Bonferroni correction

In physics, the center-of-momentum frame (zero-momentum frame, or COM frame) of a system is the unique inertial frame in which the center of mass of the system is at rest. The total momentum of the system vanishes in this reference frame. The center of momentum of a system is not a location, but usually refers to the coordinate reference frame in which the momenta of a system's components add to zero. Thus "center of momentum" already means "center-of-momentum frame" and is a short form of this phrase.^[1]

A special case of the center-of-momentum frame is the center-of-mass frame: an inertial frame in which the center of mass (which is a physical point) is at the origin at all times. In all COM frames, the center of mass is at rest, but it may not necessarily be at rest at the origin of the coordinate system.

Properties

General

The center of momentum frame is defined as the inertial frame in which the sum over the linear momentum of each particle vanishes. Let $S$ denote the lab reference system and $S^{'}$ denote the center of momentum reference frame. Quantities in $S^{'}$ are denoted by a prime. Using a galilean transformation, the particle velocity in $S^{'}$ is

$v^{'} = v - V_{c},$

where $V_{c} = \frac{\sum_{i} m_{i} v_{i}}{\sum_{i} m_{i}}$

is the velocity of the mass center. The total momentum in the center-of-mass system then vanishes:

$\sum_{i} p'_{i} = \sum_{i} m_{i} v'_{i} = \sum_{i} m_{i} (v_{i} - V_{c}) = \sum_{i} m_{i} v_{i} - \sum_{i} m_{i} \frac{\sum_{j} m_{j} v_{i}}{\sum_{j} m_{j}} = \sum_{i} m_{i} v_{i} - \sum_{j} m_{j} v_{j} = 0$

Also, the total energy of the system is the minimal energy as seen from all inertial reference frames.

Special relativity

In relativity, COM frame exists for a massive system. In the COM frame the total energy of the system is the "rest energy", and this quantity (when divided by the factor c², where c is the speed of light) therefore gives the rest mass (positive invariant mass) of the system:

m = \frac{E}{c^{2}} .

The invariant mass of the system is actually given by the relativistic invariant relation:

m^{2} = {(\frac{E}{c^{2}})}^{2} - {(\frac{p}{c})}^{2}

but for zero momentum the momentum term (p/c)² vanishes, hence the total energy coincides with the rest energy.

Systems which have energy but zero invariant mass (such as photons moving in a single direction, or equivalently, plane electromagnetic waves) do not have COM frames, because there is no frame which they have zero net momentum. Due to the invariance of the speed of light, such massless systems must travel at the speed of light in any frame, and therefore always possess a net momentum-magnitude which is equal to their energy divided by the speed of light:

p = E / c .

Two-body problem

An example of the usage of this frame is given below – in a two-body collision, not necessarily elastic (where kinetic energy is conserved). The COM frame can be used to find the momentum of the particles much easier than in a lab frame: the frame where the measurement or calculation is done. The situation is analyzed using Galilean transformations and conservation of momentum (for generality, rather than kinetic energies alone), for two particles of mass m₁ and m₂, moving at initial velocities (before collision) u₁ and u₂ respectively. The transformations are applied to take the velocity of the frame from the velocity of each particle from the lab frame (unprimed quantities) to the COM frame (primed quantities):^[2]

u_{1}^{'} = u_{1} - V, u_{2}^{'} = u_{2} - V

where V is the velocity of the COM frame. Since V is the velocity of the COM, i.e. the time derivative of the COM location R (position of the center of mass of the system):^[3]

\begin{aligned} \frac{d R}{d t} & = \frac{d}{d t} (\frac{m_{1} r_{1} + m_{2} r_{2}}{m_{1} + m_{2}}) \\ = \frac{m_{1} u_{1} + m_{2} u_{2}}{m_{1} + m_{2}} \\ = V \end{aligned}

so at the origin of the COM frame, R = 0, this implies

m_{1} u_{1}^{'} + m_{2} u_{2}^{'} = 0

The same results can be obtained by applying momentum conservation in the lab frame, where the momenta are p₁ and p₂:

V = \frac{p_{1} + p_{2}}{m_{1} + m_{2}} = \frac{m_{1} u_{1} + m_{2} u_{2}}{m_{1} + m_{2}}

and in the COM frame, where it is asserted definitively that the total momenta of the particles, p₁' and p₂', vanishes:

p_{1}^{'} + p_{2}^{'} = m_{1} u_{1}^{'} + m_{2} u_{2}^{'} = 0

Using the COM frame equation to solve for V returns the lab frame equation above, demonstrating any frame (including the COM frame) may be used to calculate the momenta of the particles. It has been established the velocity of the COM frame can be removed from the calculation using that frame, so the momenta of the particles in the COM frame can be expressed in terms of the quantities in the lab frame (i.e. the given initial values):

\begin{aligned} p_{1}^{'} & = m_{1} u_{1}^{'} \\ = m_{1} (u_{1} - V) = \frac{m_{1} m_{2}}{m_{1} + m_{2}} (u_{1} - u_{2}) \\ = - m_{2} u_{2}^{'} = - p_{2}^{'} \end{aligned}

notice the relative velocity in the lab frame of particle 1 to 2 is;

Δ u = u_{1} - u_{2}

and the 2-body reduced mass is;

μ = \frac{m_{1} m_{2}}{m_{1} + m_{2}}

so the momenta of the particles compactly reduce to

p_{1}^{'} = - p_{2}^{'} = μ Δ u

This is a substantially simpler calculation of the momenta of both particles; the reduced mass and relative velocity can be calculated from the initial velocities in the lab frame and the masses, and the momentum of one particle is simply the negative of the other. The calculation can be repeated for final velocities v₁ and v₂ in place of the initial velocities u₁ and u₂, since after the collision the velocities still satisfy the above equations:^[4]

\begin{aligned} \frac{d R}{d t} & = \frac{d}{d t} (\frac{m_{1} r_{1} + m_{2} r_{2}}{m_{1} + m_{2}}) \\ = \frac{m_{1} v_{1} + m_{2} v_{2}}{m_{1} + m_{2}} \\ = V \end{aligned}

so at the origin of the COM frame, R = 0, this implies after the collision

m_{1} v_{1}^{'} + m_{2} v_{2}^{'} = 0

In the lab frame, the conservation of momentum fully reads:

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) V

This equation does not imply that

m_{1} u_{1} = m_{1} v_{1} = m_{1} V, m_{2} u_{2} = m_{2} v_{2} = m_{2} V

instead, it simply indicates the total mass M multiplied by the velocity of the centre of mass V is the total momentum P of the system:

\begin{aligned} P & = p_{1} + p_{2} \\ = (m_{1} + m_{2}) V \\ = M V \end{aligned}

Similar analysis to the above obtains;

p_{1}^{'} = - p_{2}^{'} = μ Δ v = μ Δ u

where the final relative velocity in the lab frame of particle 1 to 2 is;

Δ v = v_{1} - v_{2} = Δ u .

References

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↑ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8
↑ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8
↑ Classical Mechanics, T.W.B. Kibble, European Physics Series, 1973, ISBN 07-084018-0
↑ An Introduction to Mechanics, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, ISBN 978-0-521-19821-9

[1] Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8

[2] Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8

[3] Classical Mechanics, T.W.B. Kibble, European Physics Series, 1973, ISBN 07-084018-0

[4] An Introduction to Mechanics, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, ISBN 978-0-521-19821-9

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Bonferroni correction

Contents

Properties

General

Special relativity

Two-body problem

See also

References

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Bonferroni correction

Properties

General

Special relativity

Two-body problem

See also

References

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