For constant force F: | Twirl/angular impulse is the change in angular momentum: For constant torque τ: |
The precession angular speed of a spinning top is given by:
where w is the weight of the spinning flywheel.
The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:
The work done W by an external agent which exerts a force F (at r) and torque τ on an object along a curved path C is:
where θ is the angle of rotation about an axis defined by a unit vector n.
The change in kinetic energy for an object initially traveling at speed and later at speed is:
For a stretched spring fixed at one end obeying Hooke's law, the elastic potential energy is
where r2 and r1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.
Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is: [10]
where I is the moment of inertia tensor.
The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,
the following general results apply to the particle.
Kinematics | Dynamics |
---|---|
Position | |
Velocity | Momentum Angular momenta |
Acceleration | The centripetal force is where again m is the mass moment, and the Coriolis force is The Coriolis acceleration and force can also be written: |
For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:
These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).
Linear motion | Angular motion |
---|---|
For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.
Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.
Motion of entities | Inertial frames | Accelerating frames |
---|---|---|
Translation V = Constant relative velocity between two inertial frames F and F'. | Relative position Relative velocity Equivalent accelerations | Relative accelerations Apparent/fictitious forces |
Rotation Ω = Constant relative angular velocity between two frames F and F'. | Relative angular position Relative velocity Equivalent accelerations | Relative accelerations Apparent/fictitious torques |
Transformation of any vector T to a rotating frame |
SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.
Physical situation | Nomenclature | Translational equations | Angular equations |
---|---|---|---|
SHM |
| Solution: | Solution: |
Unforced DHM |
| Solution (see below for ω'): Resonant frequency: Damping rate: Expected lifetime of excitation: | Solution: Resonant frequency: Damping rate: Expected lifetime of excitation: |
Physical situation | Nomenclature | Equations |
---|---|---|
Linear undamped unforced SHO |
| |
Linear unforced DHO |
| |
Low amplitude angular SHO |
| |
Low amplitude simple pendulum |
| Approximate value Exact value can be shown to be: |
Physical situation | Nomenclature | Equations |
---|---|---|
SHM energy |
| Potential energy Maximum value at x = A: Kinetic energy Total energy |
DHM energy |
Angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.
A centripetal force is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
In mechanics and physics, simple harmonic motion is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely.
In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M. Just as a linear force is a push or a pull applied to a body, a torque can be thought of as a twist applied to an object with respect to a chosen point; for example, driving a screw uses torque to force it into an object, which is applied by the screwdriver rotating around its axis to the drives on the head.
The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.
Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.
In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.
In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.
In physics, circular motion is movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, which remains at a constant distance from the axis of rotation. In circular motion, the distance between the body and a fixed point on its surface remains the same, i.e., the body is assumed rigid.
Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particular case of projectile motion on Earth, most calculations assume the effects of air resistance are passive.
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. Fictitious forces are invoked to maintain the validity and thus use of Newton's second law of motion, in frames of reference which are not inertial.
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.
Rotation around a fixed axis or axial rotation is a special case of rotational motion around an axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result.
In classical mechanics, Appell's equation of motion is an alternative general formulation of classical mechanics described by Josiah Willard Gibbs in 1879 and Paul Émile Appell in 1900.
In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.
Velocity is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.