# Relativistic impulse

In the special theory of relativity , the momentum is related to the speed differently than in Newtonian mechanics and is therefore also called relativistic momentum . The relativistic impulse is the actually effective one, e.g. B. for particles that hit target bodies in accelerators. In the case of collisions and other interactions between particles, it turns out to be an additive conservation quantity : the sum of the initial impulses corresponds to the sum of the impulses after the interaction.

In the special theory of relativity, the momentum of a particle of mass depends nonlinearly on the velocity : ${\ displaystyle {\ vec {p}}}$ ${\ displaystyle m}$ ${\ displaystyle {\ vec {v}}}$

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

Here is the relativistic factor (Lorentz factor) . The Lorentz factor increases with increasing speed, and infinite with the speed of light. ${\ displaystyle \ gamma}$

For non-relativistic velocities , approximately 1, i.e. H. for small velocities we get the classical momentum of Newtonian mechanics : ${\ displaystyle (v \ ll c)}$${\ displaystyle \ gamma}$

${\ displaystyle {\ vec {p}} _ {\ text {Newton}} = m {\ vec {v}}}$

According to Noether's theorem , conservation of momentum includes the symmetry of the effect under spatial displacements.

If a force transfers momentum to a particle in the course of time, this changes its momentum, i.e. H. Force is momentum transfer per time: ${\ displaystyle {\ vec {F}}}$

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

## Derivation

In relativistic physics, both the momentum and the energy of a particle of mass must be additive conserved quantities for every observer. The dependence of the momentum and the energy on the speed can be derived from this. ${\ displaystyle m}$${\ displaystyle {\ vec {v}}}$

A derivation also results from the effect

${\ displaystyle S [{\ mathcal {L}}] = \ int {\ mathcal {L}} \ left (t, {\ vec {x}} (t), {\ vec {v}} (t) \ right) \, \ mathrm {d} t}$

with the Lagrangian

${\ displaystyle {\ mathcal {L}} (t, {\ vec {x}}, {\ vec {v}}) = - mc ^ {2} {\ sqrt {1 - {\ frac {v ^ {2 }} {c ^ {2}}}}}.}$

Since the Lagrangian does not depend on the location (that is, the components are cyclic ), the effect is invariant under spatial displacements. The conservation quantity associated with Noether's theorem is by definition the momentum. In the present case this is the momentum to be conjugated with components ${\ displaystyle {\ vec {x}}}$${\ displaystyle x ^ {i} \ ,, i = 1,2,3 \ ,,}$${\ displaystyle {\ vec {x}}}$

${\ displaystyle p_ {i} = {\ frac {\ partial {\ mathcal {L}}} {\ partial v ^ {i}}} = {\ frac {mv ^ {i}} {\ sqrt {1-v ^ {2} / c ^ {2}}}},}$ so
${\ displaystyle {\ vec {p}} = {\ frac {m {\ vec {v}}} {\ sqrt {1-v ^ {2} / c ^ {2}}}} \ ,.}$

Since the Lagrangian does not depend on time , according to Noether's theorem is the energy ${\ displaystyle t}$

${\ displaystyle E = v ^ {i} {\ frac {\ partial {\ mathcal {L}}} {\ partial v ^ {i}}} - {\ mathcal {L}} = {\ frac {mc ^ { 2}} {\ sqrt {1-v ^ {2} / c ^ {2}}}}}$

a conservation size. The speed as a function of momentum is

${\ displaystyle {\ vec {v}} = {\ frac {\ vec {p}} {\ sqrt {m ^ {2} + p ^ {2} / c ^ {2}}}},}$

as it results from the reverse . From this follows the energy as a function of the phase space variables , the Hamilton function${\ displaystyle {\ vec {p}} ({\ vec {v}})}$

${\ displaystyle H (t, {\ vec {x}}, {\ vec {p}}) = {\ sqrt {m ^ {2} c ^ {4} + p ^ {2} c ^ {2}} }.}$

The energy and the momentum thus fulfill the energy-momentum relationship and lie on the mass shell .