Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations using Lagrange multipliers. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, time, contains the information about the dynamics of the system. No new physics are introduced in applying Lagrangian mechanics compared to Newtonian mechanics, it is, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange equations.
Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem. Lagrangian mechanics is important not just for its broad applications, but for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton developed Hamilton's principle that can be used to derive the Lagrange equation and was recognized to be applicable to much of fundamental theoretical physics as well quantum mechanics and the theory of relativity, it can be applied to other systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics and when Newton's formulation of classical mechanics is not convenient.
Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density. Lagrange's equations are used in optimization problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind. Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion. For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of independent generalized coordinates that characterize the possible motion of the particle; this choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.
For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of N point particles with masses m1, m2... mN, each particle has a position vector, denoted r1, r2... rN. Cartesian coordinates are sufficient, so r1 =, r2 = and so on. In three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system; these are all specific points in space to locate the particles. The velocity of each particle is how fast the particle moves along its path of motion, is the time derivative of its position, thusIn Newtonian mechanics, the equations of motion are given by Newton's laws; the second law "net force equals mass times acceleration",applies to each particle. For an N particle system in 3 dimensions, there are 3N second order ordinary differential equations in the positions of the particles to solve for.
Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian, it is possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles can be defined by L = T − V where T = 1 2 ∑ k = 1 N m k v k 2 is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, V is the potential energy of the system. Kinetic energy is the energy of the system's motion, vk2 = vk
In Newtonian mechanics, linear momentum, translational momentum, or momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a direction in three-dimensional space. If m is an object's mass and v is the velocity the momentum is p = m v, In SI units, it is measured in kilogram meters per second. Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, general relativity, it is an expression of one of the fundamental symmetries of time: translational symmetry. Advanced formulations of classical mechanics and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.
In these systems the conserved quantity is generalized momentum, in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function; the momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields and deformable bodies, a momentum density can be defined, a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension; the vector equations are identical to the scalar equations. The momentum of a particle is conventionally represented by the letter p.
It is the product of two quantities, the particle's mass and its velocity: p = m v. The unit of momentum is the product of the units of velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second the momentum is in gram centimeters per second. Being a vector, momentum has direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground; the momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2; the momenta of more than two particles can be added more with the following: p = ∑ i m i v i. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i.
If all the particles are moving, the center of mass will be moving as well. If the center of mass is moving at velocity vcm, the momentum is: p = m v cm; this is known as Euler's first law. If the net force applied to a particle is a constant F, is applied for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t. In differential form, this is Newton's second law. If the net force experienced by a particle changes as a function of time, F, the change in momentum between times t1 and t2 is Δ p = J = ∫ t 1
Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat & F. Cosserat in 1909; the action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to smooth symmetries over physical space. Noether's theorem is used in the calculus of variations. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics, it does not apply to systems that cannot be modeled with a Lagrangian alone. In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law; as an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is symmetric under continuous rotations: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.
The physical system itself need not be symmetric. It is the laws of its motion; as another example, if a physical process exhibits the same outcomes regardless of place or time its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. As a final example, if the behavior of a physical system does not change upon spatial or temporal reflection its Lagrangian has reflection symmetry and time reversal symmetry respectively: Noether's theorem says that these symmetries result in the conservation laws of parity and entropy, respectively. Noether's theorem is important, both because of the insight it gives into conservation laws, as a practical calculational tool, it allows investigators to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system.
As an illustration, suppose that a physical theory is proposed which conserves a quantity X. A researcher can calculate the types of Lagrangians. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory. There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the Ward–Takahashi identities. Generalizations of Noether's theorem to superspaces exist. All fine technical points aside, Noether's theorem can be stated informally If a system has a continuous symmetry property there are corresponding quantities whose values are conserved in time. A more sophisticated version of the theorem involving fields states that: To every differentiable symmetry generated by local actions there corresponds a conserved current; the word "symmetry" in the above statement refers more to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria.
The conservation law of a physical quantity is expressed as a continuity equation. The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current; the Noether current is defined up to a solenoidal vector field. In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants: If an integral I is invariant under a continuous group Gρ with ρ parameters ρ linearly independent combinations of the Lagrangian expressions are divergences. A conservation law states that some quantity X in the mathematical description of a system's evolution remains constant throughout its motion — it is an invariant. Mathematically, the rate of change of X is zero, d X d t = X ˙ = 0; such quantities are said to be conserved. For example, if the energy of a system is conserved, its energy is invariant at all times, which imposes a constraint on the system's motion and may help in solving for it.
Aside from insights that such constants of motion give into the nature of a system, they are a useful calculational tool. The earliest constants of motion discovered were momentum and energy, which were proposed in the 17th century by René Descartes and Gottfried Leibniz on the basis of collision experiments, refined by subsequent researchers. Isaac Newton was the first to enunciate the conservation of momentum in its modern form, showed that it was a consequence of Newton's third law. Accord
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object can always be rotated around an infinite number of imaginary lines called rotation axes. If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin. A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution when it is produced by gravity. The axis is called a pole. Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed; this definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two. A rotation is a progressive radial orientation to a common point; that common point lies within the axis of that motion. The axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question the body is said to orbit. There is no fundamental difference between a “rotation” and an “orbit” and or "spin".
The key distinction is where the axis of the rotation lies, either within or outside of a body in question. This distinction can be demonstrated for "non rigid" bodies. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results; the reverse of a rotation is a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, followed by a rotation around the z axis; that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as yaw and roll; this terminology is used in computer graphics. In astronomy, rotation is a observed phenomenon.
Stars and similar bodies all spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured by tracking active surface features; this rotation induces a centrifugal acceleration in the reference frame of the Earth which counteracts the effect of gravity the closer one is to the equator. One effect is that an object weighs less at the equator. Another is that the Earth is deformed into an oblate spheroid. Another consequence of the rotation of a planet is the phenomenon of precession. Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet; the tilt of the Earth's axis to its orbital plane is 23.44 degrees, but this angle changes slowly. While revolution is used as a synonym for rotation, in many fields astronomy and related fields, revolution referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis.
Moons revolve around their planet, planets revolve about their star. The motion of the components of galaxies is complex, but it includes a rotation component. Most planets in our solar system, including Earth, spin in the same direction; the exceptions are Uranus. Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. Venus may be thought of as rotating backwards; the dwarf planet Pluto is anomalous in other ways. The speed of rotation is given by period; the time-rate of change of angular frequency is angular acceleration, caused by torque. The ratio of the two is given by the moment of inertia; the angular velocity vector describes the direction of the axis of rotation. The torque is an axial vector; the physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw.
The laws of physics are believed to be invariant under any fixed rotation. In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, should, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field, laid down by the Big Bang. In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over ti
In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector p = mv; this definition can be applied to each point in physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and the orbital angular momentum; the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin; the total angular momentum of an object is the sum of orbital angular momenta.
The orbital angular momentum vector of a particle is always parallel and directly proportional to the orbital angular velocity vector ω of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. However, the spin angular momentum of the object is proportional but not always parallel to the spin angular velocity Ω, making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is additive. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density over the entire body. Torque can be defined as the rate of change of angular momentum, analogous to force; the net external torque on any system is always equal to the total torque on the system. Therefore, for a closed system, the total torque on the system must be 0, which means that the total angular momentum of the system is constant; the conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, the precession of gyroscopes.
In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is. In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning that at any time, only one component can be measured with definite precision; because of this, it turns out that the notion of an elementary particle "spinning" about an axis does not exist. For technical reasons, elementary particles still possess a spin angular momentum, but this angular momentum does not correspond to spinning motion in the ordinary sense. Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum.
Thus, where linear momentum p is proportional to mass m and linear speed v, p = m v, angular momentum L is proportional to moment of inertia I and angular speed ω, L = I ω. Unlike mass, which depends only on amount of matter, moment of inertia is dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which does not depend upon the choice of origin, angular velocity is always measured with respect to a fixed origin; therefore speaking, L should be referred to as the angular momentum relative to that center. Because I = r 2 m for a single particle and ω = v r for circular motion, angular momentum can be expanded, L = r 2 m ⋅ v r, reduced to, L = r m v, the product of the radius of rotation r and the linear momentum of the particle p = m v, where v in this case is the equivalent linear speed at the radius; this simple analysis can apply to non-circular motion if only the component of the motion, perpendicular to the radius vector is considered. In that case, L
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together. Mathematical symmetry may be observed with respect to the passage of time; this article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion; this means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry if there is a line going through it which divides it into two pieces which are mirror images of each other.
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry. An object has helical symmetry if it can be translated and rotated in three-dimensional space along a line known as a screw axis. An object contracted. Fractals exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection rotoreflection symmetry. A dyadic relation R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary Mary is the same age as Paul. Symmetric binary logical connectives are and, or, nand and nor. Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object; the set of operations that preserve a given property of the object form a group.
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include and odd functions in calculus. In statistics, it appears as symmetric probability distributions, as skewness, asymmetry of distributions. Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations; this concept has become one of the most powerful tools of theoretical physics, as it has become evident that all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his read 1972 article More is Different that "it is only overstating the case to say that physics is the study of symmetry." See Noether's theorem. Important symmetries in physics include discrete symmetries of spacetime. In biology, the notion of symmetry is used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves.
Animals that move in one direction have upper and lower sides and tail ends, therefore a left and a right. The head becomes specialized with a mouth and sense organs, the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs remain asymmetric. Plants and sessile animals such as sea anemones have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, sea lilies. In biology, the notion of symmetry is used as in physics, to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. Symmetry is important to chemistry because it undergirds all specific interactions between molecules in nature; the control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer the
The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, its discovery was a significant landmark in the development of the subject; the equation is named after Erwin Schrödinger, who derived the equation in 1925, published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. In classical mechanics, Newton's second law is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force F on the system; those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation; the concept of a wave function is a fundamental postulate of quantum mechanics.
Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, must therefore be generated by the exponential of a self-adjoint operator, the quantum Hamiltonian. This derivation is explained below. In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular and subatomic systems, but macroscopic systems even the whole universe. Schrödinger's equation is central to all applications of quantum mechanics including quantum field theory which combines special relativity with quantum mechanics. Theories of quantum gravity, such as string theory do not modify Schrödinger's equation; the Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, the path integral formulation, developed chiefly by Richard Feynman.
Paul Dirac incorporated the Schrödinger equation into a single formulation. The form of the Schrödinger equation depends on the physical situation; the most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time: where i is the imaginary unit, ℏ = h 2 π is the reduced Planck constant, Ψ is the state vector of the quantum system, t is time, H ^ is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector | r ⟩, it is a scalar function, expressed as Ψ = ⟨ r | Ψ ⟩. The momentum-space wave function can be defined as Ψ ~ = ⟨ p | Ψ ⟩, where | p ⟩ is the momentum eigenvector; the most famous example is the nonrelativistic Schrödinger equation for the wave function in position space Ψ of a single particle subject to a potential V, such as that due to an electric field. Where m is the particle's mass, ∇ 2 is the Laplacian.
This is a diffusion equation, but unlike the heat equation, this one is a wave equation given the imaginary unit present in the transient term. The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version; the general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is a classical approximation to reality and yields accurate results in many situations, but only to a certain extent. To apply the Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system insert it into the Schrödinger equation; the resulting partial differential equation is solved for the wave function, which contains information about the system. The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states.
These states are important as their individual study simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can be described by a simpler form of the Schrödinger equation, the time-independe