# Generalized impulse

The generalized momentum , also generalized , canonical , canonically conjugated , or conjugated momentum , occurs both in Hamiltonian mechanics and in Lagrange mechanics . Together with the location, it characterizes the respective state of the system, which changes over time according to Hamilton's equations of motion .

As a function of location and speed , the generalized momentum is the partial derivative of the Lagrange function with respect to speed: ${\ displaystyle q}$${\ displaystyle {\ dot {q}}}$ ${\ displaystyle L}$

${\ displaystyle p_ {j} = {\ frac {\ partial L} {\ partial {\ dot {q}} _ {j}}} \ ,, \ j = 1 .... n}$

In the transition from classical physics to quantum mechanics , the canonical momentum (in contrast to the kinetic momentum ) is replaced by the momentum operator : ${\ displaystyle {\ hat {p}}}$

${\ displaystyle p_ {j} \ rightarrow {\ hat {p}} _ {j} = - \ hbar i {\ frac {\ partial} {\ partial x_ {j}}}}$

## Examples

### Classic movement

• When a particle of mass moves in a potential without constraints in Cartesian coordinates${\ displaystyle m}$ ${\ displaystyle V (\ mathbf {x}, t)}$
${\ displaystyle L = {\ frac {1} {2}} \, m \, {\ dot {\ mathbf {x}}} ^ {2} -V (\ mathbf {x}, t)}$
the generalized momentum is equal to the kinetic momentum:
${\ displaystyle \ mathbf {p} = m {\ dot {\ mathbf {x}}}}$
• When a particle of mass moves in a potential in cylindrical coordinates${\ displaystyle m}$${\ displaystyle V (r, \ varphi, z, t)}$
${\ displaystyle L = {\ frac {1} {2}} \, m {\ bigl (} {\ dot {r}} ^ {2} + r ^ {2} {\ dot {\ varphi}} ^ { 2} + {\ dot {z}} ^ {2} {\ bigr)} - V (r, \ varphi, z, t)}$
the generalized momentum conjugated to the angle is the component of the angular momentum in the direction of the cylinder axis:
${\ displaystyle p _ {\ dot {\ varphi}} = {\ frac {\ partial L} {\ partial {\ dot {\ varphi}}}} = m \, r ^ {2} {\ dot {\ varphi} }}$
• When moving a point charge with mass in the electromagnetic field ( is the electrical potential )${\ displaystyle q}$${\ displaystyle m}$${\ displaystyle \ phi}$
${\ displaystyle L = {\ frac {1} {2}} \, m \, {\ dot {\ mathbf {x}}} ^ {2} -q \, \ phi (t, \ mathbf {x}) + q \, {\ dot {\ mathbf {x}}} \ cdot \ mathbf {A} (t, \ mathbf {x})}$
the generalized momentum has a contribution from the vector potential of the field in addition to the kinetic momentum :${\ displaystyle \ mathbf {A}}$
${\ displaystyle \ mathbf {p} = m \, {\ dot {\ mathbf {x}}} + q \, \ mathbf {A} (t, \ mathbf {x})}$

### Relativistic movement

• With the relativistic movement of a particle of mass in a potential without constraints in Cartesian coordinates${\ displaystyle m_ {0}}$${\ displaystyle V (\ mathbf {x}, t)}$
${\ displaystyle L = -m_ {0} \, c ^ {2} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x}}} ^ {2}} {c ^ {2}} }}} - V (\ mathbf {x}, t)}$
the generalized momentum is equal to the kinetic momentum:
${\ displaystyle \ mathbf {p} = {\ frac {m_ {0} \, {\ dot {\ mathbf {x}}}} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x} }} ^ {2}} {c ^ {2}}}}}}}$
• With the relativistic movement of a point charge with the mass in the electromagnetic field${\ displaystyle q}$${\ displaystyle m_ {0}}$
${\ displaystyle L = -m_ {0} \, c ^ {2} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x}}} ^ {2}} {c ^ {2}} }}} - q \, \ phi (t, \ mathbf {x}) + q \, {\ dot {\ mathbf {x}}} \ cdot \ mathbf {A} (t, \ mathbf {x})}$
the generalized momentum has a contribution from the vector potential of the field in addition to the kinetic momentum:
${\ displaystyle \ mathbf {p} = {\ frac {m_ {0} \, {\ dot {\ mathbf {x}}}} {\ sqrt {1 - {\ frac {{\ dot {\ mathbf {x} }} ^ {2}} {c ^ {2}}}}}} + q \, \ mathbf {A} (\ mathbf {x}, t)}$

## literature

• Wolfgang Nolting: Basic Course Theoretical Physics 2 Analytical Mechanics . 7th edition. Springer, Heidelberg 2006, ISBN 3-540-30660-9 .