Classical mechanics

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Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

If the present state of an object is known it is possible to predict by the laws of classical mechanics how it will move in the future (determinism) and how it has moved in the past (reversibility).

The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts employed by and the mathematical methods invented by Isaac Newton and Gottfried Wilhelm Leibniz and others in the 17th century to describe the motion of bodies under the influence of a system of forces.

Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond Newton's work, particularly through their use of analytical mechanics. They are, with some modification, also used in all areas of modern physics.

Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In case that objects become extremely massive, General relativity becomes applicable. However, a number of modern sources do include relativistic mechanics into classical physics, which in their view represents classical mechanics in its most developed and accurate form.

Description of the theory

The analysis of projectile motion is a part of classical mechanics.

The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as point particles (objects with negligible size). The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it. Each of these parameters is discussed in turn.

In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The physics of very small particles, such as the electron, is more accurately described by quantum mechanics.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom, e.g., a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles. The center of mass of a composite object behaves like a point particle.

Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously).

Position and its derivatives

The SI derived "mechanical"
(that is, not electromagnetic or thermal)
units with kg, m and s

position	m
angular position/angle	unitless (radian)
velocity	m·s−1
angular velocity	s−1
acceleration	m·s−2
angular acceleration	s−2
jerk	m·s−3
"angular jerk"	s−3
specific energy	m2·s−2
absorbed dose rate	m2·s−3
moment of inertia	kg·m2
momentum	kg·m·s−1
angular momentum	kg·m2·s−1
force	kg·m·s−2
torque	kg·m2·s−2
energy	kg·m2·s−2
power	kg·m2·s−3
pressure and energy density	kg·m−1·s−2
surface tension	kg·s−2
spring constant	kg·s−2
irradiance and energy flux	kg·s−3
kinematic viscosity	m2·s−1
dynamic viscosity	kg·m−1·s−1
density (mass density)	kg·m−3
density (weight density)	kg·m−2·s−2
number density	m−3
action	kg·m2·s−1

The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O. In cases where P is moving relative to O, r is defined as a function of t, time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval that is observed to elapse between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.

Velocity and speed

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time:

${\displaystyle \mathbf {v} ={\mathrm {d} \mathbf {r} \over \mathrm {d} t}\,\!}$.

In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at 60 − 50 = 10 km/h. However, from the perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as -10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.

Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve, where u is the speed of the first object, v is the speed of the second object, and d and e are unit vectors in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is

${\displaystyle \mathbf {u} '=\mathbf {u} -\mathbf {v} \,.}$

Similarly, the first object sees the velocity of the second object as

${\displaystyle \mathbf {v'} =\mathbf {v} -\mathbf {u} \,.}$

When both objects are moving in the same direction, this equation can be simplified to

${\displaystyle \mathbf {u} '=(u-v)\mathbf {d} \,.}$

Or, by ignoring direction, the difference can be given in terms of speed only:

${\displaystyle u'=u-v\,.}$

Acceleration

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time):

${\displaystyle \mathbf {a} ={\mathrm {d} \mathbf {v} \over \mathrm {d} t}={\mathrm {d^{2}} \mathbf {r} \over \mathrm {d} t^{2}}.}$

Acceleration represents the velocity's change over time. Velocity can change in either magnitude or direction, or both. Occasionally, a decrease in the magnitude of velocity "v" is referred to as deceleration, but generally any change in the velocity over time, including deceleration, is simply referred to as acceleration.

Frames of reference

While the position, velocity and acceleration of a particle can be described with respect to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames.

An inertial frame is a frame of reference within which an object interacting with no forces (an idealized situation) appears either at rest or moving uniformly in a straight line. This is the fundamental definition of an inertial frame. These are characterized by the requirement that all forces entering the observer's physical laws originate from identifiable sources caused by fields, such as electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), gravitational field (caused by mass), and so forth.

A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that do not accelerate with respect to distant stars (an extremely distant point) are regarded as good approximations to inertial frames. Non-inertial reference frames accelerate in relation to an existing inertial frame. They form the basis for Einstein's relativity. Due to the relative motion, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration. These forces are referred to as fictitious forces, inertia forces, or pseudo-forces.

Consider two reference frames S and S'. For observers in each of the reference frames an event has space-time coordinates of (x,y,z,t) in frame S and (x',y',z',t') in frame S'. Assuming time is measured the same in all reference frames, and if we require x = x' when t = 0, then the relation between the space-time coordinates of the same event observed from the reference frames S' and S, which are moving at a relative velocity of u in the x direction is:

${\displaystyle x'=x-ut\,}$

${\displaystyle y'=y\,}$

${\displaystyle z'=z\,}$

${\displaystyle t'=t\,.}$

This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). This group is a limiting case of the Poincaré group used in special relativity. The limiting case applies when the velocity u is very small compared to c, the speed of light.

The transformations have the following consequences:

• v′ = v − u (the velocity v′ of a particle from the perspective of S′ is slower by u than its velocity v from the perspective of S)
• a′ = a (the acceleration of a particle is the same in any inertial reference frame)
• F′ = F (the force on a particle is the same in any inertial reference frame)
• the speed of light is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics have a counterpart in classical mechanics.

For some problems, it is convenient to use rotating coordinates (reference frames). Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious centrifugal force and Coriolis force.

Forces; Newton's second law

Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":

${\displaystyle \mathbf {F} ={\mathrm {d} \mathbf {p} \over \mathrm {d} t}={\mathrm {d} (m\mathbf {v} ) \over \mathrm {d} t}.}$

The quantity mv is called the (canonical) momentum. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is a = dv/dt, the second law can be written in the simplified and more familiar form:

${\displaystyle \mathbf {F} =m\mathbf {a} \,.}$

So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion.

As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example:

${\displaystyle \mathbf {F} _{\rm {R}}=-\lambda \mathbf {v} \,,}$

where λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is

${\displaystyle -\lambda \mathbf {v} =m\mathbf {a} =m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}\,.}$

This can be integrated to obtain

${\displaystyle \mathbf {v} =\mathbf {v} _{0}e^{\frac {-\lambda t}{m}}}$

where v₀ is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the conservation of energy), and the particle is slowing down. This expression can be further integrated to obtain the position r of the particle as a function of time.

Important forces include the gravitational force and the Lorentz force for electromagnetism. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, −F, on A. The strong form of Newton's third law requires that F and −F act along the line connecting A and B, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.

Work and energy

If a constant force F is applied to a particle that makes a displacement Δr, the work done by the force is defined as the scalar product of the force and displacement vectors:

${\displaystyle W=\mathbf {F} \cdot \Delta \mathbf {r} \,.}$

More generally, if the force varies as a function of position as the particle moves from r₁ to r₂ along a path C, the work done on the particle is given by the line integral

${\displaystyle W=\int _{C}\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} \,.}$

If the work done in moving the particle from r₁ to r₂ is the same no matter what path is taken, the force is said to be conservative. Gravity is a conservative force, as is the force due to an idealized spring, as given by Hooke's law. The force due to friction is non-conservative.

The kinetic energy E_k of a particle of mass m travelling at speed v is given by

${\displaystyle E_{\mathrm {k} }={\tfrac {1}{2}}mv^{2}\,.}$

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.

The work–energy theorem states that for a particle of constant mass m, the total work W done on the particle as it moves from position r₁ to r₂ is equal to the change in kinetic energy E_k of the particle:

${\displaystyle W=\Delta E_{\mathrm {k} }=E_{\mathrm {k,2} }-E_{\mathrm {k,1} }={\tfrac {1}{2}}m\left(v_{2}^{\,2}-v_{1}^{\,2}\right)\,.}$

Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted E_p:

${\displaystyle \mathbf {F} =-\mathbf {\nabla } E_{\mathrm {p} }\,.}$

If all the forces acting on a particle are conservative, and E_p is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force

${\displaystyle \mathbf {F} \cdot \Delta \mathbf {r} =-\mathbf {\nabla } E_{\mathrm {p} }\cdot \Delta \mathbf {r} =-\Delta E_{\mathrm {p} }\,.}$

The decrease in the potential energy is equal to the increase in the kinetic energy

${\displaystyle -\Delta E_{\mathrm {p} }=\Delta E_{\mathrm {k} }\Rightarrow \Delta (E_{\mathrm {k} }+E_{\mathrm {p} })=0\,.}$

This result is known as conservation of energy and states that the total energy,

${\displaystyle \sum E=E_{\mathrm {k} }+E_{\mathrm {p} }\,,}$

is constant in time. It is often useful, because many commonly encountered forces are conservative.

Beyond Newton's laws

Classical mechanics also describes the more complex motions of extended non-pointlike objects. Euler's laws provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. The rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass".

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates.

The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector divided by c², where c is the speed of light in free space.

Limits of validity

Domain of validity for Classical Mechanics

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. Geometric optics is an approximation to the quantum theory of light, and does not have a superior "classical" form.

When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory (QFT) is of use. QFT deals with small distances and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level, statistical mechanics becomes useful. Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in thermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. In case that objects become extremely heavy (i.e. their Schwarzschild radius is not negligibly small for a given application), deviations from Newtonian mechanics become apparent and can be quantified by using the Parameterized post-Newtonian formalism. In that case, General relativity (GR) becomes applicable. However, until now there is no theory of Quantum gravity unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.

The Newtonian approximation to special relativity

In special relativity, the momentum of a particle is given by

${\displaystyle \mathbf {p} ={\frac {m\mathbf {v} }{\sqrt {1-v^{2}/c^{2}}}}\,,}$

where m is the particle's rest mass, v its velocity, v is the modulus of v, and c is the speed of light.

If v is very small compared to c, v²/c² is approximately zero, and so

${\displaystyle \mathbf {p} \approx m\mathbf {v} \,.}$

Thus the Newtonian equation p = mv is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light.

For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by

${\displaystyle f=f_{\mathrm {c} }{\frac {m_{0}}{m_{0}+T/c^{2}}}\,,}$

where f_c is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m₀ circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage.

The classical approximation to quantum mechanics

The ray approximation of classical mechanics breaks down when the de Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is

${\displaystyle \lambda ={\frac {h}{p}}}$

where h is Planck's constant and p is the momentum.

Again, this happens with electrons before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 V, had a wavelength of 0.167 nm, which was long enough to exhibit a single diffraction side lobe when reflecting from the face of a nickel crystal with atomic spacing of 0.215 nm. With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution from around a radian to a milliradian and see quantum diffraction from the periodic patterns of integrated circuit computer memory.

More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated circuits.

Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.

History

The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology,

Some Greek philosophers of antiquity, among them Aristotle, founder of Aristotelian physics, may have been the first to maintain the idea that "everything happens for a reason" and that theoretical principles can assist in the understanding of nature. While to a modern reader, many of these preserved ideas come forth as eminently reasonable, there is a conspicuous lack of both mathematical theory and controlled experiment, as we know it. These later became decisive factors in forming modern science, and their early application came to be known as classical mechanics.

In his Elementa super demonstrationem ponderum, medieval mathematician Jordanus de Nemore introduced the concept of "positional gravity" and the use of component forces.

Three stage Theory of impetus according to Albert of Saxony.

The first published causal explanation of the motions of planets was Johannes Kepler's Astronomia nova, published in 1609. He concluded, based on Tycho Brahe's observations on the orbit of Mars, that the planet's orbits were ellipses. This break with ancient thought was happening around the same time that Galileo was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannonballs of different weights from the tower of Pisa, showing that they both hit the ground at the same time. The reality of that particular experiment is disputed, but he did carry out quantitative experiments by rolling balls on an inclined plane. His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics.

Sir Isaac Newton (1643–1727), an influential figure in the history of physics and whose three laws of motion form the basis of classical mechanics

Newton founded his principles of natural philosophy on three proposed laws of motion: the law of inertia, his second law of acceleration (mentioned above), and the law of action and reaction; and hence laid the foundations for classical mechanics. Both Newton's second and third laws were given the proper scientific and mathematical treatment in Newton's Philosophiæ Naturalis Principia Mathematica. Here they are distinguished from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. Newton also enunciated the principles of conservation of momentum and angular momentum. In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of gravity in Newton's law of universal gravitation. The combination of Newton's laws of motion and gravitation provide the fullest and most accurate description of classical mechanics. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of Kepler's laws of motion of the planets.

Newton had previously invented the calculus, of mathematics, and used it to perform the mathematical calculations. For acceptability, his book, the Principia, was formulated entirely in terms of the long-established geometric methods, which were soon eclipsed by his calculus. However, it was Leibniz who developed the notation of the derivative and integral preferred today.

Hamilton's greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics.

Newton, and most of his contemporaries, with the notable exception of Huygens, worked on the assumption that classical mechanics would be able to explain all phenomena, including light, in the form of geometric optics. Even when discovering the so-called Newton's rings (a wave interference phenomenon) he maintained his own corpuscular theory of light.

After Newton, classical mechanics became a principal field of study in mathematics as well as physics. Several re-formulations progressively allowed finding solutions to a far greater number of problems. The first notable re-formulation was in 1788 by Joseph Louis Lagrange. Lagrangian mechanics was in turn re-formulated in 1833 by William Rowan Hamilton.

Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. Some of these difficulties related to compatibility with electromagnetic theory, and the famous Michelson–Morley experiment. The resolution of these problems led to the special theory of relativity, often still considered a part of classical mechanics.

A second set of difficulties were related to thermodynamics. When combined with thermodynamics, classical mechanics leads to the Gibbs paradox of classical statistical mechanics, in which entropy is not a well-defined quantity. Black-body radiation was not explained without the introduction of quanta. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms and the photo-electric effect. The effort at resolving these problems led to the development of quantum mechanics.

Since the end of the 20th century, classical mechanics in physics has no longer been an independent theory. Instead, classical mechanics is now considered an approximate theory to the more general quantum mechanics. Emphasis has shifted to understanding the fundamental forces of nature as in the Standard model and its more modern extensions into a unified theory of everything. Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields. Also, it has been extended into the complex domain where complex classical mechanics exhibits behaviors very similar to quantum mechanics.

Quantum predictions

November 2, 2018, US Department of Energy
Original link: https://phys.org/news/2018-11-quantum.html?utm_source=menu&utm_medium=link&utm_campaign=item-menu

Mechanical strain, pressure or temperature changes or adding chemical doping agents can prompt an abrupt switch from insulator to conductor in materials such as nickel oxide (pictured here). Nickel ions (blue) and oxygen ions (red) surround a dopant ion of potassium (yellow). Quantum Monte Carlo methods can accurately predict regions where charge density (purple) will accumulate in these materials. Credit: Anouar Benali, Argonne National Laboratory

Read more at: https://phys.org/news/2018-11-quantum.html#jCp

Solving a complex problem quickly requires careful tradeoffs – and simulating the behavior of materials is no exception. To get answers that predict molecular workings feasibly, scientists must swap in mathematical approximations that speed computation at accuracy's expense.

But magnetism, electrical conductivity and other properties can be quite delicate, says Paul R.C. Kent of the Department of Energy's (DOE's) Oak Ridge National Laboratory. These properties depend on quantum mechanics, the movements and interactions of myriad electrons and atoms that form materials and determine their properties. Researchers who study such features must model large groups of atoms and molecules rather than just a few. This problem's complexity demands boosting computational tools' efficiency and accuracy.

That's where a method called quantum Monte Carlo (QMC) modeling comes in. Many other techniques approximate electrons' behavior as an overall average, for example, rather than considering them individually. QMC enables accounting for the individual behavior of all of the electrons without major approximations, reducing systematic errors in simulations and producing reliable results, Kent says.

Kent's interest in QMC dates back to his Ph.D. research at Cambridge University in the 1990s. At ORNL, he recently returned to the method because advances in both supercomputer hardware and in algorithms had allowed researchers to improve its accuracy.

"We can do new materials and a wider fraction of elements across the periodic table," Kent says. "More importantly, we can start to do some of the materials and properties where the more approximate methods that we use day to day are just unreliable."

Even with these advances, simulations of these types of materials, ones that include up to a few hundred atoms and thousands of electrons, requires computational heavy lifting. Kent leads a DOE Basic Energy Sciences Center, the Center for Predictive Simulations of Functional Materials (CPSFM) that includes researchers from ORNL, Argonne National Laboratory, Sandia National Laboratories, Lawrence Livermore National Laboratory, the University of California, Berkeley and North Carolina State University.

Their work is supported by a DOE Innovative and Novel Computational Impact on Theory and Experiments (INCITE) allocation of 140 million processor hours, split between Oak Ridge Leadership Computing Facility's Titan and Argonne Leadership Computing Facility's Mira supercomputers. Both computing centers are DOE Office of Science user facilities.

To take QMC to the next level, Kent and colleagues start with materials such as vanadium dioxide that display unusual electronic behavior. At cooler temperatures, this material insulates against the flow of electricity. But at just above room temperature, vanadium dioxide abruptly changes its structure and behavior.

Suddenly this material becomes metallic and conducts electricity efficiently. Scientists still don't understand exactly how and why this occurs. Factors such as mechanical strain, pressure or doping the materials with other elements also induce this rapid transition from insulator to conductor.

However, if scientists and engineers could control this behavior, these materials could be used as switches, sensors or, possibly, the basis for new electronic devices. "This big change in conductivity of a material is the type of thing we'd like to be able to predict reliably," Kent says.

Laboratory researchers also are studying these insulator-to-conductors with experiments. That validation effort lends confidence to the predictive power of their computational methods in a range of materials. The team has built open-source software, known as QMCPACK, that is now available online and on all of the DOE Office of Science computational facilities.

Kent and his colleagues hope to build up to high-temperature superconductors and other complex and mysterious materials. Although scientists know these materials' broad properties, Kent says, "we can't relate those to the actual structure and the elements in the materials yet. So that's a really grand challenge for the condensed-matter physics field."

The most accurate quantum mechanical modeling methods restrict scientists to examining just a few atoms or molecules. When scientists want to study larger systems, the computation costs rapidly become unwieldy. QMC offers a compromise: a calculation's size increases cubically relative to the number of electrons, a more manageable challenge. QMC incorporates only a few controlled approximations and can be applied to the numerous atoms and electrons needed. It's well suited for today's petascale supercomputers – capable of one quadrillion calculations or more each second – and tomorrow's exascale supercomputers, which will be at least a thousand times faster. The method maps simulation elements relatively easily onto the compute nodes in these systems.

The CPSFM team continues to optimize QMCPACK for ever-faster supercomputers, including OLCF's Summit, which will be fully operational in January 2019. The higher memory capacity on that machine's Nvidia Volta GPUs – 16 gigabytes per graphics processing unit compared with 6 gigabytes on Titan – already boosts computation speed. With the help of OLCF's Ed D'Azevedo and Andreas Tillack, the researchers have implemented improved algorithms that can double the speed of their larger calculations.

QMCPACK is part of DOE's Exascale Computing Project, and the team is already anticipating additional scaling challenges for running QMCPACK on future machines. To perform the desired simulations within roughly 12 hours on an exascale supercomputer, Kent estimates that they'll need algorithms that are 30 times more scalable than those within the current version.

Even with improved hardware and algorithms, QMC calculations will always be expensive. So Kent and his team would like to use QMCPACK to understand where cheaper methods go wrong so that they can improve them. Then they can save QMC calculations for the most challenging problems in materials science, Kent says. "Ideally we will learn what's causing these materials to be very tricky to model and then improve cheaper approaches so that we can do much wider scans of different materials."

The combination of improved QMC methods and a suite of computationally cheaper modeling approaches could lead the way to new materials and an understanding of their properties. Designing and testing new compounds in the laboratory is expensive, Kent says. Scientists could save valuable time and resources if they could first predict the behavior of novel materials in a simulation.

Plus, he notes, reliable computational methods could help scientists understand properties and processes that depend on individual atoms that are extremely difficult to observe using experiments. "That's a place where there's a lot of interest in going after the fundamental science, predicting new materials and enabling technological applications."

Orbital elements

From Wikipedia, the free encyclopedia

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

A real orbit (and its elements) changes over time due to gravitational perturbations by other objects and the effects of relativity. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time.

Keplerian elements

In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For Earth-orbiting satellites, the reference plane is usually the Earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. The reference plane, together with the vernal point (♈︎), establishes a reference frame.

The traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion.

When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. When viewed from a non-inertial frame centred on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body is called the primary, the other body is called the secondary. The primary does not necessarily possess more mass than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary.

The main two elements that define the shape and size of the ellipse:

• Eccentricity (e)—shape of the ellipse, describing how much it is elongated compared to a circle (not marked in diagram).
• Semimajor axis (a)—the sum of the periapsis and apoapsis distances divided by two. For circular orbits, the semimajor axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass.

Two elements define the orientation of the orbital plane in which the ellipse is embedded:

• Inclination (i)—vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane, the green angle i in the diagram). Tilt angle is measured perpendicular to line of intersection between orbital plane and reference plane. Any three points on an ellipse will define the ellipse orbital plane. The plane and the ellipse are both two-dimensional objects defined in three-dimensional space.
• Longitude of the ascending node (Ω)—horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane, symbolized by ☊) with respect to the reference frame's vernal point (symbolized by ♈︎). This is measured in the reference plane, and is shown as the green angle Ω in the diagram.

And finally:

• Argument of periapsis (ω) defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis (the closest point the satellite object comes to the primary object around which it orbits, the blue angle ω in the diagram).
• True anomaly (ν, θ, or f) at epoch (M₀) defines the position of the orbiting body along the ellipse at a specific time (the "epoch").

The mean anomaly is a mathematically convenient "angle" which varies linearly with time, but which does not correspond to a real geometric angle. It can be converted into the true anomaly ν, which does represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the central body) and the position of the orbiting object at any given time. Thus, the true anomaly is shown as the red angle ν in the diagram, and the mean anomaly is not shown.

The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system.

Note that non-elliptic trajectories also exist, but are not closed, and are thus not orbits. If the eccentricity is greater than one, the trajectory is a hyperbola. If the eccentricity is equal to one and the angular momentum is zero, the trajectory is radial. If the eccentricity is one and there is angular momentum, the trajectory is a parabola.

Required parameters

Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

This is because the problem contains six degrees of freedom. These correspond to the three spatial dimensions which define position (x, y, z in a Cartesian coordinate system), plus the velocity in each of these dimensions. These can be described as orbital state vectors, but this is often an inconvenient way to represent an orbit, which is why Keplerian elements are commonly used instead.

Sometimes the epoch is considered a "seventh" orbital parameter, rather than part of the reference frame.

If the epoch is defined to be at the moment when one of the elements is zero, the number of unspecified elements is reduced to five. (The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.)

Alternative parametrizations

Keplerian elements can be obtained from orbital state vectors (a three-dimensional vector for the position and another for the velocity) by manual transformations or with computer software.

Other orbital parameters can be computed from the Keplerian elements such as the period, apoapsis, and periapsis. (When orbiting the Earth, the last two terms are known as the apogee and perigee.) It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter, GM, is given for the central body.

Instead of the mean anomaly at epoch, the mean anomaly M, mean longitude, true anomaly ν₀, or (rarely) the eccentric anomaly might be used.

Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a seventh orbital element. Sometimes it is assumed that mean anomaly is zero at the epoch (by choosing the appropriate definition of the epoch), leaving only the five other orbital elements to be specified.

Different sets of elements are used for various astronomical bodies. The eccentricity, e, and either the semi-major axis, a, or the distance of periapsis, q, are used to specify the shape and size of an orbit. The angle of the ascending node, Ω, the inclination, i, and the argument of periapsis, ω, or the longitude of periapsis, ϖ, specify the orientation of the orbit in its plane. Either the longitude at epoch, L₀, the mean anomaly at epoch, M₀, or the time of perihelion passage, T₀, are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or the node are used as the primary reference. The semi-major axis is known if the mean motion and the gravitational mass are known.

It is also quite common to see either the mean anomaly (M) or the mean longitude (L) expressed directly, without either M₀ or L₀ as intermediary steps, as a polynomial function with respect to time. This method of expression will consolidate the mean motion (n) into the polynomial as one of the coefficients. The appearance will be that L or M are expressed in a more complicated manner, but we will appear to need one fewer orbital element.

Mean motion can also be obscured behind citations of the orbital period P.

Sets of orbital elements

Object	Elements used
Major planet	e, a, i, Ω, ϖ, L0
Comet	e, q, i, Ω, ω, T0
Asteroid	e, a, i, Ω, ω, M0
Two-line elements	e, i, Ω, ω, n, M0

Euler angle transformations

The angles Ω, i, ω are the Euler angles (α, β, γ with the notations of that article) characterizing the orientation of the coordinate system

x̂, ŷ, ẑ from the inertial coordinate frame Î, Ĵ, K̂

where:

• Î, Ĵ is in the equatorial plane of the central body. Î is in the direction of the vernal equinox. Ĵ is perpendicular to Î and with Î defines the reference plane. K̂ is perpendicular to the reference plane. Orbital elements of bodies (planets, comets, asteroids,...) in the solar system usually use the ecliptic as that plane.
• x̂, ŷ are in the orbital plane and with x̂ in the direction to the pericenter (periapsis). ẑ is perpendicular to the plane of the orbit. ŷ is mutually perpendicular to x̂ and ẑ.

Then, the transformation from the Î, Ĵ, K̂ coordinate frame to the x̂, ŷ, ẑ frame with the Euler angles Ω, i, ω is:

${\displaystyle {\begin{aligned}x_{1}&=\cos \Omega \cdot \cos \omega -\sin \Omega \cdot \cos i\cdot \sin \omega ;\\x_{2}&=\sin \Omega \cdot \cos \omega +\cos \Omega \cdot \cos i\cdot \sin \omega ;\\x_{3}&=\sin i\cdot \sin \omega ;\\y_{1}&=-\cos \Omega \cdot \sin \omega -\sin \Omega \cdot \cos i\cdot \cos \omega ;\\y_{2}&=-\sin \Omega \cdot \sin \omega +\cos \Omega \cdot \cos i\cdot \cos \omega ;\\y_{3}&=\sin i\cdot \cos \omega ;\\z_{1}&=\sin i\cdot \sin \Omega ;\\z_{2}&=-\sin i\cdot \cos \Omega ;\\z_{3}&=\cos i;\end{aligned}}}$

${\displaystyle \left({\begin{array}{ccc}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\end{array}}\right)=\left({\begin{array}{ccc}\cos \omega &\sin \omega &0\\-\sin \omega &\cos \omega &0\\0&0&1\end{array}}\right)\cdot \left({\begin{array}{ccc}1&0&0\\0&\cos i&\sin i\\0&-\sin i&\cos i\end{array}}\right)\cdot \left({\begin{array}{ccc}\cos \Omega &\sin \Omega &0\\-\sin \Omega &\cos \Omega &0\\0&0&1\end{array}}\right);}$

where

${\displaystyle {\begin{aligned}{\hat {x}}&=x_{1}{\hat {I}}+x_{2}{\hat {J}}+x_{3}{\hat {K}};\\{\hat {y}}&=y_{1}{\hat {I}}+y_{2}{\hat {J}}+y_{3}{\hat {K}};\\{\hat {z}}&=z_{1}{\hat {I}}+z_{2}{\hat {J}}+z_{3}{\hat {K}}.\end{aligned}}}$

The inverse transformation, which computes the 3 coordinates in the I-J-K system given the 3 (or 2) coordinates in the x-y-z system, is represented by the inverse matrix. According to the rules of matrix algebra, the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles.

The transformation from x̂, ŷ, ẑ to Euler angles Ω, i, ω is:

${\displaystyle {\begin{aligned}\Omega &=\operatorname {arg} \left(-z_{2},z_{1}\right)\\i&=\operatorname {arg} \left(z_{3},{\sqrt {{z_{1}}^{2}+{z_{2}}^{2}}}\right)\\\omega &=\operatorname {arg} \left(y_{3},x_{3}\right)\end{aligned}}}$

where arg(x,y) signifies the polar argument that can be computed with the standard function atan2(y,x) available in many programming languages.

Orbit prediction

Under ideal conditions of a perfectly spherical central body and zero perturbations, all orbital elements except the mean anomaly are constants. The mean anomaly changes linearly with time, scaled by the mean motion,

${\displaystyle n={\sqrt {\frac {\mu }{a^{3}}}}.}$

Hence if at any instant t₀ the orbital parameters are [e₀, a₀, i₀, Ω₀, ω₀, M₀], then the elements at time t₀ + δt is given by [e₀, a₀, i₀, Ω₀, ω₀, M₀ + n δt]

Perturbations and elemental variance

Unperturbed, two-body, Newtonian orbits are always conic sections, so the Keplerian elements define an ellipse, parabola, or hyperbola. Real orbits have perturbations, so a given set of Keplerian elements accurately describes an orbit only at the epoch. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, the nonsphericity of the primary, atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, and so on.
Keplerian elements can often be used to produce useful predictions at times near the epoch. Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations, differential equations which come in different forms developed by Lagrange, Gauss, Delaunay, Poincaré, or Hill.

Two-line elements

Keplerian elements parameters can be encoded as text in a number of formats. The most common of them is the NASA/NORAD "two-line elements" (TLE) format, originally designed for use with 80-column punched cards, but still in use because it is the most common format, and can be handled easily by all modern data storages as well.

Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through the SGP/SGP4/SDP4/SGP8/SDP8 algorithms.

Example of a two-line element:

 1 27651U 03004A   07083.49636287  .00000119  00000-0  30706-4 0  2692
 2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249

Delaunay variables

The Delaunay orbital elements, commonly referred to as Delaunay variables, are action-angle coordinates consisting of the argument of periapsis, the mean anomaly and the longitude of the ascending node, along with their conjugate momenta. They are used to simplify perturbative calculations in celestial mechanics, for example while investigating the Kozai–Lidov oscillations in hierarchical triple systems. They were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon.