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Path Of Least Resistance
The path of least resistance is the physical or metaphorical pathway that provides the least resistance to forward motion by a given object or entity, among a set of alternative paths. The concept is often used to describe why an object or entity takes a given path. The way in which water flows is often given as an example for the idea. Description In physics, the "path of least resistance" is a heuristic from folk physics that can sometimes, in very simple situations, describe approximately what happens. It is an approximation of the tendency to the least energy state. Other examples are "what goes up must come down" (gravity) and "heat goes from hot to cold" (second law of thermodynamics). But these simple descriptions are not derived from laws of physics and in more complicated cases these heuristics will fail to give even approximately correct results. In electrical circuits, for example, the current always follows all available paths, and in some simple cases the "path of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartoon Mountain Pass Symbolizing Path Of Least Resistance
A cartoon is a type of visual art that is typically drawn, frequently animated, in an unrealistic or semi-realistic style. The specific meaning has evolved over time, but the modern usage usually refers to either: an image or series of images intended for satire, caricature, or humor; or a motion picture that relies on a sequence of illustrations for its animation. Someone who creates cartoons in the first sense is called a ''cartoonist'', and in the second sense they are usually called an ''animator''. The concept originated in the Middle Ages, and first described a preparatory drawing for a piece of art, such as a painting, fresco, tapestry, or stained glass window. In the 19th century, beginning in ''Punch'' magazine in 1843, cartoon came to refer – ironically at first – to humorous artworks in magazines and newspapers. Then it also was used for political cartoons and comic strips. When the medium developed, in the early 20th century, it began to refer to animated films ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captured in the local minimum of a potential well. Therefore, a body may not proceed to the global minimum of potential energy, as it would naturally tend to do due to entropy. Overview Energy may be released from a potential well if sufficient energy is added to the system such that the local maximum is surmounted. In quantum physics, potential energy may escape a potential well without added energy due to the probabilistic characteristics of quantum particles; in these cases a particle may be imagined to tunnel ''through'' the walls of a potential well. The graph of a 2D potential energy function is a potential energy surface that can be imagined as the Earth's surface in a landscape of hills and valleys. Then a potential well would be a val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Lines Of Drift
Natural lines of drift are those paths across terrain that are the most likely to be used when going from one place to another. These paths are paths of least resistance: those that offer the greatest ease while taking into account obstacles (e.g. rivers, cliffs, dense unbroken woodland, etc.) and modes of transit (e.g. pedestrian, automobile, horses.). Common endpoints or fixed points may include water sources, food sources, and obstacle passages such as fords or bridges. Local paths may be derived from game trails or from artificial paths created by utility lines or political boundaries. Property ownership and land use may also be factors in determining local variation. Improved paths may also be partially defined by the logistics necessary to build roads or railways. See also * Footpath *Trail A trail, also known as a path or track, is an unpaved lane or small road usually passing through a natural area. In the United Kingdom and the Republic of Ireland, a path o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on the observation that if the multi-variable function F(\mathbf) is def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain. Overview Any physical law which can be expressed as a variational principle describes a self-adjoint operator. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation. History Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Least Action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are ''stationary points'' of the system's ''action functional''. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal (even locally) on the trajectories. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). In relativity, a different action must be minimized or maximized. The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics. In 1933, the physicist Paul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mountain Pass Theorem
The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points. Statement The assumptions of the theorem are: * I is a functional from a Hilbert space ''H'' to the reals, * I\in C^1(H,\mathbb) and I' is Lipschitz continuous on bounded subsets of ''H'', * I satisfies the Palais–Smale compactness condition, * I 0, * there exist positive constants ''r'' and ''a'' such that I geq a if \Vert u\Vert =r, and * there exists v\in H with \Vert v\Vert >r such that I leq 0. If we define: :\Gamma=\ and: :c=\inf_\max_ I mathbf(t) then the conclusion of the theorem is that ''c'' is a critical value of ''I''. Visualization The intuition behind the theorem is in the name "mountain pass." ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field. The unit for energy in the International System of Units (SI) is the joule, which has the symbol J. The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, that are called ''conservative forces'', can be represented at every point in space by vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siphon
A siphon (from grc, σίφων, síphōn, "pipe, tube", also spelled nonetymologically syphon) is any of a wide variety of devices that involve the flow of liquids through tubes. In a narrower sense, the word refers particularly to a tube in an inverted "U" shape, which causes a liquid to flow upward, above the surface of a reservoir, with no pump, but powered by the fall of the liquid as it flows down the tube under the pull of gravity, then discharging at a level lower than the surface of the reservoir from which it came. There are two leading theories about how siphons cause liquid to flow uphill, against gravity, without being pumped, and powered only by gravity. The traditional theory for centuries was that gravity pulling the liquid down on the exit side of the siphon resulted in reduced pressure at the top of the siphon. Then atmospheric pressure was able to push the liquid from the upper reservoir, up into the reduced pressure at the top of the siphon, like in a baromet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Of Least Resistance
The path of least resistance is the physical or metaphorical pathway that provides the least resistance to forward motion by a given object or entity, among a set of alternative paths. The concept is often used to describe why an object or entity takes a given path. The way in which water flows is often given as an example for the idea. Description In physics, the "path of least resistance" is a heuristic from folk physics that can sometimes, in very simple situations, describe approximately what happens. It is an approximation of the tendency to the least energy state. Other examples are "what goes up must come down" (gravity) and "heat goes from hot to cold" (second law of thermodynamics). But these simple descriptions are not derived from laws of physics and in more complicated cases these heuristics will fail to give even approximately correct results. In electrical circuits, for example, the current always follows all available paths, and in some simple cases the "path of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superfluid
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two isotopes of helium (helium-3 and helium-4) when they are liquefied by cooling to cryogenic temperatures. It is also a property of various other exotic states of matter theorized to exist in astrophysics, high-energy physics, and theories of quantum gravity. The theory of superfluidity was developed by Soviet theoretical physicists Lev Landau and Isaak Khalatnikov. Superfluidity is often coincidental with Bose–Einstein condensation, but neither phenomenon is directly related to the other; not all Bose–Einstein condensates can be regarded as superfluids, and not all superfluids are Bose–Einstein condensates. Superfluidity of liquid helium Superfluidity was discovered in helium-4 by Pyotr Kapitsa and independently by John F. Al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
