Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a quantitative measure of the force, and the third asserts that a single isolated force doesn't exist. These three laws have been expressed in several ways, over nearly three centuries,^{[i]} and can be summarised as follows:
First law: | In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.^{[2]}^{[3]} |
Second law: | In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma. (It is assumed here that the mass m is constant – see below.) |
Third law: | When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. |
The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.^{[4]} Newton used them to explain and investigate the motion of many physical objects and systems.^{[5]} For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.
A fourth law is often also described in the bibliography, which states that forces add up like vectors, that is, that forces obey the principle of superposition.^{[6]}^{[7]}^{[8]}
Newton's laws are applied to objects which are idealised as single point masses,^{[9]} in the sense that the size and shape of the object's body are neglected to focus on its motion more easily. This can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star.
In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newton's laws of motion for rigid bodies called Euler's laws of motion, later applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton's laws of motion, then Euler's laws can be derived from Newton's laws. Euler's laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.^{[10]}
Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Some authors interpret the first law as defining what an inertial reference frame is; from this point of view, the second law holds only when the observation is made from an inertial reference frame, and therefore the first law cannot be proved as a special case of the second. Other authors do treat the first law as a corollary of the second.^{[11]}^{[12]} The explicit concept of an inertial frame of reference was not developed until long after Newton's death.
In the given interpretation mass, acceleration, momentum, and (most importantly) force are assumed to be externally defined quantities. This is the most common, but not the only interpretation of the way one can consider the laws to be a definition of these quantities.
Newtonian mechanics has been superseded by special relativity, but it is still useful as an approximation when the speeds involved are much slower than the speed of light.^{[13]}
The first law states that if the net force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Velocity is a vector quantity which expresses both the object's speed and the direction of its motion; therefore, the statement that the object's velocity is constant is a statement that both its speed and the direction of its motion are constant.
The first law can be stated mathematically when the mass is a non-zero constant, as,
Consequently,
This is known as uniform motion. An object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a tablecloth is skilfully whipped from under dishes on a tabletop and the dishes remain in their initial state of rest). If an object is moving, it continues to move without turning or changing its speed. This is evident in space probes that continuously move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely.
Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. The first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed.^{[11]}^{[14]} Newton's first law is often referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero. In this sense, the first law can be restated as:
In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it.^{[15]}
Newton's first and second laws are valid only in an inertial reference frame. Any reference frame that is in uniform motion with respect to an inertial frame is also an inertial frame, i.e. Galilean invariance or the principle of Newtonian relativity.^{[16]}
The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.
The second law can also be stated in terms of an object's acceleration. Since Newton's second law is valid only for constant-mass systems,^{[17]}^{[18]}^{[19]} m can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,
where F is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it. An application of this notation is the derivation of G Subscript C.
Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude; such is the case with uniform circular motion. The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum.
Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below).
Newton's second law is an approximation that is increasingly worse at high speeds because of relativistic effects.
An impulse J occurs when a force F acts over an interval of time Δt, and it is given by^{[20]}^{[21]}
Since force is the time derivative of momentum, it follows that
This relation between impulse and momentum is closer to Newton's wording of the second law.^{[22]}
Impulse is a concept frequently used in the analysis of collisions and impacts.^{[23]}
Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law;^{[18]} that is, the following formula is wrong:^{[19]}
The falsehood of this formula can be seen by noting that it does not respect Galilean invariance: a variable-mass object with F = 0 in one frame will be seen to have F ≠ 0 in another frame.^{[17]} The correct equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is^{[17]}
where u is the velocity of the escaping or incoming mass relative to the body. From this equation one can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket equation. Under some conventions, the quantity u dm/dt on the left-hand side, which represents the advection of momentum, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes F = ma.
The third law states that all forces between two objects exist in equal magnitude and opposite direction: if one object A exerts a force F_{A} on a second object B, then B simultaneously exerts a force F_{B} on A, and the two forces are equal in magnitude and opposite in direction: F_{A} = −F_{B}.^{[24]} The third law means that all forces are interactions between different bodies,^{[25]}^{[26]} or different regions within one body, and thus that there is no such thing as a force that is not accompanied by an equal and opposite force. In some situations, the magnitude and direction of the forces are determined entirely by one of the two bodies, say Body A; the force exerted by Body A on Body B is called the "action", and the force exerted by Body B on Body A is called the "reaction". This law is sometimes referred to as the action-reaction law, with F_{A} called the "action" and F_{B} the "reaction". In other situations the magnitude and directions of the forces are determined jointly by both bodies and it isn't necessary to identify one force as the "action" and the other as the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.^{[24]}
The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road).
From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car push against the road while the road pushes back on the tires—the tires and road simultaneously push against each other. In swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forward—both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.^{[27]}
From the original Latin of Newton's Principia:
Translated to English, this reads:
The ancient Greek philosopher Aristotle had the view that all objects have a natural place in the universe: that heavy objects (such as rocks) wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. He thought that a body was in its natural state when it was at rest, and for the body to move in a straight line at a constant speed an external agent was needed continually to propel it, otherwise it would stop moving. Galileo Galilei, however, realised that a force is necessary to change the velocity of a body, i.e., acceleration, but no force is needed to maintain its velocity. In other words, Galileo stated that, in the absence of a force, a moving object will continue moving. (The tendency of objects to resist changes in motion was what Johannes Kepler had called inertia.) This insight was refined by Newton, who made it into his first law, also known as the "law of inertia"—no force means no acceleration, and hence the body will maintain its velocity. As Newton's first law is a restatement of the law of inertia which Galileo had already described, Newton appropriately gave credit to Galileo.
The law of inertia apparently occurred to several different natural philosophers and scientists independently, including Thomas Hobbes in his Leviathan.^{[29]} The 17th-century philosopher and mathematician René Descartes also formulated the law, although he did not perform any experiments to confirm it.^{[30]}^{[31]}
Newton's original Latin reads:
This was translated quite closely in Motte's 1729 translation as:
According to modern ideas of how Newton was using his terminology,^{[32]} this is understood, in modern terms, as an equivalent of:
The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.
This may be expressed by the formula F = p', where p' is the time derivative of the momentum p. This equation can be seen clearly in the Wren Library of Trinity College, Cambridge, in a glass case in which Newton's manuscript is open to the relevant page.
Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading:
If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.
The sense or senses in which Newton used his terminology, and how he understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern formulations.^{[33]}
Translated to English, this reads:
Newton's Scholium (explanatory comment) to this law:
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.^{[34]}
In the above, as usual, motion is Newton's name for momentum, hence his careful distinction between motion and velocity.
Newton used the third law to derive the law of conservation of momentum;^{[35]} from a deeper perspective, however, conservation of momentum is the more fundamental idea (derived via Noether's theorem from Galilean invariance), and holds in cases where Newton's third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.
Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena.
These three laws hold to a good approximation for macroscopic objects under everyday conditions. However, Newton's laws (combined with universal gravitation and classical electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, very high speeds (in special relativity, the Lorentz factor must be included in the expression for momentum along with the rest mass and velocity) or very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theories, including general relativity and quantum field theory.
In quantum mechanics, concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton's laws are just as exact for these operators as they are for classical objects. At speeds comparable to the speed of light, the second law holds in the original form F = dp/dt, where F and p are four-vectors.
In modern physics, the laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics.
This can be stated simply, "Momentum, energy and angular momentum cannot be created or destroyed."
Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories (e.g., quantum mechanics, quantum electrodynamics, general relativity, etc.). The standard model explains in detail how the three fundamental forces known as gauge forces originate out of exchange by virtual particles. Other forces, such as gravity and fermionic degeneracy pressure, also arise from the momentum conservation. Indeed, the conservation of 4-momentum in inertial motion via curved space-time results in what we call gravitational force in general relativity theory. The application of the space derivative (which is a momentum operator in quantum mechanics) to the overlapping wave functions of a pair of fermions (particles with half-integer spin) results in shifts of maxima of compound wavefunction away from each other, which is observable as the "repulsion" of the fermions.
Newton stated the third law within a world-view that assumed instantaneous action at a distance between material particles. However, he was prepared for philosophical criticism of this action at a distance, and it was in this context that he stated the famous phrase "I feign no hypotheses". In modern physics, action at a distance has been completely eliminated, except for subtle effects involving quantum entanglement. (In particular, this refers to Bell's theorem—that no local model can reproduce the predictions of quantum theory.) Despite only being an approximation, in modern engineering and all practical applications involving the motion of vehicles and satellites, the concept of action at a distance is used extensively.
The discovery of the second law of thermodynamics by Carnot in the 19th century showed that not every physical quantity is conserved over time, thus disproving the validity of inducing the opposite metaphysical view from Newton's laws. Hence, a "steady-state" worldview based solely on Newton's laws and the conservation laws does not take entropy into account.
[...] while Newton had used the word 'body' vaguely and in at least three different meanings, Euler realized that the statements of Newton are generally correct only when applied to masses concentrated at isolated points;
It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass.[Emphasis as in the original]
Recall that F = dP/dt was established for a system composed of a certain set of particles[. ... I]t is essential to deal with the same set of particles throughout the time interval[. ...] Consequently, the mass of the system can not change during the time of interest.
Quoting Newton in the Principia: It is not one action by which the Sun attracts Jupiter, and another by which Jupiter attracts the Sun; but it is one action by which the Sun and Jupiter mutually endeavour to come nearer together.
Any single force is only one aspect of a mutual interaction between two bodies.
That when a thing lies still, unless somewhat else stir it, it will lie still forever, is a truth that no man doubts. But [the proposition] that when a thing is in motion it will eternally be in motion unless somewhat else stay it, though the reason be the same (namely that nothing can change itself), is not so easily assented to. For men measure not only other men but all other things by themselves. And because they find themselves subject after motion to pain and lassitude, [they] think every thing else grows weary of motion and seeks repose of its own accord, little considering whether it be not some other motion wherein that desire of rest they find in themselves, consists.
In classical mechanics, analytical dynamics, or more briefly dynamics, is concerned with the relationship between motion of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies, particularly mass and moment of inertia. The foundation of modern-day dynamics is Newtonian mechanics and its reformulation as Lagrangian mechanics and Hamiltonian mechanics.
Angular accelerationAngular acceleration is the time rate of change of angular velocity. In three dimensions, it is a pseudovector. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (α).
Angular frequencyIn physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function.
Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity.
One revolution is equal to 2π radians, hence
where:
Applied mechanics (also engineering mechanics) is a branch of the physical sciences and the practical application of mechanics. Pure mechanics describes the response of bodies (solids and fluids) or systems of bodies to external forces. Some examples of mechanical systems include the flow of a liquid under pressure, the fracture of a solid from an applied force, or the vibration of an ear in response to sound. A practitioner of the discipline is known as a mechanician.
Applied mechanics describes the behavior of a body, in either a beginning state of rest or of motion, subjected to the action of forces. Applied mechanics, bridges the gap between physical theory and its application to technology. It is used in many fields of engineering, especially mechanical engineering and civil engineering. In this context, it is commonly referred to as Engineering Mechanics. Much of modern engineering mechanics is based on Isaac Newton's laws of motion while the modern practice of their application can be traced back to Stephen Timoshenko, who is said to be the father of modern engineering mechanics.
Within the practical sciences, applied mechanics is useful in formulating new ideas and theories, discovering and interpreting phenomena, and developing experimental and computational tools. In the application of the natural sciences, mechanics was said to be complemented by thermodynamics, the study of heat and more generally energy, and electromechanics, the study of electricity and magnetism.
CFD-DEM modelA CFD-DEM model is suitable for the modeling or simulation of fluid-solids or fluid-particles systems. In a typical CFD-DEM model, the phase motion of discrete solids or particles is obtained by the Discrete Element Method (DEM) which applies Newton's laws of motion to every particle and the flow of continuum fluid is described by the local averaged Navier–Stokes equations that can be solved by the traditional Computational Fluid Dynamics (CFD). The model is first proposed by Tsuji et al. The interactions between the fluid phase and solids phase is better modeled according to Newton's third law.
Circular motionIn physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.
Examples of circular motion include: an artificial satellite orbiting the Earth at a constant height, a ceiling fan's blades rotating around a hub, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.
Classical physicsClassical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the realm of "classical physics".
As such, the definition of a classical theory depends on context. Classical physical concepts are often used when modern theories are unnecessarily complex for a particular situation. Most usually classical physics refers to pre-1900 physics, while modern physics refers to post-1900 physics which incorporates elements of quantum mechanics and relativity.
Dynamics (mechanics)Dynamics is the branch of classical mechanics concerned with the study of forces and their effects on motion. Isaac Newton defined the fundamental physical laws which govern dynamics in physics, especially his second law of motion.
Euler's laws of motionIn classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.
History of classical mechanicsThis article deals with the history of classical mechanics.
InertiaInertia is the resistance, of any physical object, to any change in its velocity. This includes changes to the object's speed, or direction of motion.
An aspect of this property is the tendency of objects to keep moving in a straight line at a constant speed, when no forces are upon them—and this aspect in particular is also called inertia.
The principle of inertia is one of the fundamental principles in classical physics that are still used today to describe the motion of objects and how they are affected by the applied forces on them.
Inertia comes from the Latin word, iners, meaning idle, sluggish. Inertia is one of the primary manifestations of mass, which is a quantitative property of physical systems. Isaac Newton defined inertia as his first law in his Philosophiæ Naturalis Principia Mathematica, which states:
The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.
In common usage, the term "inertia" may refer to an object's "amount of resistance to change in velocity" (which is quantified by its mass), or sometimes to its momentum, depending on the context. The term "inertia" is more properly understood as shorthand for "the principle of inertia" as described by Newton in his First Law of Motion: an object not subject to any net external force moves at a constant velocity. Thus, an object will continue moving at its current velocity until some force causes its speed or direction to change.
On the surface of the Earth, inertia is often masked by the effects of friction and air resistance, both of which tend to decrease the speed of moving objects (commonly to the point of rest), and gravity. This misled the philosopher Aristotle to believe that objects would move only as long as force was applied to them:
...it [body] stops when the force which is pushing the travelling object has no longer power to push it along...
Laws of infernal dynamicsThe laws of infernal dynamics are an adage about the cursedness of the universe. Attributed to Science fiction author David Gerrold, the laws are as follows:
An object in motion will be moving in the wrong direction.
An object at rest will be in the wrong place.
The energy required to move an object in the correct direction, or put it in the right place, will be more than you wish to expend but not so much as to make the task impossible.The laws are a parody on the first and second of Newton's laws of motion in the spirit of Murphy's law. Newton's first law of motion has here been split into two parts, the first two laws. Newton's third law of motion is left unparodied, though a separate adage states that "for every action, there is an equal and opposite criticism."
MechanicsMechanics (Greek μηχανική) is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.
The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo, Kepler, and Newton laid the foundation for what is now known as classical mechanics.
It is a branch of classical physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light.
It can also be defined as a branch of science which deals with the motion of and forces on objects. The field is yet less widely understood in terms of quantum theory.
Second lawSecond law may refer to:
the second of Newton's laws of motion
Second law of thermodynamics
Second Law, or in Greek Deuterosis, the talmudic Mishna(h), which comes after the Law of Moses
The 2nd Law, a music album by the English rock band Muse
Mendel's second law of independent assortment
Statically indeterminateIn statics, a structure is statically indeterminate (or hyperstatic) when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure.
Based on Newton's laws of motion, the equilibrium equations available for a two-dimensional body are:
In the beam construction on the right, the four unknown reactions are V_{A}, V_{B}, V_{C} and H_{A}. The equilibrium equations are:
Since there are four unknown forces (or variables) (V_{A}, V_{B}, V_{C} and H_{A}) but only three equilibrium equations, this system of simultaneous equations does not have a unique solution. The structure is therefore classified as statically indeterminate. Considerations in the material properties and compatibility in deformations are taken to solve statically indeterminate systems or structures.
Structural engineering theoryStructural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully structural engineers will need a detailed knowledge of mathematics and of relevant empirical and theoretical design codes. They will also need to know about the corrosion resistance of the materials and structures, especially when those structures are exposed to the external environment.
The criteria which govern the design of a structure are either serviceability (criteria which define whether the structure is able to adequately fulfill its function) or strength (criteria which define whether a structure is able to safely support and resist its design loads). A structural engineer designs a structure to have sufficient strength and stiffness to meet these criteria.
Loads imposed on structures are supported by means of forces transmitted through structural elements. These forces can manifest themselves as tension (axial force), compression (axial force), shear, and bending, or flexure (a bending moment is a force multiplied by a distance, or lever arm, hence producing a turning effect or torque).
Time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of classical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion. These principles can also be equivalently expressed more abstractly by Hamiltonian mechanics or Lagrangian mechanics.
The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is discrete.
Stateful systems often have dual descriptions in terms of states or in terms of observable values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in quantum mechanics where the Schrödinger picture and Heisenberg picture are (mostly) equivalent descriptions of time evolution.
This page is based on a Wikipedia article written by authors
(here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.