Relativistic Momentum

Let’s know about Relativistic Momentum. In Newtonian mechanics , linear motion , translational motion , or simply motion ( pl. Momenta) is the product of the mass and velocity of an object. It is a vector quantity, which has magnitude and direction. If m is the mass of an object and v is its velocity (also a vector quantity), then the object’s momentum is: in SI units , speed is measured in kilogram meters per second ( kg m /s ). (Relativistic Momentum)

\mathbf{p} = m \mathbf{v}.

Newton’s second law of motion states that the rate of change of momentum of a body is equal to the net force acting on it. Momentum depends on the frame of reference , but it is a conserved quantity in any inertial frame, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also used in special relativity (with a modified formula) and, in a modified form, in electrodynamics , quantum mechanics , quantum field theory and general relativity .is protected in. It is an expression of one of the fundamental symmetries of space and time: translational symmetry .(Relativistic Momentum)

Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics , allow one to choose coordinate systems that incorporate symmetries and constraints. The conserved quantity in these systems is the normalized momentum , and in general it 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 to the Heisenberg uncertainty principle . In continuous systems such as electromagnetic fields , fluid dynamics , and deformed bodies , a momentum density can be defined, and a constant version of conservation of momentum leads to equations such as the Navier–Stokes equation for liquids or for deformed solids. Cauchy equation for motion or fluids.(Relativistic Momentum)

Newtonian

Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resultant direction and speed after objects collide. Below, the basic properties of momentum are described in one dimension. Vector equations are almost the same as scalar equations ( see multiple dimensions ).

single particle

The speed of a particle is traditionally denoted by the letter p . It is the product of two quantities, the particle’s mass ( denoted by the letter m ) and its velocity ( v ):(Relativistic Momentum)

p=mv.

The unit of momentum is the product of the units of mass and velocity. In SI units , if mass is in kilograms and velocity is in meters per second then speed is in kilogram meters per second (kg⋅m/s). In cgs units , if mass is in grams and velocity in centimeters per second, then speed is in grams per second in centimeters (g⋅cm/s).

Momentum being a vector has magnitude and direction. For example, a 1 kg model airplane, traveling north at 1 m/s in straight and flat flight, has a speed of 1 kg⋅m/s due north as measured with reference to the ground.

many particles

The momentum of a system of particles is the vector sum of their momentum. If two particles have mass m ₁ and m ₂ , and velocities v ₁ and v ₂ , then the total momentum is

{\displaystyle {\begin{aligned}p&=p_{1}+p_{2}\\&=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}}}

The momentum of more than two particles can be associated more generally with the following:

{\displaystyle p=\sum _{i}m_{i}v_{i}.}

A system of particles has a center of mass , which is determined by the weighted sum of their positions:

{\displaystyle r_{\text{cm}}={\frac {m_{1}r_{1}+m_{2}r_{2}+\cdots }{m_{1}+m_{2}+\cdots }}={\frac {\sum _{i}m_{i}r_{i}}{\sum _{i}m_{i}}}.}

If one or more particles are in motion, the center of mass of the system will generally also be in motion (unless the body is in pure rotation around it). If the total mass of the particles is , and the center of mass is moving with a velocity v _{cm , the speed of the system is:} m

p=mv_{\text{cm}}.

This is called Euler’s first law .

relation to force

If the net force F applied on a particle is constant, and is applied for a time interval t , then the speed of the particle changes by an amount (Relativistic Momentum)

\Delta p=F\Delta t\,.

In differential form, this is Newton’s second law ; The rate of change of momentum of a particle is equal to the instantaneous force F acting on it,

{\displaystyle F={\frac {dp}{dt}}.}

If the net force experienced by a particle changes as a function of time, F ( t ) , is the change in momentum (or impulse J ) between times t₁ and t₂

{\displaystyle \Delta p=J=\int _{t_{1}}^{t_{2}}F(t)\,dt\,.}

Impulse is measured in derived units of newton second (1 N⋅s = 1 kg⋅m / s) or dyne second (1 dyne⋅s = 1 g⋅cm / s)

Under the assumption of a constant mass m , this is equivalent to writing

{\displaystyle F={\frac {d(mv)}{dt}}=m{\frac {dv}{dt}}=ma,}

Therefore, the net force is equal to the mass of the particle multiplied by its acceleration. ^[1]

Example : A model airplane of mass 1 kg accelerates from rest in 2 seconds towards north with a velocity of 6 m/s. The net force required to produce this acceleration is 3 Newtons to the north. The change in momentum northwards is 6 kg⋅m/s. The rate of change of momentum northwards is 3 (kg⋅m/s)/s which is numerically equal to 3 Newtons. (Relativistic Momentum)

Protection

In a closed system (which does not exchange any substance with its surroundings and is not acted upon by external forces) the total momentum remains constant. This fact, known as the law of conservation of momentum, is implied by Newton’s laws of motion. ^[4] [5] Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If the number of particles is 1 and 2, then the second law states that F ₁ =dp ₁/dt and f ₂ =dp ₂/dt, therefore,

{\frac {dp_{1}}{dt}}=-{\frac {dp_{2}}{dt}},

With a negative sign indicating that the forces are opposing. uniformly,

{\frac {d}{dt}}\left(p_{1}+p_{2}\right)=0.

The velocities of the particles are u ₁ and u ₂ before the interaction, and after they are v ₁ and v ₂ , then

m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}.

This law holds that no matter how complex the force between the particles is. Similarly, if there are many particles, the speed of exchange between each pair of particles becomes zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separations caused by explosive forces. ^[4] It can also be generalized to situations where Newton’s laws do not apply, for example in the theory of relativity and electrodynamics. (Relativistic Momentum)

Dependency on reference frame

Speed ​​is a measurable quantity, and the measurement depends on the motion of the observer. For example: If an apple is sitting in a glass elevator that is descending, then an outside observer, looking into the elevator, sees the apple moving, therefore, for that observer, the apple has zero-zero speed. it occurs. For someone inside the elevator, the apple does not move, therefore, its momentum is zero. The two observers each have a frame of reference in which they observe motion, and, if the elevator is descending rapidly, they will see behavior that conforms to the same physical laws. (Relativistic Momentum)

Let the position of a particle in a fixed reference frame be x . Moving at a uniform speed u , from the point of view of the second frame of reference , the position (represented by a primed coordinate) changes with time

x'=x-ut\,.

This is called the Galilean transformation. If the particle is moving with a speed dx/dt= v In the first frame of reference, in the second, it is moving at

v'={\frac {dx'}{dt}}=v-u\,.

Since u does not change, the accelerations are the same:

a'={\frac {dv'}{dt}}=a\,.

Thus, the momentum is conserved in both the reference frames. Furthermore, as long as the force has the same form, in both frames, Newton’s second law remains unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of the reference frame is called Newtonian relativity or Galilean invariance. (Relativistic Momentum)

Changes in the reference frame can, often, simplify the calculation of momentum. For example, in a collision of two particles, a reference frame may be chosen, where, a particle starts from rest. Another, more commonly used reference frame, is the center of mass frame – the one that is moving with the center of mass. The total momentum in this frame is zero. (Relativistic Momentum)

application for collision

By itself, the law of conservation of momentum is not sufficient to determine the motion of particles after a collision. Another property of motion, kinetic energy, must be known. It is not necessarily protected. If it is conserved, the collision is called an elastic collision ; If not, it’s an inelastic collision . (Relativistic Momentum)

elastic collision

An elastic collision is one in which no kinetic energy is absorbed in the collision. Perfectly elastic “collision” can occur when objects do not touch each other, for example in nuclear or nuclear scattering where electrical repulsion keeps them apart. A catapult maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls is a good example of an almost perfectly elastic collision due to their high rigidity , but there is always some dissipation when the bodies come into contact. ^[8]

A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. The velocities U ₁ and U ₂ before the collision and V ₁ and V ₂ after, are the equations expressing the conservation of momentum and kinetic energy:

{\begin{aligned}m_{1}u_{1}+m_{2}u_{2}&=m_{1}v_{1}+m_{2}v_{2}\\{\tfrac {1}{2}}m_{1}u_{1}^{2}+{\tfrac {1}{2}}m_{2}u_{2}^{2}&={\tfrac {1}{2}}m_{1}v_{1}^{2}+{\tfrac {1}{2}}m_{2}v_{2}^{2}\,.\end{aligned}}

A change of reference frame can simplify collision analysis. For example, suppose there are two bodies of equal mass m , one stationary and one approaching the other with a speed v (as shown in the figure). The center of mass is moving at a speed V/2and both the bodies are moving towards it with speed V/2, Because of symmetry, both must move away from the center of mass at the same speed after the collision. Adding the speed of the center of mass to the two, we find that the body which was moving has now stopped and the other is moving away with a speed v . Bodies exchange their velocities. Switching to the center of mass frame leads us to the same conclusion, regardless of the bodies’ velocities. Therefore, the final velocities are given by

{\begin{aligned}v_{1}&=u_{2}\\v_{2}&=u_{1}\,.\end{aligned}}

In general, when the initial velocities are known, the final velocities are given by

v_{1}=\left({\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}\right)u_{1}+\left({\frac {2m_{2}}{m_{1}+m_{2}}}\right)u_{2}\,

v_{2}=\left({\frac {m_{2}-m_{1}}{m_{1}+m_{2}}}\right)u_{2}+\left({\frac {2m_{1}}{m_{1}+m_{2}}}\right)u_{1}\,.

If the mass of one body is much greater than that of the other, its velocity will be slightly affected by the collision while the other body will experience a large change. (Relativistic Momentum)

Elastic bump

In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as heat or sound). Examples include traffic collisions, ^[10] in which the effects of loss of kinetic energy can be seen in damage to vehicles; electrons lose some of their energy to atoms (as in the Frank-Hertz experiment); ^[11] and particle accelerators in which kinetic energy is converted into mass in the form of new particles. (Relativistic Momentum)

In a completely inelastic collision (such as a bug hitting a windshield), the latter has the same speed of the two bodies. A head-on inelastic collision between two objects can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are u1 and u2 before the _{collision then in} a perfectly stable collision both the bodies are traveling with velocity v after the collision. The equation expressing the conservation of momentum is:

{\displaystyle {\begin{aligned}m_{1}u_{1}+m_{2}u_{2}&=\left(m_{1}+m_{2}\right)v\,.\end{aligned}}}

If a body is motionless to begin with (eg ), the equation for conservation of momentum is

{\displaystyle u_ {2} = 0}

m_{1}u_{1}=\left(m_{1}+m_{2}\right)v\,,

So

v={\frac {m_{1}}{m_{1}+m_{2}}}u_{1}\,.

In an isolated case, if the frame of reference is moving at a final velocity such that , the objects will be brought to rest by a completely inelastic collision and 100% of the kinetic energy is converted into other forms of energy. In this example the initial velocities of the bodies would not be zero, or the bodies would have to be massless. v= 0

One measure of collision invulnerability is the coefficient of restitution CR , _which is defined as the ratio of the relative velocity of separation and the relative velocity of the approach. In applying this measurement to a ball bouncing off a solid surface, it can be easily measured using the following formula: (Relativistic Momentum)

C_{\text{R}}={\sqrt {\frac {\text{bounce height}}{\text{drop height}}}}\,.

The equations of motion and energy also apply to the motion of objects that start together and then move apart. For example, an explosion is the result of a chain reaction that converts stored potential energy in chemical, mechanical or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. Rockets also use conservation of momentum: the propellant is pushed outward, gains momentum, and an equal and opposite motion is imparted to the rocket. (Relativistic Momentum)

multiple dimensions

Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with x , y , z axes, velocity has components v in _x – direction, v _y in y – direction, v in _{z –} direction . The vector is represented by the boldface symbol:

\mathbf {v} =\left(v_{x},v_{y},v_{z}\right).

Similarly, momentum is a vector quantity and is represented by a boldface symbol:

\mathbf {p} =\left(p_{x},p_{y},p_{z}\right).

The equations in the previous sections work in vector form if the scalars p and v are replaced by the vectors p and v . Each vector equation represents three scalar equations. for example,

\mathbf {p} =m\mathbf {v} 

Represents three equations: 

{\begin{aligned}p_{x}&=mv_{x}\\p_{y}&=mv_{y}\\p_{z}&=mv_{z}.\end{aligned}}

The kinetic energy equations are an exception to the above substitution rule. The equations are still one-dimensional, but each represents the magnitude of a scalar vector, for example,

v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,.

Each vector equation represents three scalar equations. Often the coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results are combined to produce a vector result. ^[14]

A simple construction involving the center of mass frame can be used to show that if a stationary elastic field collides with a moving sphere, both will emerge at right angles to the collision (as in the figure). 

objects of variable mass

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as fuel ejecting from a rocket or a star accreting gas. When analyzing such an object, one considers the mass of the object as a function that varies with time: m ( t ) . Therefore , the momentum of the object at time t is p ( t ) = m ( t ) v ( t ) . Then one can try to apply Newton’s second law of motion by saying that the external force F on the object will change from its momentum p ( t ) to F.= is related by. dp/dt, but this is incorrect, as is the expression related to applying the product ruled ( mv )/dt:

{\displaystyle F=m(t){\frac {dv}{dt}}+v(t){\frac {dm}{dt}}.}(wrong)

This equation does not accurately describe the motion of variable-mass objects. the correct equation is

F=m(t){\frac {dv}{dt}}-u{\frac {dm}{dt}},

where u is the velocity of the ejected/integrated mass observed in the object’s rest frame . ^[16] This is different from V , which is the velocity of the object as seen in an inertial frame. (Relativistic Momentum)

This equation is derived by tracking both the momentum of the object as well as the momentum of the ejected/integrated mass ( dm ). When considered together, the object and mass ( dm ) form a closed system in which total momentum is conserved.

{\displaystyle P(t+dt)=(m-dm)(v+dv)+dm(v-u)=mv+mdv-udm=P(t)+mdv-udm}

Relativist

lorentz invariance

Newtonian physics holds that absolute time and space exist outside of any observer; This gives rise to the Galilean invasion. It also results in a prediction that the speed of light can vary from one reference frame to another. This is the opposite of observation. In the special theory of relativity, Einstein holds the postulate that the equations of motion do not depend on the reference frame, but postulates that the speed of light c is irreversible. Consequently, position and time in the two reference frames are related to the Lorentz transformation rather than the Galilean transformation. ^[17]

For example, one reference frame is moving with a velocity v in the x direction relative to another. The Galilean transform gives the coordinates of the moving frame as

{\displaystyle {\begin{aligned}t'&=t\\x'&=x-vt\end{aligned}}}

Whereas the Lorentz transformation gives

{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\,\end{aligned}}}

where is the Lorentz factor:

{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}

Newton’s second law, with mass constant, is not irreversible under the Lorentz transformation. However, this can be made invariant by making the inertial mass m of an object a function of velocity:

m=\gamma m_{0}\,;

m ₀ is the invariant mass of the object.

modified speed,

\mathbf {p} =\gamma m_{0}\mathbf {v} \,,

Follows Newton’s second law:

\mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.

Within the field of classical mechanics, relativistic motion closely approximates Newton’s motion: at low velocities, m ₀ V is approximately equal to m ₀ V , the Newton expression for momentum.

four-vector formulation

In the theory of special relativity, physical quantities are expressed as four-vectors consisting of three space coordinates with time as the fourth coordinate. These vectors are usually represented by capital letters, for example r for position . The expression of the four-speed depends on how the coordinates are expressed. Time can be given in its usual units or multiplied by the speed of light so that all components of a four-vector have dimensions of length. After scaling is done, the interval of the proper time, , is defined by 

c^{2}d\tau ^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}\,,

Is invariant under Lorentz transformations (what follows in this expression and in – – (+ -) metric signatures have been used, different authors use different conventions). Mathematically this irreversibility can be ensured in one of two ways: by treating the four-dimensional vectors as Euclidean vectors and by multiplying the time by −1 ; Or by keeping the time in real quantities and embedding the vectors in Minkowski space. ^[21] In a Minkowski space, two four-vector U = ( U ₀ , U ₁ , U ₂ , U ₃ ) and V = ( V₀ , V ₁ , V ₂ , V ₃ ) is defined as

\mathbf {U} \cdot \mathbf {V} =U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\, .

In all coordinate systems, the (contravariant) relativistic four-velocity is defined as

\mathbf {U} \equiv {\frac {d\mathbf {R} }{d\tau }}=\gamma {\frac {d\mathbf {R} }{dt}}\,,

and is (contravariant) four-speed

\mathbf {P} =m_{0}\mathbf {U} \,,

where M0 is the invariant mass _. If R = ( ct , x , y , z ) (in Minkowski space), then

{\displaystyle \mathbf {P} =\gamma m_{0}\left(c,\mathbf {v} \right)=(mc,\mathbf {p} )\,.}

Using Einstein’s mass-energy equivalence, E = mc ² , this can be written as:

\mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)\,.

Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

The magnitude of the four-vector momentum m ₀ c is equal to:

{\displaystyle \|\mathbf {P} \|^{2}=\mathbf {P} \cdot \mathbf {P} =\gamma ^{2}m_{0}^{2}\left(c^{2}-v^{2}\right)=(m_{0}c)^{2}\,,}

and is immutable in all reference frames.

The relativistic energy-momentum relationship holds even for massless particles such as photons; By setting m0 = 0 _it follows that

E=pc\,.

In a game of relativistic “billiards”, if a stationary particle collides with a moving particle in an elastic collision, the subsequent paths created by the two will form an acute angle. This is in contrast to the non-relativistic case where they travel at right angles. ^[22]

The four-momentum of a planar wave can be related to a wave four-vector

{\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)=\hbar \mathbf {K} =\hbar \left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)}

For a particle, the relationship between the temporal components, E = H , is the Planck -Einstein relation, and the relationship between the spatial components, P = H K , describes a de Broglie matter wave.

normalized

Newton’s laws can be difficult to apply to many types of motion because speed is limited by constraints . For example, on an abacus a bead is forced to move along its string and a pendulum bob is forced to swing a certain distance from the axis. By transforming the normal Cartesian coordinates into a set of normalized coordinates, many such constraints can be included that may be reduced in number. ^[24] Sophisticated mathematical methods have been developed to solve mechanics problems in generalized coordinates. They introduce a generalized motion , which can be referred to as canonical or conjugate motion., which extends the concepts of both linear momentum and angular momentum. To distinguish it from generalized motion, the product of mass and velocity is also known as mechanical , kinetic or kinetic motion . ^{[6] [25] [26]} The two main methods are described below.

Lagrangian mechanics

In Lagrangian mechanics, a Lagrangian is defined as the difference between kinetic energy T and potential energy V :

{\mathcal {L}} = TV \ ,.

If the normalized coordinates are represented as the vector q = ( q ₁ , q ₂ , … , q _N ) and the time differential is represented by a point above the variable, then the equation of motion ( Lagrange or Euler’s (known as the Lagrange equation) is a set of N equations:

{\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial {\mathcal {L}}}{\partial q_{j}}}=0\,.

If a coordinate q _i is not a Cartesian coordinate, then the dimensions of the linear momentum in the associated generalized momentum component _pi are not necessary. Even though q _{i is a Cartesian} coordinate, pi will not be the same as mechanical motion if the potential depends on velocity. ^[6] The representation of sciatic motion by the symbol of some formulas . ^[28]

In this mathematical framework, normalized motion is associated with normalized coordinates. Its components are defined as:

p_{j}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\,.

Each component p _j is called the conjugate momentum for the coordinate q _{j .}

Now if the given coordinate q does not appear in the Lagrangian (although its time derivative may appear), then

p_{j}={\text{constant}}\,.

This is a generalization of conservation of momentum. ^[6]

Even though the normalized coordinates are just normal spatial coordinates, the conjugate momentum is not necessarily the normal momentum coordinates. An example is found in the section on electromagnetism.

hamiltonian mechanics

In Hamiltonian mechanics, the Lagrangian (a function of normalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as:

{\mathcal {H}}\left(\mathbf {q} ,\mathbf {p} ,t\right)=\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {L}}\left(\mathbf {q} ,{\dot {\mathbf {q} }},t\right)\,,

where the momentum is obtained by varying the Lagrangian as above. Hamilton’s equations of motion are

{\begin{aligned}{\dot {q}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\\-{\dot {p}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial q_{i}}}\\-{\frac {\partial {\mathcal {L}}}{\partial t}}&={\frac {d{\mathcal {H}}}{dt}}\,.\end{aligned}}

Like in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved. ^[30]

symmetry and protection

Conservation of momentum is a mathematical consequence of the uniformity of space (shift symmetry) (position in space is the canonical conjugate quantity for momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; This is a special case of Noether’s theorem. ^[31] For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include curved spacetimes in general relativity ^[32] or time crystals in condensed matter physics.

Electromagnetic

particles in a field

In Maxwell’s equations, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force ( Lorentz force ) on a particle with charge q due to the combination of electric field E and magnetic field B is

\mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).

(in SI units). ^[37] : 2 It has electric potential ( r , t ) and magnetic vector potential A ( r , t ) . ^[28] In the non-relativistic regime, its generalized momentum is

{\displaystyle \mathbf {P} =m\mathbf {\mathbf {v} } +q\mathbf {A} ,}

Whereas in relativistic mechanics it becomes

{\displaystyle \mathbf {P} =\gamma m\mathbf {\mathbf {v} } +q\mathbf {A} .}

The quantity is sometimes called potential momentum . ^{[38] [ 39] [40]} This is the motion due to the interaction of the particle with the electromagnetic field. The name is an analogy with potential energy , which is energy due to the interaction of a particle with electromagnetic fields. These quantities form a four-vector, so the analogy is consistent; In addition, the concept of potential momentum is important in explaining the so-called hidden motion of electromagnetic fields

{\displaystyle V=q\mathbf {A} }{\displaystyle U=q\varphi }

In Newtonian mechanics, the law of conservation of momentum can be derived from the law of action and reaction, which states that every force has an equal and opposite force. Under certain circumstances, moving charged particles can exert forces on each other in opposite directions. ^[42] Nevertheless, the combined motion of the particles and the electromagnetic field is conserved.

Protection

Vacuum

The Lorentz force imparts a motion to the particle, so according to Newton’s second law the particle must accelerate the electromagnetic fields. ^[43]

In vacuum, the momentum per unit volume is

\mathbf {g} ={\frac {1}{\mu _{0}c^{2}}}\mathbf {E} \times \mathbf {B} \,,

where μ is the vacuum permittivity and _c is the speed of light. The momentum density is proportional to the Poynting vector S which gives the directional rate of energy transfer per unit area:

\mathbf {g} ={\frac {\mathbf {S} }{c^{2}}}\,.

If momentum is to be conserved over volume V over field Q , then the change in momentum of matter through the Lorentz force must be balanced by a change in momentum of the electromagnetic field and an outflow of momentum. If P _{mech is the momentum of all particles in} Q and the particles are considered to be a continuum, then Newton’s second law gives

{\displaystyle {\frac {d\mathbf {P} _{\text{mech}}}{dt}}=\iiint \limits _{Q}\left(\rho \mathbf {E} +\mathbf {J } \times\mathbf{B}\right)dV\,.}

electromagnetic speed is

{\displaystyle\mathbf{P}={\frac{1}{\mu_{0}c^{2}}}\iiint\limits_{Q}\mathbf{E} \times\mathbf{B}\,dV\,,}

and the equation for conservation of momentum i for each component is

{\displaystyle {\frac {d}{dt}}\left(\mathbf {P} _{\text{mech}}+\mathbf {P} _{\text{field}}\right)_{i}=\iint \limits _{\sigma }\left(\sum \limits _{j}T_{ij}n_{j}\right)d\Sigma \,.}

The term on the right is an integral over the surface area representing the momentum flow in and out of the volume of the surface, and nj is a component of the surface _normal s . The quantity T _ij is called the Maxwell stress tensor, which is defined as

T_{ij}\equiv \epsilon _{0}\left(E_{i}E_{j}-{\frac {1}{2}}\delta _{ij}E^{2}\right)+{\frac {1}{\mu _{0}}}\left(B_{i}B_{j}-{\frac {1}{2}}\delta _{ij}B^{2}\right)\,.

Media

The above results hold for the microscopic Maxwell equations, which apply to electromagnetic forces in vacuum (or on very small scales in media). Momentum density in media is more difficult to define because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density has been revised to

\mathbf {g} ={\frac {1}{c^{2}}}\mathbf {E} \times \mathbf {H} ={\frac {\mathbf {S} }{c^{2}}}\,,

where H-field H is related to B-field and magnetization M

\mathbf {B} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,.

The electromagnetic stress tensor depends on the properties of the media. ^[43]

quantum mechanical

In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the motion and state of a given observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.

The momentum conductor for a particle described on the basis of position can be written as

\mathbf {p} ={\hbar \over i}\nabla =-i\hbar \nabla \,,

where is the gradient operator, h is the lesser Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, although the momentum operator may take other forms in other bases. For example, a momentum conductor in momentum space is represented as

\mathbf {p} \psi (p)=p\psi (p)

where the operator p acts on a wave function ( p ) , multiplying that wave function by the value p , in a similar fashion to the way the position operator acts on a wave function ( x ) , that wave function Gets the value by multiplying by x .

For both massless and massless objects, the relative motion is related to the phase constant by 

\beta 

{\displaystyle p=\hbar \beta }

Electromagnetic radiation (including visible light, ultraviolet light and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still move. This leads to applications such as solar sails. The calculation of the speed of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy). ^[46] [47]

In deformed bodies and fluids

Conservation in Continuity

In fields such as fluid dynamics and solid mechanics, it is not possible to follow the motion of individual atoms or molecules. Instead, the material should be approximated by a continuum in which at each point a particle or fluid parcel is assigned the average of the properties of atoms in a small nearby region. In particular, it is a density and velocity V that depend on time t and position r . The speed per unit volume is v . ^[48]

Consider a column of water in hydrostatic equilibrium. All the forces on the water are in equilibrium and the water is motionless. Two forces are balanced on any drop of water. The first is gravity, which acts directly on each atom and the molecule inside. The gravitational force per unit volume is g , where g is the gravitational acceleration. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by an amount needed to balance gravity. The normal force per unit area is the pressure p . The average force per unit volume inside the droplet is the gradient of the pressure, so the force balancing equation is

-\nabla p+\rho \mathbf {g} =0\,.

If the forces are not balanced, the drop accelerates. This acceleration is not just a partial derivative∂ v/t Because the fluid in a given volume changes with time. Instead, the content derivation is required:

{\frac {D}{Dt}}\equiv {\frac {\partial }{\partial t}}+\mathbf {v} \cdot {\boldsymbol {\nabla }}\,.

Applied to any physical quantity, the physical derivative includes the rate of change at a point and the changes caused by convection in the form of a fluid. The rate of change in speed per unit volume is equal to DV _/dt, It is equal to the total force acting on the droplet.

Forces that can change the speed of a droplet include pressure and the gradient of gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a shear stress , exerted by a force parallel to the droplet surface, is proportional to the rate of deformation or the strain rate. Such shear stress occurs when the fluid has a velocity gradient because the fluid is moving faster on one side than the other. If the speed in the x direction varies with z , then the tangential force per unit area of ​​x normal to the z direction is

{\displaystyle \sigma _{\text{zx}}=-\mu {\frac {\partial v_{x}}{\partial z}}\,,}

where μ is viscosity. It is also a flow of X-momentum, or flow per unit area, through a surface. ^[51]

The momentum equilibrium equations for the incompressible flow of a Newtonian fluid, including the effect of viscosity, are

\rho {\frac {D \mathbf {v}} {Dt}} = - {\boldsymbol {\nabla}} p + \mu \nabla ^ {2} \mathbf {v} + \rho \mathbf {g}. \,

These are known as Navier-Stokes equations. ^[52]

Momentum equilibrium equations can be extended to more general materials, including solids. For a surface with normal to the direction i and the force in the direction j , there _is a stress component ij . Nine components make up the Cauchy stress tensor , which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation:

\rho {\frac {D\mathbf {v} }{Dt}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {f} \,,

where f is the body force. ^[53]

The Cauchy Momentum Equation is broadly applicable to deformation of solids and liquids. The relationship between strain and strain rate depends on the properties of the material (see Types of viscosity).

acoustic waves

A disturbance in a medium gives rise to oscillations or waves, which propagate away from their source. In a fluid, small changes in pressure P can often be described by the acoustic wave equation:

{\frac {\partial ^{2}p}{\partial t^{2}}}=c^{2}\nabla ^{2}p\,,

where c is the speed of sound. In a solid, similar equations can be obtained for the propagation of pressure (p-waves) and shear (s-waves). ^[54]

The velocity v _i of a momentum component v j is equal to the flux, or transport per unit area, v _j v _{j .} In the linear approximation leading to the above acoustic equation, the time average of this flow is zero. However, non-linear effects can give rise to a non-zero average. ^[55] It is possible to have a flow of momentum even though the wave itself does not have a mean momentum. ^[56]

history of concept

In about 530 AD, working in Alexandria, the Byzantine philosopher John Philoponus developed the concept of motion in his Commentaries on Aristotle’s Physics . Aristotle claimed that whatever is going on must be carried on by something or the other. For example, a thrown ball must keep moving at the speed of the wind. Most writers continued to accept Aristotle’s theory until the time of Galileo, but some were skeptical. Philoponus pointed to the absurdity in Aristotle’s claim that the motion of an object is accelerated by the same wind that is opposing its path. He proposed instead that an incentive was given to the act of throwing the object. ^[57] Ibn Sina (also known by his Latin name Avicenna) read Philoponus and published The Book of Healing in 1020.published his theory of motion. They agreed that an impetus is given to the projectile by the thrower; But unlike Philoponus, who believed that it was a temporary property that would decrease even in a vacuum, he saw it as a constant, requiring external forces such as air resistance to destroy it. ^{[58] [ 59] [60]} The work of Philoponus and possibly Ibn Sina, ^[60]was read and refined by the European philosophers Peter Olivi and Jean Buridan. Buridan, who was made rector of the University of Paris in about 1350, noted the incentive for motion to be proportional to weight. Furthermore, Buridan’s theory differed from its predecessor in that he did not consider the incentive for self-dissolution, claiming that a body would be arrested by the forces of air resistance and gravity that would oppose its stimulus. can. ^[61] [62]

René Descartes believed that the total “quantity of motion” (Latin: quantitas motus ) is conserved in the universe, ^[63] where the quantity of motion is understood as the product of size and momentum. This should not be read as a statement of the modern law of momentum, as he had no concept of mass as distinct from weight and size, and more important, he believed that it was motion rather than motion. which is protected. So for Descartes if a moving object bounces off a surface, its direction changes but not its speed, its amount of motion will not change. ^{[64] [65] [66]} Galileo in his To New Sciences used the Italian term impato to similarly describe the amount of motion of Descartes.used.

Leibniz, in his “Lectures on Metaphysics”, made an argument against Descartes’ construction of a conservation of “quantity of momentum”, using the example of leaving blocks of different distances of different sizes. He explains that force is conserved but the amount of momentum, which is considered as the product of the size and momentum of an object, is not conserved. ^[67]

Christiaan Huygens long ago concluded that Descartes’ laws for elastic collisions of two bodies must be false, and he formulated the correct laws. ^[68] An important step was his recognition of the Galilean irreversibility of problems. ^[69] It then took many years for his ideas to be disseminated. He personally handed them over to William Bronker and Christopher Wren in London in 1661. ^[70] What Spinoza wrote about him to Henry Oldenburg in 1666 was preserved during the Second Anglo-Dutch War. ^[71] Huygens actually published a manuscript De Motu Corporum X Percussion in the period 1652–6.I had prepared them. The war ended in 1667, and Huygens announced its results to the Royal Society in 1668. He published them in the Journal des Scavans in 1669. ^[72]

The first correct statement of the law of conservation of momentum was given by the English mathematician John Wallis in his 1670 work, Mechanica sive de motu, Tractatus geometricus : “The initial state of a body, of rest, or of motion, will remain” and “if the force resists greater than that, speed will result”. ^[73] Wallis used viz for momentum and force for the amount of motion . Newton’s Philosophiae Naturalis Principia Mathematica , when it was first published in 1687, showed a similar casting to use words for mathematical motion. His Definition II Quantitas Motus, defines “a quantity of motion” as “the amount of velocity and matter produced jointly”, which it identifies as motion. ^[74] Thus when in Law II he refers to mutable motion , “change in motion”, which is proportional to the force affected, it is generally taken to mean motion and not motion. ^[75] It remained only to specify a standard term for the amount of motion. The first time “motion” was used in its proper mathematical sense is not clear, but by the time of Jennings’ Collection of Objects in 1721, five years before the final edition of Newton’s Principia Mathematica , momentum M or “quantity of motion”.Product of Q and V , where Q is “volume of material” and V is “velocity”,