Time translation symmetry From Wikipedia, the free encyclopedia Hypothesis that physics experiments will behave the same regardless of when they are conducted Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged (i.e. invariant) under such a transformation. There are many symmetries in nature besides time translation, such as spatial translation or rotational symmetries. Overview[edit] Newtonian mechanics[edit] To formally describe time translation symmetry we say the equations, or laws, that describe a system at times and are the same for any value of For example, considering Newton's equation: One finds for its solutions the combination: does not depend on the variable . . Quantum mechanics[edit] The invariance of a Hamiltonian of an isolated system under time translation implies its energy does not change with the passage of time. or:
Rotational invariance Mathematics[edit] Functions[edit] For example, the function is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle θ the function, after some cancellation of terms, takes exactly the same form or symbolically x′ = Rx. In words, the function of the rotated coordinates takes exactly the same form as it did with the initial coordinates, the only difference is the rotated coordinates replace the initial ones. The concept also extends to a vector-valued function f of one or more variables; In all the above cases, the arguments (here called "coordinates" for concreteness) are rotated, not the function itself. Operators[edit] which acts on a function f to obtain another function ∇2f. If g is the function g(p) = f(R(p)), where R is any rotation, then (∇2g)(p) = (∇2f )(R(p)); that is, rotating a function merely rotates its Laplacian. Physics[edit] Application to quantum mechanics[edit] for any rotation R. then thus See also[edit]
Translational symmetry Invariance of operations under geometric translation if the result after applying A doesn't change if the argument function is translated. More precisely it must hold that Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law. Translational symmetry of an object means that a particular translation does not change the object. Geometry[edit] Translational invariance implies that, at least in one direction, the object is infinite: for any given point p, the set of points with the same properties due to the translational symmetry form the infinite discrete set {p + na | n ∈ Z} = p + Z a. Examples[edit] See also[edit] References[edit] Stenger, Victor J. (2000) and MahouShiroUSA (2007).
Vis-viva equation In astrodynamics, the vis-viva equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight. Vis viva (Latin for "living force") is a term from the history of mechanics, and it survives in this sole context. It represents the principle that the difference between the total work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done. Equation[edit] For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the vis-viva equation[1] is as follows:[2] where: The product of GM can also be expressed as the standard gravitational parameter using the Greek letter μ. Derivation for elliptic orbits (0 ≤ eccentricity < 1)[edit] Rearranging, , thus Thus, or and , and
Particle physics and representation theory The connection between particle physics and representation theory is a natural connection, first noted in the 1930s by Eugene Wigner,[1] between the properties of elementary particles and the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe. General picture[edit] In quantum mechanics, any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is written as a vector (or "ket") in a Hilbert space H. Let G be the symmetry group of the universe – that is, the set of symmetries under which the laws of physics are invariant. . Poincaré group[edit] Other symmetries[edit] Approximate symmetries[edit] Hypothetical example[edit]
Zilch (electromagnetism) In particular, first, Daniel M. Lipkin observed that if he defined the quantities then the free Maxwell equations imply that which implies that the quantity is constant. is the spatial density of optical chirality, while The zilch(es) are often described in terms of the zilch tensor, as and The conservation law means that the following ten quantities are time-independent: ). is known as the run from 0 to 3) and it is clear that there are ten such quantities (nine independent). What are the symmetries of the standard Maxwell action functional (with , where is the dynamical field variable) that give rise to the conservation of all zilches using Noether's theorem?
Parity (physics) Symmetry of spatially mirrored systems In physics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity. By contrast, in interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotation, which has a determinant equal to 1. In quantum mechanics, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are odd functions. Simple symmetry relations[edit]
Noether's theorem Statement relating differentiable symmetries to conserved quantities Noether's theorem is used in theoretical physics and the calculus of variations. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law. Briefly, the relationships between symmetries and conservation laws are as follows: 1) Uniformity of space distance-wise ⟹ conservation of linear momentum; 2) Isotropy of space direction-wise ⟹ conservation of angular momentum; Basic illustrations and background[edit]
Noether's second theorem From Wikipedia, the free encyclopedia Physics theorem for symmetries of action Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations. The theorem is named after its discoverer, Emmy Noether. See also[edit] Notes[edit] References[edit] Further reading[edit]
Mutability From Wikipedia, the free encyclopedia The principle of mutability is the notion that any physical property which appears to follow a conservation law may undergo some physical process that violates its conservation.[1][2][3] John Archibald Wheeler offered this speculative principle after Stephen Hawking predicted the evaporation of black holes which violates baryon number conservation.[4] See also[edit] Philosophy of physics References[edit]