# pulse

Physical size
Surname pulse
Formula symbol ${\ displaystyle {\ vec {p}}}$ Size and
unit system
unit dimension
SI N · s
kg · m · s −1
M · L · T −1

The momentum is a fundamental physical quantity that characterizes the mechanical state of motion of a physical object. The momentum of a physical object is greater, the faster it moves and the more massive it is. The impulse thus stands for what is vaguely referred to in everyday language as “ momentum ” and “ force ”.

The symbol of the impulse is mostly (from the Latin pellere ' to push, to drive' ). The unit in the International System of Units is kg · m · s −1  = N · s . ${\ displaystyle p}$ In contrast to kinetic energy , the momentum is a vector quantity and therefore has a magnitude and a direction. Its direction is the direction of movement of the object. In classical mechanics, its amount is given by the product of the mass of the body and the speed of its center of mass . In relativistic mechanics , another formula ( four-pulse ) applies , which approximately corresponds to the classical formula for speeds far below the speed of light . But it also attributes an impulse to massless objects moving at the speed of light, e.g. B. classical electromagnetic waves or photons .

The momentum of a body exclusively characterizes the translational movement of its center of mass. Any additional rotation around the center of mass is described by the angular momentum . The momentum is an additive quantity. The total momentum of an object with several components is the vector sum of the momentum of its parts.

The momentum, like the speed and the kinetic energy, depends on the choice of the reference system . In a firmly selected inertial system , the momentum is a conservation quantity , that is: an object on which no external forces act, retains its total momentum in terms of magnitude and direction. Do two objects exert force on each other, e.g. B. in a collision process , their two pulses change in opposite ways so that their vectorial sum is retained. The amount by which the momentum changes for one of the objects is called the momentum transfer . In the context of classical mechanics, the momentum transfer is independent of the choice of the inertial system.

The concept of impulses developed out of the search for the measure for the "amount of movement" present in a physical object, which experience has shown to be retained in all internal processes. This explains the now obsolete terms “quantity of movement” or “amount of movement” for the impulse. Originally, these terms could also refer to kinetic energy ; it was not until the beginning of the 19th century that the terms were clearly distinguished. In English, the impulse is called momentum , while impulse describes the momentum transfer (impulse of force).

## Definition, relationships with mass and energy

### Classic mechanics

The concept of impulse was introduced by Isaac Newton : He writes in Principia Mathematica :

"Quantitas motus est mensura ejusdem orta ex velocitate et quantitate materiae conjunctim."

"The size of the movement is measured by the speed and the size of the matter combined."

"Size of matter" means mass, "size of movement" means impulse. Expressed in today's formula language, this definition is:

${\ displaystyle {\ vec {p}} = m \ cdot {\ vec {v}}}$ Since the mass is a scalar quantity, momentum and velocity are vectors with the same direction. Their amounts cannot be compared with one another because they have different physical dimensions. ${\ displaystyle m}$ ${\ displaystyle {\ vec {p}}}$ ${\ displaystyle {\ vec {v}}}$ In order to change the speed of a body (by direction and / or amount), its momentum has to be changed. The transmitted impulse divided by the time required is the force : ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ frac {\ mathrm {d} {\ vec {p}}} {\ mathrm {d} t}} = {\ vec {F}}}$ The connection between the momentum of a body and the force acting on it results in a connection to the momentum for the acceleration work performed :

${\ displaystyle W = \ int \ limits _ {C} {\ vec {F}} ({\ vec {s}}) \ cdot \ mathrm {d} {\ vec {s}} = \ int \ limits _ { C} {\ vec {F}} ({\ vec {s}}) \, \ mathrm {d} t \ cdot {\ frac {\ mathrm {d} {\ vec {s}}} {\ mathrm {d } t}} = \ int \ limits _ {C} \ mathrm {d} {\ vec {p}} \ cdot {\ vec {v}} = {\ frac {1} {m}} \ int \ limits _ {C} \ mathrm {d} {\ vec {p}} \ cdot {\ vec {p}}}$ This acceleration work is the kinetic energy . It follows

${\ displaystyle E _ {\ text {kinetic}} = {\ frac {{\ vec {p}} ^ {\, 2}} {2 \, m}} = {\ frac {m \; {\ vec {v }} ^ {\, 2}} {2}}}$ .

### Special theory of relativity

According to the theory of relativity , the momentum of a body with mass moving with speed is through ${\ displaystyle v}$ ${\ displaystyle m> 0}$ ${\ displaystyle {\ vec {p}} = {\ frac {m \ cdot {\ vec {v}}} {\ sqrt {1- {v ^ {2} \ over c ^ {2}}}}}}$ given. In it is the speed of light and always . The momentum depends nonlinearly on the speed; it increases towards infinity as the speed of light approaches . ${\ displaystyle c}$ ${\ displaystyle v The energy-momentum relationship is generally valid

${\ displaystyle E ^ {2} -p ^ {2} \ cdot c ^ {2} = m ^ {2} \ cdot c ^ {4}.}$ For objects with mass it follows:

${\ displaystyle E = {\ frac {m \ cdot c ^ {2}} {\ sqrt {1- {v ^ {2} \ over c ^ {2}}}}}}$ For follows and ( rest energy ). ${\ displaystyle v = 0}$ ${\ displaystyle p = 0}$ ${\ displaystyle E = m \, c ^ {2}}$ Objects without mass always move at the speed of light. For this it follows from the energy-momentum relation

${\ displaystyle E = p \, c}$ and that gives them the impulse

${\ displaystyle p = {E \ over c}.}$ ### Electromagnetic field

An electromagnetic field with electric field strength and magnetic field strength has the energy density${\ displaystyle {\ vec {E}}}$ ${\ displaystyle {\ vec {H}}}$ ${\ displaystyle u = {\ frac {1} {2}} \ varepsilon _ {0} E ^ {2} + {\ frac {1} {2}} \ mu _ {0} H ^ {2}.}$ These include the energy flux density ( Poynting vector )

${\ displaystyle {\ vec {S}} = {\ vec {E}} \ times {\ vec {H}}}$ and the momentum density

${\ displaystyle \ qquad {\ vec {g}} = {\ frac {1} {c ^ {2}}} {\ vec {E}} \ times {\ vec {H}}.}$ Integrated over a certain volume, these three expressions give the energy , the energy flow and the momentum associated with the entire field in this volume. For advancing plane waves it results again . ${\ displaystyle E}$ ${\ displaystyle p}$ ${\ displaystyle E = p \, c}$ ## Conservation of momentum

In an inertial system the momentum is a conserved quantity . In a physical system on which no external forces act (in this context also referred to as a closed system), the sum of all impulses of the components belonging to the system remains constant.

The initial total impulse is then also equal to the vector sum of the individual impulses present at any later point in time. Impacts and other processes within the system, in which the speeds of the components change, always end in such a way that this principle is not violated (see kinematics (particle processes) ).

Conservation of momentum also applies to inelastic collisions . The kinetic energy decreases through plastic deformation or other processes, but the law of conservation of momentum is independent of the law of conservation of energy and applies to both elastic and inelastic collisions.

## Impulse

The force on a body and its duration of action result in a change in momentum, which is referred to as a force impulse . Both the amount and the direction of the force play a role. The impulse is often referred to with the symbol , its SI unit is 1 N · s. ${\ displaystyle {\ vec {I}}}$ If the force is (with ) constant in the time interval , the impulse can be calculated as: ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle \ Delta t_ {1,2}: = t_ {2} -t_ {1}}$ ${\ displaystyle t_ {1} ${\ displaystyle {\ vec {I}} (t_ {1}, t_ {2}) = \ Delta {\ vec {p}} = {\ vec {F}} \ cdot \ Delta t_ {1,2}}$ If, on the other hand, it is not constant, but still without a change in sign (in each individual force component), one can calculate with an average force using the mean value theorem of the integral calculus . ${\ displaystyle {\ vec {F}}}$ In the general case it is time-dependent and the impulse is defined by integration: ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {I}} (t_ {1}, t_ {2}) = \ Delta {\ vec {p}} (t_ {1}, t_ {2}) = \ int _ {t_ {1 }} ^ {t_ {2}} {\ vec {F}} (t) \ cdot \ mathrm {d} t}$ ## Momentum in the Lagrange and Hamilton formalism

In the Lagrange and Hamilton formalism is generalized pulse introduced; the three components of the momentum vector count towards the generalized momentum; but also, for example, the angular momentum .

In the Hamilton formalism and in quantum mechanics , the momentum is the variable canonically conjugated to the position. The (generalized) impulse is also called canonical impulse in this context . The possible pairs of generalized position coordinates and canonical impulses of a physical system form the phase space in Hamiltonian mechanics . ${\ displaystyle (q, p)}$ ${\ displaystyle q}$ ${\ displaystyle p}$ In magnetic fields , the canonical momentum of a charged particle contains an additional term that is related to the vector potential of the B-field (see generalized momentum ).

## Impulse in flowing media

In the case of continuously distributed mass, such as in fluid mechanics , a small area around the point contains the mass, where the volume of the area. is the mass density and the position vector (components numbered). It can change over time . ${\ displaystyle {\ vec {x}}}$ ${\ displaystyle \ rho (t, {\ vec {x}}) \, \ mathrm {d} ^ {3} x.}$ ${\ displaystyle \ mathrm {d} ^ {3} x}$ ${\ displaystyle \ rho (t, {\ vec {x}})}$ ${\ displaystyle {\ vec {x}} = (x_ {1}, x_ {2}, x_ {3})}$ ${\ displaystyle t}$ When this mass moves with speed , it has momentum . Divided by the volume of the pulse density is mass density times velocity: . ${\ displaystyle {\ vec {v}} (t, {\ vec {x}})}$ ${\ displaystyle \ rho (t, {\ vec {x}}) \, {\ vec {v}} (t, {\ vec {x}}) \, \ mathrm {d} ^ {3} x}$ ${\ displaystyle \ rho \, {\ vec {v}}}$ Because of the conservation of momentum, the continuity equation applies to the momentum density at a fixed location

${\ displaystyle {\ frac {\ partial (\ rho \, {\ vec {v}})} {\ partial t}} = {\ vec {f}} - \ sum _ {i = 1} ^ {3} {\ frac {\ partial} {\ partial x_ {i}}} (\ rho \, {\ vec {v}} \, v_ {i}),}$ which states that the temporal change in the momentum density is made up of the force density acting on the volume element (for example the gradient of the pressure or the weight ) and the momentum flow into and out of the area. ${\ displaystyle {\ vec {f}} _ {\ text {Gravitation}} = \ rho \, {\ vec {g}}}$ The Euler's equations are the system of partial differential equations, which allows together with conservation of momentum, and conservation of energy, the development of a continuous time system. The Navier-Stokes equations extend these equations by additionally describing viscosity.

What is remarkable about Euler's equation is that there is a conservation equation for momentum, but not for velocity. This does not play a special role in classical mechanics, since there is a simple scalar relationship . In the relativistic Euler equations, on the other hand, the Lorentz factor , which depends on , mixes in every vector component . Therefore, the reconstruction of the velocity vector (primitive variables) from the system of relativistic mass, momentum and energy density (conserved variables) is usually connected with the solution of a nonlinear system of equations. ${\ displaystyle {\ vec {p}} = m {\ vec {v}}}$ ${\ displaystyle {\ vec {v}} ^ {2}}$ ## Momentum in quantum mechanics

The momentum plays a crucial role in quantum mechanics . Heisenberg's uncertainty principle applies to the determination of momentum and position , according to which a particle cannot have an exact momentum and an exact position at the same time. The wave-particle dualism requires quantum mechanical objects to take into account their wave and particle nature at the same time. While a well-defined location, but a little defined momentum, fits better intuitively to the understanding of particles, a well-defined momentum (the wave vector ) is more a property of the wave. The duality is represented mathematically in the fact that canonical quantum mechanics can be operated either in space or momentum space (also called position representation and momentum representation). Depending on the representation, the momentum operator is then a normal measurement operator or it is a differential operator. In both cases, the measurement of the impulse ensures that it is then exactly determined; there is a collapse of the wave function , which leads to the total delocalization of the object. Colloquially this is sometimes expressed by the fact that “no specific momentum belongs to a physical state of a particle” or “only the probability can be given that the momentum of a particle lies in this or that range”. These statements are, however, characterized by a particle or location-centered thinking and can also be turned around: “A physical state of a wave does not have a specific location” or “only the probability can be given that the location of a wave is in this or that area lies ”.

The states with well-determined momentum are called eigen-states of the momentum operator . Their wave functions are plane waves with wavelength

${\ displaystyle \ lambda = {\ frac {h} {p}},}$ where is Planck's constant and the momentum. The De Broglie wavelength of matter waves of free particles is determined by the momentum. ${\ displaystyle h}$ ${\ displaystyle p}$ ${\ displaystyle \ lambda}$ 