# Momentum is a scalar quantity

## pulse This article describes the physical quantity of momentum. For other meanings, see Impulse (disambiguation).

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

The symbol of the impulse is usually \$ p \$ (from Latin pellere, German `` push, drive ''). The unit in the International System of Units is kg · m · s−1 = N s.

In contrast to kinetic energy, the momentum is a vector quantity and therefore has a magnitude and a direction. The direction of the impulse is the direction of movement of the object. In the area of ​​validity of classical mechanics, the magnitude of the momentum is the product of the mass of the object and the speed of its center of mass. The momentum exclusively characterizes the translational movement of the center of mass. Any additional rotation of the object around the center of mass is described by the angular momentum.

The momentum is an additive quantity. For an object with several components, the total impulse is the vector sum of the impulses of all its parts.

In relativistic mechanics, a different formula applies to the momentum (four-part momentum). In the event that the speed is much smaller than the speed of light, it approximately agrees with the classical formula. But it also ascribes an impulse to massless objects moving at the speed of light, e.g. B. electromagnetic waves or photons.

The momentum, like the speed and the kinetic energy, depends on the choice of the reference system.

In relation to a fixed selected inertial system, the momentum is a conserved quantity. An object, on which no external forces act, retains its total momentum in terms of magnitude and direction regardless of any internal processes. Do two objects exert force on each other, e.g. B. in a collision process, their two impulses 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 Impulse transfer designated. 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 of the "amount of movement" present in a physical object, which experience has shown to be retained in all internal processes. This explains the names that are outdated today "Movement size" 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 differentiated. In English the impulse becomes momentum called while impulses describes the momentum transfer (impulse of force).

### 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:

\$ \ vec p = m \ cdot \ vec v \$

Since the mass \$ m \$ is a scalar quantity, momentum \$ \ vec p \$ and velocity \$ \ vec v \$ are vectors with the same direction. Their amounts cannot be compared with one another because they have different physical dimensions.

The following applies to the kinetic energy:

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

In order to change the speed of a body (by direction and / or amount), its momentum must be changed. The transmitted momentum divided by the time required is the force \$ \ vec {F} \$:

\$ \ frac {\ mathrm d \ vec {p}} {\ mathrm d t} = \ vec {F} \$

### Special theory of relativity

According to the theory of relativity, the momentum of a body moving with velocity \$ v \$ with mass \$ m> 0 \$ is through

\$ \ vec p = \ frac {m \ cdot \ vec v} {\ sqrt {1- {v ^ 2 \ over c ^ 2}}} \$

given. Here \$ c \$ is the speed of light and always \$ v

The energy-momentum relationship is generally valid

\$ E ^ 2 - p ^ 2 \ times c ^ 2 = m ^ 2 \ times c ^ 4. \$

For objects with mass it follows:

\$ E = \ frac {m \ cdot c ^ 2} {\ sqrt {1- {v ^ 2 \ over c ^ 2}}} \$

For \$ v = 0 \$ it follows that \$ p = 0 \$ and \$ E = m \, c ^ 2 \$ (rest energy).

Objects without mass always move at the speed of light. For them it follows from the energy-momentum relationship

\$ E = p \, c. \$

### Electromagnetic field

An electromagnetic field with electric field strength \$ \ vec E \$ and magnetic field strength \$ \ vec H \$ has the energy density

\$ u = \ frac {1} {2} \ varepsilon_0 E ^ 2 + \ frac {1} {2} \ mu_0 H ^ 2. \$

These include the energy flux density (Poynting vector)

\$ \ vec {S} = \ vec {E} \ times \ vec {H} \$

and the momentum density

\$ \ qquad \ vec {g} = \ frac {1} {c ^ 2} \ vec {E} \ times \ vec {H}. \$

Integrated over a certain volume, these three expressions result in the energy \$ E \$, the energy flow and the momentum \$ p \$, which are connected to the entire field in this volume. For advancing plane waves we get \$ E = p \, c \$.

### Conservation of momentum Kick-off at pool: The momentum of the white ball is distributed over all balls.

In an inertial system the momentum is a conserved quantity. In a physical system on which no external forces act (in this context also called 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)).

The conservation of momentum also applies to inelastic collisions. Kinetic energy is lost 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 Change in momentum and force-time area

A change in momentum results from the force on a body and its duration of action, which is called Impulse referred to as. Both the amount and the direction of the force play a role. The impulse of force is often denoted by the symbol \$ \ vec I \$, its SI unit is 1 N · s.

If the force \$ \ vec F \$ is constant in the time interval \$ \ Delta t_ {1,2}: = t_2-t_1 \$ (with \$ t_1

\$ \ vec I (t_1, t_2) = \ Delta \ vec p = \ vec F \ cdot \ Delta t_ {1,2} \$

If, on the other hand, \$ \ vec F \$ 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.

In the most general case, \$ \ vec F \$ is time-dependent and the impulse is defined by integration:

\$ \ 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

The generalized momentum is introduced in the Lagrange and Hamilton formalism; 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 in this context canonical impulse designated. The possible pairs \$ (q, p) \$ of generalized spatial coordinates \$ q \$ and canonical impulses \$ p \$ of a physical system form the phase space in Hamiltonian mechanics.

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

With continuously distributed mass, such as in fluid mechanics, a small area around the point \$ \ vec {x} \$ contains the mass \$ \ rho (t, \ vec {x}) \, \ mathrm d ^ 3 x. \$ Where \$ \ mathrm d ^ 3 x \$ is the volume of the area. \$ \ rho (t, \ vec {x}) \$ is the mass density and \$ \ vec {x} = (x ^ 1, x ^ 2, x ^ 3) \$ the position vector (components numbered). It can change over time \$ t \$.

If this mass moves with the velocity \$ \ vec {v} (t, \ vec {x}) \$, it has the momentum \$ \ rho (t, \ vec {x}) \, \ vec {v} (t , \ vec {x}) \, \ mathrm d ^ 3 x \$. Divided by the volume, the momentum density results as mass density multiplied by speed: \$ \ rho \, \ vec {v} \$.

Because of the conservation of momentum, the continuity equation applies to the momentum density at a fixed location

\$ \ frac {\ partial (\ rho \, \ vec {v})} {\ partial t} = \ vec {f} - \ sum_ {i = 1} ^ 3 \ frac {\ partial} {\ partial x ^ i} (\ rho \, \ vec {v} \, v ^ i), \$

which says that the change in momentum density over time is composed of the force density acting on the volume element (for example the gradient of the pressure or the weight, \$ \ vec {f} _ {\ text {gravitation}} = \ rho \, \ vec {g} \$) and the momentum current in and out of the area.

Euler's equations are the system of partial differential equations which, together with conservation of momentum and conservation of energy, allow the time evolution of a continuous 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 the simple scalar relationship \$ \ vec p = m \ vec v \$. In the relativistic Euler equations, however, the Lorentz factor mixes into every vector component, which depends on \$ \ vec v ^ 2 \$. 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 non-linear system of equations.

### Momentum in quantum mechanics

The momentum plays a decisive 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 place, but a little defined momentum, fits better intuitively to the understanding of particles, a well-defined momentum (the wave vector) is more of 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 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 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

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

where \$ h \$ is Planck's quantum and \$ p \$ is the momentum. The De Broglie wavelength \$ \ lambda \$ of matter waves of free particles is thus determined by the momentum.