Teorema de noether

En matemáticas y física , el teorema de Noether , también llamado teorema de simetría , debido a Emmy Noether , destaca el vínculo entre las simetrías de un sistema físico y las cantidades conservadas . Ejemplos importantes son el impulso si el sistema tiene una simetría para traslaciones espaciales , el momento angular para sistemas invariantes para rotaciones y la energía para simetrías de tiempo.

Generalidad

Más específicamente, el teorema de Noether establece que en cada simetría del Lagrangiano , es decir, en cada transformación continua de las coordenadas generalizadas $q_{i}$ ${\ Displaystyle q_ {i}}$ $q_ {i}$ Y ${\dot {q}}_{i}$ ${\ Displaystyle {\ dot {q}} _ {i}}$ ${\ dot q} _ {i}$ y posiblemente tiempo $t$ ${\ Displaystyle t}$ $t$ , que deja el Lagrangiano sin cambios ${\mathcal {L}}({\dot {\mathbf {q} }},\mathbf {q} ,t)$ ${\ Displaystyle {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t)}$ ${\ Displaystyle {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t)}$ , corresponde a una cantidad conservada . Por ejemplo, si sigue la transformación $q(t)\to q(t)+\varepsilon$ ${\ Displaystyle q (t) \ to q (t) + \ varepsilon}$ ${\ Displaystyle q (t) \ to q (t) + \ varepsilon}$ , Dónde está $\varepsilon$ ${\ Displaystyle \ varepsilon}$ $\ varepsilon$ es una cantidad infinitesimal, tenemos que:

{\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}=0

{\ displaystyle {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} = 0}

{\ displaystyle {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} = 0}

es decir $\mathbf {q}$ ${\ Displaystyle \ mathbf {q}}$ $\ mathbf q$ es una coordenada cíclica , es decir, el Lagrangiano no depende explícitamente de ella, entonces $\mathbf {p}$ ${\ Displaystyle \ mathbf {p}}$ ${\ mathbf p}$ se conserva:

{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}=\mathbf {p} ={\text{costante}}

{\ Displaystyle {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} = \ mathbf {p} = {\ text {constante}}}

{\ Displaystyle {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} = \ mathbf {p} = {\ text {constante}}}

Dónde está $\mathbf {p}$ ${\ Displaystyle \ mathbf {p}}$ ${\ mathbf p}$ es el momento conjugado a la coordenada $\mathbf {q}$ ${\ Displaystyle \ mathbf {q}}$ $\ mathbf q$ .

El teorema, que también está formulado para las simetrías de la acción funcional, fue publicado por Emmy Noether en 1918 en el artículo "Invariante Variationsprobleme", que apareció en el Gottinger Nachrichten . ^[1] ^[2]

Introducción

En el caso más simple, se puede considerar un punto material de masa. $metro$ ${\ Displaystyle m}$ $metro$ en una dimensión con posición $\mathbf {q} (t)$ ${\ Displaystyle \ mathbf {q} (t)}$ ${\ mathbf q} (t)$ y velocidad ${\dot {\mathbf {q} }}=d\mathbf {q} /dt$ ${\ Displaystyle {\ dot {\ mathbf {q}}} = d \ mathbf {q} / dt}$ ${\ Displaystyle {\ dot {\ mathbf {q}}} = d \ mathbf {q} / dt}$ , descrito por el Lagrangiano ${\mathcal {L}}({\dot {\mathbf {q} }},\mathbf {q} )$ ${\ Displaystyle {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q})}$ ${\ Displaystyle {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q})}$ . El impulso $\mathbf {p} =\partial {\mathcal {L}}/\partial {\dot {\mathbf {q} }}$ ${\ Displaystyle \ mathbf {p} = \ parcial {\ mathcal {L}} / \ parcial {\ dot {\ mathbf {q}}}}$ ${\ Displaystyle \ mathbf {p} = \ parcial {\ mathcal {L}} / \ parcial {\ dot {\ mathbf {q}}}}$ del punto material y la fuerza $\mathbf {F}$ ${\ Displaystyle \ mathbf {F}}$ $\ mathbf F$ agente en él:

\mathbf {F} ={\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}

{\ Displaystyle \ mathbf {F} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}}}

{\ Displaystyle \ mathbf {F} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}}}

están relacionados por la ecuación de Euler-Lagrange :

{F}={\dot {p}}

{\ Displaystyle {F} = {\ dot {p}}}

{\ Displaystyle {F} = {\ dot {p}}}

que constituye la ecuación de movimiento del sistema. Supongamos que trasladamos la posición del punto de $\mathbf {q}$ ${\ Displaystyle \ mathbf {q}}$ $\ mathbf q$ para $\mathbf {q} ^{\prime }$ ${\ Displaystyle \ mathbf {q} ^ {\ prime}}$ ${\ Displaystyle \ mathbf {q} ^ {\ prime}}$ con una transformación espacial parametrizada por la variable $s$ ${\ Displaystyle s}$ $s$ , es decir $\mathbf {q} ^{\prime }=\mathbf {q} (s)$ ${\ Displaystyle \ mathbf {q} ^ {\ prime} = \ mathbf {q} (s)}$ ${\ Displaystyle \ mathbf {q} ^ {\ prime} = \ mathbf {q} (s)}$ . Si el lagrangiano permanece sin cambios después de la transformación, entonces su derivada con respecto a $s$ ${\ Displaystyle s}$ $s$ No es nada:

{\frac {d}{ds}}{\mathcal {L}}({\dot {\mathbf {q} }}(s),\mathbf {q} (s))=0

{\ Displaystyle {\ frac {d} {ds}} {\ mathcal {L}} ({\ dot {\ mathbf {q}}} (s), \ mathbf {q} (s)) = 0}

{\ Displaystyle {\ frac {d} {ds}} {\ mathcal {L}} ({\ dot {\ mathbf {q}}} (s), \ mathbf {q} (s)) = 0}

El teorema de Noether establece que en este caso la cantidad $J=\mathbf {p} \cdot d\mathbf {q} (s)/ds$ ${\ Displaystyle J = \ mathbf {p} \ cdot d \ mathbf {q} (s) / ds}$ ${\ Displaystyle J = \ mathbf {p} \ cdot d \ mathbf {q} (s) / ds}$ se conserva, es decir ${\dot {J}}=0$ ${\ Displaystyle {\ dot {J}} = 0}$ ${\ dot J} = 0$ . Se dice que $J$ ${\ Displaystyle J}$ $J$ es una constante de movimiento .

De manera equivalente, si el punto material tiene una posición $\mathbf {q} =(q_{1},\dots ,q_{n})$ ${\ Displaystyle \ mathbf {q} = (q_ {1}, \ dots, q_ {n})}$ ${\ mathbf q} = (q_ {1}, \ dots, q_ {n})$ y si el lagrangiano no depende de alguna variable $q_{i}$ ${\ Displaystyle q_ {i}}$ $q_ {i}$ las ecuaciones de Euler-Lagrange:

{\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}\right)-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}=0,\quad i=1,\dots ,n

{\ Displaystyle {\ frac {d} {dt}} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {q}} _ {i}}} \ derecha) - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial q_ {i}}} = 0, \ quad i = 1, \ puntos, n}

{\ Displaystyle {\ frac {d} {dt}} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {q}} _ {i}}} \ derecha) - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial q_ {i}}} = 0, \ quad i = 1, \ dots, n}

muestra que si $\partial {\mathcal {L}}/\partial {q}_{i}=0$ ${\ estilo de visualización \ parcial {\ mathcal {L}} / \ parcial {q} _ {i} = 0}$ ${\ estilo de visualización \ parcial {\ mathcal {L}} / \ parcial {q} _ {i} = 0}$ entonces la cantidad $p_{i}=\partial {\mathcal {L}}/\partial {\dot {q}}_{i}$ ${\ Displaystyle p_ {i} = \ parcial {\ mathcal {L}} / \ parcial {\ dot {q}} _ {i}}$ ${\ Displaystyle p_ {i} = \ parcial {\ mathcal {L}} / \ parcial {\ dot {q}} _ {i}}$ se conserva , teniendo una derivada temporal nula.

Cuando una función es invariante con respecto a una transformación continua que involucra una o más variables, se dice que la función tiene una o más simetrías . El teorema de Noether también puede enunciarse considerando, en lugar de directamente el Lagrangiano, las simetrías de la acción asociada al movimiento del sistema, que es la integral del Lagrangiano con respecto al tiempo. ^[3]

Declaración

Dado un sistema de coordenadas generalizado $\mathbf {q} =(q_{1},\dots ,q_{n})$ ${\ Displaystyle \ mathbf {q} = (q_ {1}, \ dots, q_ {n})}$ ${\ Displaystyle \ mathbf {q} = (q_ {1}, \ dots, q_ {n})}$ para $norte$ ${\ Displaystyle n}$ $norte$ grados de libertad con velocidad $\mathbf {\dot {q}} =({\dot {q}}_{1},\dots ,{\dot {q}}_{n})$ ${\ Displaystyle \ mathbf {\ dot {q}} = ({\ dot {q}} _ {1}, \ dots, {\ dot {q}} _ {n})}$ ${\ Displaystyle \ mathbf {\ dot {q}} = ({\ dot {q}} _ {1}, \ dots, {\ dot {q}} _ {n})}$ y una función $\mathbf {f} (t)$ ${\ Displaystyle \ mathbf {f} (t)}$ ${\ Displaystyle \ mathbf {f} (t)}$ , si sigue la transformación infinitesimal:

t\to t,\quad q_{i}(t)\to q_{i}(t)+\varepsilon f_{i}(t),\quad {\dot {q}}_{i}(t)\to {\dot {q}}_{i}(t)+\varepsilon {\dot {f}}_{i}(t)

{\ Displaystyle t \ a t, \ quad q_ {i} (t) \ to q_ {i} (t) + \ varepsilon f_ {i} (t), \ quad {\ dot {q}} _ {i} (t) \ a {\ dot {q}} _ {i} (t) + \ varepsilon {\ dot {f}} _ {i} (t)}

{\ Displaystyle t \ a t, \ quad q_ {i} (t) \ to q_ {i} (t) + \ varepsilon f_ {i} (t), \ quad {\ dot {q}} _ {i} (t) \ a {\ dot {q}} _ {i} (t) + \ varepsilon {\ dot {f}} _ {i} (t)}

el lagrangiano ${\mathcal {L}}({\dot {\mathbf {q} }},\mathbf {q} ,t)$ ${\ Displaystyle {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t)}$ ${\ Displaystyle {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t)}$ no cambia, entonces las cantidades:

\sum _{i=1}^{n}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}f_{i}

{\ Displaystyle \ sum _ {i = 1} ^ {n} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} f_ {i}}

{\ Displaystyle \ sum _ {i = 1} ^ {n} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} f_ {i}}

son constantes de movimiento, es decir, se conservan . ^[4]

En el caso de una transformación que también involucra tiempo, es decir $t\to t+\varepsilon$ ${\ Displaystyle t \ to t + \ varepsilon}$ ${\ Displaystyle t \ to t + \ varepsilon}$ , tenemos eso:

{\frac {d{\mathcal {L}}}{dt}}={\frac {\partial {\mathcal {L}}}{\partial t}}+\sum _{i=1}^{n}\left[{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}{\ddot {q}}_{i}\right]

{\ Displaystyle {\ frac {d {\ mathcal {L}}} {dt}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial t}} + \ sum _ {i = 1} ^ {n} \ izquierda [{\ frac {\ parcial {\ mathcal {L}}} {\ parcial q_ {i}}} {\ dot {q}} _ {i} + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} {\ ddot {q}} _ {i} \ right]}

{\ Displaystyle {\ frac {d {\ mathcal {L}}} {dt}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial t}} + \ sum _ {i = 1} ^ {n} \ izquierda [{\ frac {\ parcial {\ mathcal {L}}} {\ parcial q_ {i}}} {\ dot {q}} _ {i} + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} {\ ddot {q}} _ {i} \ right]}

y dado que la ecuación de movimiento tiene la forma ( ecuación de Euler-Lagrange ):

{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}-{\frac {\partial {\mathcal {L}}}{\partial q_{i}}}=0,\quad \forall i

{\ estilo de visualización {\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {q}} _ {i}}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial q_ {i}}} = 0, \ quad \ forall i}

{\ estilo de visualización {\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {q}} _ {i}}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial q_ {i}}} = 0, \ quad \ forall i}

el primer término entre paréntesis se puede reescribir para tener:

{\frac {d{\mathcal {L}}}{dt}}={\frac {\partial {\mathcal {L}}}{\partial t}}+\sum _{i=1}^{n}\left[\left({\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}\right){\dot {q}}_{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}{\ddot {q}}_{i}\right]

{\ Displaystyle {\ frac {d {\ mathcal {L}}} {dt}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial t}} + \ sum _ {i = 1} ^ {n} \ left [\ left ({\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} \ right) {\ dot {q}} _ {i} + {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}} _ {i}}} {\ ddot {q }} _ {i} \ right]}

{\ Displaystyle {\ frac {d {\ mathcal {L}}} {dt}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial t}} + \ sum _ {i = 1} ^ {n} \ left [\ left ({\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} \ right) {\ dot {q}} _ {i} + {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}} _ {i}}} {\ ddot {q }} _ {i} \ right]}

es decir:

{\frac {d{\mathcal {H}}}{dt}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}

{\ Displaystyle {\ frac {d {\ mathcal {H}}} {dt}} = - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial t}}}

{\ Displaystyle {\ frac {d {\ mathcal {H}}} {dt}} = - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial t}}}

Dónde está ${\mathcal {H}}$ ${\ Displaystyle {\ mathcal {H}}}$ ${\ mathcal {H}}$ es el hamiltoniano , la transformada de Legendre del lagrangiano:

{\mathcal {H}}=\sum _{i=1}^{n}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}-{\mathcal {L}}

{\ Displaystyle {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i }}} {\ dot {q}} _ {i} - {\ mathcal {L}}}

{\ Displaystyle {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i }}} {\ dot {q}} _ {i} - {\ mathcal {L}}}

Si entonces ${\mathcal {L}}$ ${\ Displaystyle {\ mathcal {L}}}$ $\ mathcal {L}$ no depende explícitamente del tiempo ( $-\partial {\mathcal {L}}/\partial t=0$ ${\ estilo de visualización - \ parcial {\ mathcal {L}} / \ parcial t = 0}$ ${\ estilo de visualización - \ parcial {\ mathcal {L}} / \ parcial t = 0}$ ) asi que ${\mathcal {H}}$ ${\ Displaystyle {\ mathcal {H}}}$ ${\ mathcal {H}}$ se conserva $d{\mathcal {H}}/dt=0$ ${\ Displaystyle d {\ mathcal {H}} / dt = 0}$ ${\ Displaystyle d {\ mathcal {H}} / dt = 0}$ , es decir ${\mathcal {H}}={\text{costante}}$ ${\ Displaystyle {\ mathcal {H}} = {\ text {constante}}}$ ${\ Displaystyle {\ mathcal {H}} = {\ text {constante}}}$ ).

Simetrías de acción

El teorema de Noether puede enunciarse considerando, en lugar del lagrangiano, la acción funcional integral ${\mathcal {S}}$ ${\ Displaystyle {\ mathcal {S}}}$ ${\ mathcal {S}}$ :

{\mathcal {S}}=\int {\mathcal {L}}({\dot {\mathbf {q} }},\mathbf {q} ,t)\,\mathrm {d} t

{\ Displaystyle {\ mathcal {S}} = \ int {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t) \, \ mathrm {d} t}

{\ Displaystyle {\ mathcal {S}} = \ int {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t) \, \ mathrm {d} t}

Asumir que ${\mathcal {S}}$ ${\ Displaystyle {\ mathcal {S}}}$ ${\ mathcal {S}}$ es invariante con respecto a la transformación:

t\to {\bar {t}}(\mathbf {q} ,t,\lambda )

{\ Displaystyle t \ to {\ bar {t}} (\ mathbf {q}, t, \ lambda)}

{\ Displaystyle t \ to {\ bar {t}} (\ mathbf {q}, t, \ lambda)}

q_{i}\to {\bar {q}}_{i}(\mathbf {q} ,t,\lambda )\qquad \mathbf {q} \to \mathbf {\bar {q}} (\mathbf {q} ,t,\lambda )

{\ Displaystyle q_ {i} \ to {\ bar {q}} _ {i} (\ mathbf {q}, t, \ lambda) \ qquad \ mathbf {q} \ to \ mathbf {\ bar {q}} (\ mathbf {q}, t, \ lambda)}

{\ Displaystyle q_ {i} \ to {\ bar {q}} _ {i} (\ mathbf {q}, t, \ lambda) \ qquad \ mathbf {q} \ to \ mathbf {\ bar {q}} (\ mathbf {q}, t, \ lambda)}

Dónde está $\lambda$ ${\ Displaystyle \ lambda}$ $\ lambda$ es un parámetro continuo, es decir, ocurre:

\int _{t_{1}}^{t_{2}}{\mathcal {L}}({\dot {\mathbf {q} }},\mathbf {q} ,\tau )\,\mathrm {d} \tau =\int _{t_{1}'}^{t_{2}'}{\mathcal {L}}({\dot {\mathbf {\bar {q}} }},\mathbf {\bar {q}} ,\tau )\,\mathrm {d} \tau

{\ Displaystyle \ int _ {t_ {1}} ^ {t_ {2}} {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, \ tau) \, \ mathrm {d} \ tau = \ int _ {t_ {1} '} ^ {t_ {2}'} {\ mathcal {L}} ({\ dot {\ mathbf {\ bar {q}}}}, \ mathbf {\ bar {q}}, \ tau) \, \ mathrm {d} \ tau}

{\ Displaystyle \ int _ {t_ {1}} ^ {t_ {2}} {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, \ tau) \, \ mathrm {d} \ tau = \ int _ {t_ {1} '} ^ {t_ {2}'} {\ mathcal {L}} ({\ dot {\ mathbf {\ bar {q}}}}, \ mathbf {\ bar {q}}, \ tau) \, \ mathrm {d} \ tau}

donde los extremos de integración varían durante la transformación. Considerando una variación $\delta \lambda$ ${\ Displaystyle \ delta \ lambda}$ ${\ Displaystyle \ delta \ lambda}$ infinitesimal:

\qquad \delta t={\bar {t}}-t=A(\mathbf {q} ,t)\delta \lambda \qquad \delta \mathbf {q} =\mathbf {\bar {q}} ({\bar {t}})-\mathbf {q} (t)=B(\mathbf {q} ,t)\delta \lambda

{\ Displaystyle \ qquad \ delta t = {\ bar {t}} - t = A (\ mathbf {q}, t) \ delta \ lambda \ qquad \ delta \ mathbf {q} = \ mathbf {\ bar {q }} ({\ bar {t}}) - \ mathbf {q} (t) = B (\ mathbf {q}, t) \ delta \ lambda}

{\ Displaystyle \ qquad \ delta t = {\ bar {t}} - t = A (\ mathbf {q}, t) \ delta \ lambda \ qquad \ delta \ mathbf {q} = \ mathbf {\ bar {q }} ({\ bar {t}}) - \ mathbf {q} (t) = B (\ mathbf {q}, t) \ delta \ lambda}

la cantidad almacenada es:

\left({\mathcal {L}}-{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right)A(\mathbf {q} ,t)+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{i}}}B(\mathbf {q} ,t)=-{\mathcal {H}}A(\mathbf {q} ,t)+p_{i}B(\mathbf {q} ,t)

{\ Displaystyle \ left ({\ mathcal {L}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} {\ dot {q} } _ {i} \ derecha) A (\ mathbf {q}, t) + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} B (\ mathbf {q}, t) = - {\ mathcal {H}} A (\ mathbf {q}, t) + p_ {i} B (\ mathbf {q}, t)}

{\ Displaystyle \ left ({\ mathcal {L}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} {\ dot {q} } _ {i} \ derecha) A (\ mathbf {q}, t) + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {q}} _ {i}}} B (\ mathbf {q}, t) = - {\ mathcal {H}} A (\ mathbf {q}, t) + p_ {i} B (\ mathbf {q}, t)}

Dónde está ${\mathcal {H}}$ ${\ Displaystyle {\ mathcal {H}}}$ ${\ mathcal {H}}$ se llama hamiltoniano y $p_{i}$ ${\ Displaystyle p_ {i}}$ $Pi}$ es el momento lineal conjugado con la coordenada $q_{i}$ ${\ Displaystyle q_ {i}}$ $q_ {i}$ . ^[5]

Demostración

Demostración 1

Considere un sistema físico descrito por un campo $\psi$ ${\ Displaystyle \ psi}$ $\ psi$ . Cuando una cierta cantidad es invariante bajo una transformación del sistema, entonces el Lagrangiano correspondiente es simétrico, es decir, si $\psi$ ${\ Displaystyle \ psi}$ $\ psi$ se transforma por una transformación infinitesimal $\alpha$ ${\ Displaystyle \ alpha}$ $\ alpha$ igual que:

\psi \rightarrow \psi +\alpha \Delta \psi

{\ Displaystyle \ psi \ rightarrow \ psi + \ alpha \ Delta \ psi}

\ psi \ rightarrow \ psi + \ alpha \ Delta \ psi

el lagrangiano ${\mathcal {L}}$ ${\ Displaystyle {\ mathcal {L}}}$ ${\ mathcal {L}}$ , teniendo que ser invariante, debe convertirse en:

{\mathcal {L}}\rightarrow {\mathcal {L}}+\alpha \partial _{\mu }{\mathcal {J}}^{\mu }

{\ Displaystyle {\ mathcal {L}} \ rightarrow {\ mathcal {L}} + \ alpha \ partial _ {\ mu} {\ mathcal {J}} ^ {\ mu}}

{\ mathcal {L}} \ rightarrow {\ mathcal {L}} + \ alpha \ partial _ {\ mu} {\ mathcal {J}} ^ {\ mu}

Dónde está ${\mathcal {J}}$ ${\ Displaystyle {\ mathcal {J}}}$ ${\ mathcal {J}}$ representa una corriente de alguna cantidad que fluye a través de la superficie de la integral que define la acción.

En general, la variación de ${\mathcal {L}}$ ${\ Displaystyle {\ mathcal {L}}}$ ${\ mathcal {L}}$ Se puede escribir como:

\alpha \Delta {\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial \psi }}(\alpha \Delta \psi )+{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\partial _{\mu }(\alpha \Delta \psi )

{\ Displaystyle \ alpha \ Delta {\ mathcal {L}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ psi}} (\ alpha \ Delta \ psi) + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ parcial _ {\ mu} (\ alpha \ Delta \ psi)}

\ alpha \ Delta {\ mathcal {L}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ psi}} (\ alpha \ Delta \ psi) + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ parcial _ {\ mu} (\ alpha \ Delta \ psi)

Considerando la derivada de un producto, el segundo término se puede reescribir como:

\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\alpha \Delta \psi \right)-\alpha \Delta \psi \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)

{\ estilo de visualización \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ derecha) - \ alpha \ Delta \ psi \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ Derecha)}

\ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ derecha) - \ alpha \ Delta \ psi \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ derecha)

Sustituyendo y tomando un factor común $\alpha \Delta \psi$ ${\ Displaystyle \ alpha \ Delta \ psi}$ $\ alpha \ Delta \ psi$ usted obtiene:

-\alpha \Delta \psi \left(\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}-{\frac {\partial {\mathcal {L}}}{\partial \psi }}\right)+\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\alpha \Delta \psi \right)

{\ Displaystyle - \ alpha \ Delta \ psi \ left (\ parcial _ {\ mu} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ psi}} \ derecha) + \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ right)}

{\ Displaystyle - \ alpha \ Delta \ psi \ left (\ parcial _ {\ mu} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ psi}} \ derecha) + \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ right)}

Recordando la ecuación de Euler-Lagrange , lo anterior se convierte en:

\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\alpha \Delta \psi \right)

{\ estilo de visualización \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ Derecha)}

\ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ derecha)

o:

\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\alpha \Delta \psi \right)=\alpha \partial _{\mu }{\mathcal {J}}^{\mu }

{\ estilo de visualización \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ derecha) = \ alpha \ partial _ {\ mu} {\ mathcal {J}} ^ {\ mu}}

\ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ alpha \ Delta \ psi \ derecha) = \ alpha \ partial _ {\ mu} {\ mathcal {J}} ^ {\ mu}

Al reescribir el conjunto, podemos ver cómo hay una conservación de la corriente ${\mathcal {J}}$ ${\ Displaystyle {\ mathcal {J}}}$ ${\ mathcal {J}}$ señalando que:

\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\Delta \psi -{\mathcal {J}}^{\mu }\right)=0

{\ estilo de visualización \ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ Delta \ psi - {\ mathcal {J}} ^ {\ mu} \ right) = 0}

\ parcial _ {\ mu} \ izquierda ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial (\ parcial _ {\ mu} \ psi)}} \ Delta \ psi - {\ mathcal {J }} ^ {\ mu} \ right) = 0

Demostración 2

Asume variables dependientes $\mathbf {q}$ ${\ Displaystyle \ mathbf {q}}$ $\ mathbf q$ son tales que la acción , dada por la integral del Lagrangiano :

{\mathcal {S}}=\int {\mathcal {L}}({\dot {\mathbf {q} }},\mathbf {q} ,t)\,\mathrm {d} t

{\ Displaystyle {\ mathcal {S}} = \ int {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t) \, \ mathrm {d} t}

{\ Displaystyle {\ mathcal {S}} = \ int {\ mathcal {L}} ({\ dot {\ mathbf {q}}}, \ mathbf {q}, t) \, \ mathrm {d} t}

es invariante con respecto a variaciones infinitesimales de ellos. En otras palabras, debe satisfacerse la ecuación de Euler-Lagrange :

{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}-{\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}=0

{\ estilo de visualización {\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} = 0}

{\ estilo de visualización {\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} = 0}

Suponga que la integral de acción es invariante con respecto a una simetría continua. Tal simetría está representada por un flujo $\phi$ ${\ Displaystyle \ phi}$ $\ phi$ que actúa sobre las variables de la siguiente manera:

t\rightarrow t'=t+\varepsilon \tau

{\ Displaystyle t \ rightarrow t '= t + \ varepsilon \ tau}

{\ Displaystyle t \ rightarrow t '= t + \ varepsilon \ tau}

\mathbf {q} (t)\rightarrow \mathbf {q} '(t')=\phi (\mathbf {q} (t),\varepsilon )=\phi (\mathbf {q} (t'-\varepsilon \tau ),\varepsilon )

{\ Displaystyle \ mathbf {q} (t) \ rightarrow \ mathbf {q} '(t') = \ phi (\ mathbf {q} (t), \ varepsilon) = \ phi (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon)}

{\ Displaystyle \ mathbf {q} (t) \ rightarrow \ mathbf {q} '(t') = \ phi (\ mathbf {q} (t), \ varepsilon) = \ phi (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon)}

Dónde está $\varepsilon$ ${\ Displaystyle \ varepsilon}$ $\ varepsilon$ es una variable real que cuantifica el aumento de caudal, mientras que $\tau$ ${\ Displaystyle \ tau}$ $\ tau$ es una constante real relacionada con la traslación del flujo en el tiempo (puede ser cero). Tenemos:

{\dot {\mathbf {q} }}(t)\rightarrow {\dot {\mathbf {q} }}'(t')={\frac {d}{dt}}\phi (\mathbf {q} (t),\varepsilon )={\frac {\partial \phi }{\partial \mathbf {q} }}(\mathbf {q} (t'-\varepsilon \tau ),\varepsilon )\cdot {\dot {\mathbf {q} }}(t'-\varepsilon \tau )

{\ Displaystyle {\ dot {\ mathbf {q}}} (t) \ rightarrow {\ dot {\ mathbf {q}}} '(t') = {\ frac {d} {dt}} \ phi (\ mathbf {q} (t), \ varepsilon) = {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon) \ cdot {\ dot {\ mathbf {q}}} (t '- \ varepsilon \ tau)}

{\ Displaystyle {\ dot {\ mathbf {q}}} (t) \ rightarrow {\ dot {\ mathbf {q}}} '(t') = {\ frac {d} {dt}} \ phi (\ mathbf {q} (t), \ varepsilon) = {\ frac {\ partial \ phi} {\ partial \ mathbf {q}}} (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon) \ cdot {\ dot {\ mathbf {q}}} (t '- \ varepsilon \ tau)}

y la acción integral se convierte en:

{\begin{aligned}{\mathcal {S}}'(\varepsilon )=\int _{t_{1}+\varepsilon \tau }^{t_{2}+\varepsilon \tau }{\mathcal {L}}[{\dot {\mathbf {q} }}'(t'),\mathbf {q} '(t'),t']\,\mathrm {d} t'=\int _{t_{1}+\varepsilon \tau }^{t_{2}+\varepsilon \tau }{\mathcal {L}}\left[{\frac {\partial \phi }{\partial \mathbf {q} }}(\mathbf {q} (t'-\varepsilon \tau ),\varepsilon )\cdot {\dot {\mathbf {q} }}(t'-\varepsilon \tau ),\ \phi (\mathbf {q} (t'-\varepsilon \tau ),\varepsilon ),\ t'\right]\,\mathrm {d} t'\end{aligned}}

{\ Displaystyle {\ begin {alineado} {\ mathcal {S}} '(\ varepsilon) = \ int _ {t_ {1} + \ varepsilon \ tau} ^ {t_ {2} + \ varepsilon \ tau} {\ mathcal {L}} [{\ dot {\ mathbf {q}}} '(t'), \ mathbf {q} '(t'), t '] \, \ mathrm {d} t' = \ int _ {t_ {1} + \ varepsilon \ tau} ^ {t_ {2} + \ varepsilon \ tau} {\ mathcal {L}} \ left [{\ frac {\ parcial \ phi} {\ parcial \ mathbf {q} }} (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon) \ cdot {\ dot {\ mathbf {q}}} (t' - \ varepsilon \ tau), \ \ phi (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon), \ t' \ right] \, \ mathrm {d} t '\ end {alineado}}}

{\ Displaystyle {\ begin {alineado} {\ mathcal {S}} '(\ varepsilon) = \ int _ {t_ {1} + \ varepsilon \ tau} ^ {t_ {2} + \ varepsilon \ tau} {\ mathcal {L}} [{\ dot {\ mathbf {q}}} '(t'), \ mathbf {q} '(t'), t '] \, \ mathrm {d} t' = \ int _ {t_ {1} + \ varepsilon \ tau} ^ {t_ {2} + \ varepsilon \ tau} {\ mathcal {L}} \ left [{\ frac {\ parcial \ phi} {\ parcial \ mathbf {q} }} (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon) \ cdot {\ dot {\ mathbf {q}}} (t' - \ varepsilon \ tau), \ \ phi (\ mathbf {q} (t '- \ varepsilon \ tau), \ varepsilon), \ t' \ right] \, \ mathrm {d} t '\ end {alineado}}}

La acción solo puede considerarse en función de $\varepsilon$ ${\ Displaystyle \ varepsilon}$ $\ varepsilon$ . Calcular la derivada en $\varepsilon =0$ ${\ Displaystyle \ varepsilon = 0}$ $\ varepsilon = 0$ y explotando la simetría obtenemos:

{\begin{aligned}0&={\frac {d{\mathcal {S}}'}{d\varepsilon }}(0)={\mathcal {L}}[{\dot {\mathbf {q} }}(t_{2}),\mathbf {q} (t_{2}),t_{2}]\tau -{\mathcal {L}}[{\dot {\mathbf {q} }}(t_{1}),\mathbf {q} (t_{1}),t_{1}]\tau \\[6pt]&{}+\int _{t_{1}}^{t_{2}}{\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}\left(-{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}\tau +{\frac {\partial \phi }{\partial \varepsilon }}\right)+{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}\left(-{\frac {\partial ^{2}\phi }{\partial \mathbf {q} ^{2}}}{\dot {\mathbf {q} }}^{2}\tau +{\frac {\partial ^{2}\phi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}-{\frac {\partial \phi }{\partial \mathbf {q} }}{\ddot {\mathbf {q} }}\tau \right)\,\mathrm {d} t\end{aligned}}

{\ Displaystyle {\ begin {alineado} 0 & = {\ frac {d {\ mathcal {S}} '} {d \ varepsilon}} (0) = {\ mathcal {L}} [{\ dot {\ mathbf {q}}} (t_ {2}), \ mathbf {q} (t_ {2}), t_ {2}] \ tau - {\ mathcal {L}} [{\ dot {\ mathbf {q}} } (t_ {1}), \ mathbf {q} (t_ {1}), t_ {1}] \ tau \\ [6pt] & {} + \ int _ {t_ {1}} ^ {t_ {2 }} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} \ izquierda (- {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} { \ dot {\ mathbf {q}}} \ tau + {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} \ derecha) + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} \ left (- {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ mathbf {q} ^ {2}}} {\ dot {\ mathbf {q}}} ^ {2} \ tau + {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} - {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ ddot {\ mathbf {q}}} \ tau \ right) \, \ mathrm {d} t \ end {alineado}} }

{\ Displaystyle {\ begin {alineado} 0 & = {\ frac {d {\ mathcal {S}} '} {d \ varepsilon}} (0) = {\ mathcal {L}} [{\ dot {\ mathbf {q}}} (t_ {2}), \ mathbf {q} (t_ {2}), t_ {2}] \ tau - {\ mathcal {L}} [{\ dot {\ mathbf {q}} } (t_ {1}), \ mathbf {q} (t_ {1}), t_ {1}] \ tau \\ [6pt] & {} + \ int _ {t_ {1}} ^ {t_ {2 }} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} \ izquierda (- {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} { \ dot {\ mathbf {q}}} \ tau + {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} \ derecha) + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} \ left (- {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ mathbf {q} ^ {2}}} {\ dot {\ mathbf {q}}} ^ {2} \ tau + {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} - {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ ddot {\ mathbf {q}}} \ tau \ right) \, \ mathrm {d} t \ end {alineado}} }

La ecuación de Euler-Lagrange implica que:

{\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}\tau \right)=\left({\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}\right){\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}\left({\frac {d}{dt}}{\frac {\partial \phi }{\partial \mathbf {q} }}\right){\dot {\mathbf {q} }}\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\ddot {\mathbf {q} }}\tau ={\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}\left({\frac {\partial ^{2}\phi }{\partial \mathbf {q} ^{2}}}{\dot {\mathbf {q} }}\right){\dot {\mathbf {q} }}\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\ddot {\mathbf {q} }}\tau

{\ Displaystyle {\ frac {d} {dt}} \ left ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac { \ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} \ tau \ right) = \ left ({\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {\ mathbf {q}}}}} \ derecha) {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ punto {\ mathbf {q}}} \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} \ left ({\ frac {d} {dt}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} \ right) {\ dot {\ mathbf {q}}} \ tau + {\ frac {\ partial {\ mathcal { L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ ddot {\ mathbf {q}}} \ tau = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot { \ mathbf {q}}} \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} \ left ({\ frac {\ parcial ^ {2} \ phi} {\ parcial \ mathbf {q} ^ {2}}} {\ dot {\ mathbf {q}}} \ right) {\ dot {\ mathbf {q}}} \ tau + {\ frac {\ parcial {\ mathcal {L}}} { \ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ ddot {\ mathbf {q}}} \ tau}

{\ Displaystyle {\ frac {d} {dt}} \ left ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac { \ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} \ tau \ right) = \ left ({\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {\ mathbf {q}}}}} \ derecha) {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ punto {\ mathbf {q}}} \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} \ left ({\ frac {d} {dt}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} \ right) {\ dot {\ mathbf {q}}} \ tau + {\ frac {\ partial {\ mathcal { L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ ddot {\ mathbf {q}}} \ tau = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot { \ mathbf {q}}} \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} \ left ({\ frac {\ parcial ^ {2} \ phi} {\ parcial \ mathbf {q} ^ {2}}} {\ dot {\ mathbf {q}}} \ right) {\ dot {\ mathbf {q}}} \ tau + {\ frac {\ parcial {\ mathcal {L}}} { \ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ ddot {\ mathbf {q}}} \ tau}

y sustituyendo en la ecuación anterior llegamos a:

{\begin{aligned}0&={\frac {d{\mathcal {S}}'}{d\varepsilon }}(0)={\mathcal {L}}[\mathbf {q} (t_{2}),{\dot {\mathbf {q} }}(t_{2}),t_{2}]\tau -{\mathcal {L}}[\mathbf {q} (t_{1}),{\dot {\mathbf {q} }}(t_{1}),t_{1}]\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}(t_{2})\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}(t_{1})\tau \\[6pt]&+\int _{t_{1}}^{t_{2}}{\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}{\frac {\partial \phi }{\partial \varepsilon }}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\phi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}\,\mathrm {d} t\end{aligned}}

{\ displaystyle {\ begin {alineado} 0 & = {\ frac {d {\ mathcal {S}} '} {d \ varepsilon}} (0) = {\ mathcal {L}} [\ mathbf {q} ( t_ {2}), {\ dot {\ mathbf {q}}} (t_ {2}), t_ {2}] \ tau - {\ mathcal {L}} [\ mathbf {q} (t_ {1} ), {\ dot {\ mathbf {q}}} (t_ {1}), t_ {1}] \ tau - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} (t_ {2}) \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ punto { \ mathbf {q}}} (t_ {1}) \ tau \\ [6pt] & + \ int _ {t_ {1}} ^ {t_ {2}} {\ frac {\ partial {\ mathcal {L} }} {\ parcial \ mathbf {q}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot { \ mathbf {q}}}}} {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} \, \ mathrm {d} t \ end {alineado}}}

{\ displaystyle {\ begin {alineado} 0 & = {\ frac {d {\ mathcal {S}} '} {d \ varepsilon}} (0) = {\ mathcal {L}} [\ mathbf {q} ( t_ {2}), {\ dot {\ mathbf {q}}} (t_ {2}), t_ {2}] \ tau - {\ mathcal {L}} [\ mathbf {q} (t_ {1} ), {\ dot {\ mathbf {q}}} (t_ {1}), t_ {1}] \ tau - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} (t_ {2}) \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ punto { \ mathbf {q}}} (t_ {1}) \ tau \\ [6pt] & + \ int _ {t_ {1}} ^ {t_ {2}} {\ frac {\ partial {\ mathcal {L} }} {\ parcial \ mathbf {q}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot { \ mathbf {q}}}}} {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} \, \ mathrm {d} t \ end {alineado}}}

Luego, usando la ecuación de Euler-Lagrange nuevamente:

{\frac {d}{dt}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}\right)=\left({\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}\right){\frac {\partial \phi }{\partial \varepsilon }}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\phi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}={\frac {\partial {\mathcal {L}}}{\partial \mathbf {q} }}{\frac {\partial \phi }{\partial \varepsilon }}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\phi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}

{\ Displaystyle {\ frac {d} {dt}} \ left ({\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac { \ parcial \ phi} {\ parcial \ varepsilon}} \ derecha) = \ izquierda ({\ frac {d} {dt}} {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot { \ mathbf {q}}}}} \ derecha) {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot { \ mathbf {q}}}}} {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} = {\ frac {\ parcial {\ mathcal {L}}} {\ parcial \ mathbf {q}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} + {\ frac {\ parcial {\ mathcal {L }}} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial \ mathbf {q}}} {\ dot { \ mathbf {q}}}}

{\ frac {d} {dt}} \ izquierda ({\ frac {\ parcial {\ mathcal L}} {\ parcial {\ dot {{\ mathbf {q}}}}}} {\ frac {\ parcial \ phi} {\ partial \ varepsilon}} \ right) = \ left ({\ frac {d} {dt}} {\ frac {\ partial {\ mathcal L}} {\ partial {\ dot {{\ mathbf {q }}}}}} \ derecha) {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} + {\ frac {\ parcial {\ mathcal L}} {\ parcial {\ dot {{\ mathbf {q }}}}}} {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial {\ mathbf {q}}}} {\ dot {{\ mathbf {q}}}} = { \ frac {\ parcial {\ mathcal L}} {\ parcial {\ mathbf {q}}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} + {\ frac {\ parcial {\ mathcal L }} {\ parcial {\ punto {{\ mathbf {q}}}}}} {\ frac {\ parcial ^ {2} \ phi} {\ parcial \ varepsilon \ parcial {\ mathbf {q}}}} { \ dot {{\ mathbf {q}}}}

e insertando en el informe anterior se puede escribir:

0={\mathcal {L}}[\mathbf {q} (t_{2}),{\dot {\mathbf {q} }}(t_{2}),t_{2}]\tau -{\mathcal {L}}[\mathbf {q} (t_{1}),{\dot {\mathbf {q} }}(t_{1}),t_{1}]\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}(t_{2})\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}(t_{1})\tau +{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}(t_{2})-{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}(t_{1})

{\ displaystyle 0 = {\ mathcal {L}} [\ mathbf {q} (t_ {2}), {\ dot {\ mathbf {q}}} (t_ {2}), t_ {2}] \ tau - {\ mathcal {L}} [\ mathbf {q} (t_ {1}), {\ dot {\ mathbf {q}}} (t_ {1}), t_ {1}] \ tau - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ punto { \ mathbf {q}}} (t_ {2}) \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac { \ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} (t_ {1}) \ tau + {\ frac {\ parcial {\ mathcal {L}}} { \ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} (t_ {2}) - {\ frac {\ parcial {\ mathcal {L} }} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} (t_ {1})}

{\ Displaystyle 0 = {\ mathcal {L}} [\ mathbf {q} (t_ {2}), {\ dot {\ mathbf {q}}} (t_ {2}), t_ {2}] \ tau - {\ mathcal {L}} [\ mathbf {q} (t_ {1}), {\ dot {\ mathbf {q}}} (t_ {1}), t_ {1}] \ tau - {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ mathbf {q}}} {\ punto { \ mathbf {q}}} (t_ {2}) \ tau + {\ frac {\ parcial {\ mathcal {L}}} {\ parcial {\ dot {\ mathbf {q}}}}} {\ frac { \ parcial \ phi} {\ parcial \ mathbf {q}}} {\ dot {\ mathbf {q}}} (t_ {1}) \ tau + {\ frac {\ parcial {\ mathcal {L}}} { \ parcial {\ dot {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} (t_ {2}) - {\ frac {\ parcial {\ mathcal {L} }} {\ parcial {\ punto {\ mathbf {q}}}}} {\ frac {\ parcial \ phi} {\ parcial \ varepsilon}} (t_ {1})}

da cui si evince che la quantità:

\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}-{\mathcal {L}}\right)\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}

\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}-{\mathcal {L}}\right)\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}

\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}-{\mathcal {L}}\right)\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}

è una costante del moto, ovvero è una quantità conservata.

Dato che $\phi [\mathbf {q} ,0]=\mathbf {q}$ $\phi [\mathbf {q} ,0]=\mathbf {q}$ $\phi [{\mathbf q},0]={\mathbf q}$ si ha:

{\frac {\partial \phi }{\partial \mathbf {q} }}=1

{\frac {\partial \phi }{\partial \mathbf {q} }}=1

{\frac {\partial \phi }{\partial {\mathbf {q}}}}=1

e la quantità conservata si semplifica assumendo la forma:

\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\dot {\mathbf {q} }}-{\mathcal {L}}\right)\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}

\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\dot {\mathbf {q} }}-{\mathcal {L}}\right)\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}

\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\dot {\mathbf {q} }}-{\mathcal {L}}\right)\tau -{\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \phi }{\partial \varepsilon }}

Nella derivazione si è assunto che il flusso non varia nel tempo, e un risultato più generale si ottiene in un modo equivalente.

Dimostrazione 3

Si consideri una varietà liscia $M$ ${\displaystyle M}$ $M$ e una varietà bersaglio $T$ ${\displaystyle T}$ $T$ . Sia ${\mathcal {C}}$ ${\mathcal {C}}$ ${\mathcal {C}}$ lospazio delle configurazioni delle funzioni lisce da $M$ ${\displaystyle M}$ $M$ a $T$ ${\displaystyle T}$ $T$ . In modo più generale si possono considerare sezioni del fibrato lungo $M$ ${\displaystyle M}$ $M$ . In meccanica classica , ad esempio, $M$ ${\displaystyle M}$ $M$ è la varietà monodimensionale $\mathbb {R}$ $\mathbb {R}$ $\mathbb{R}$ che rappresenta il tempo, e lo spazio bersaglio è lo spazio delle fasi , il fibrato cotangente dello spazio delle posizioni generalizzate .

L' azione è un funzionale del tipo:

{\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb {R}

{\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb {R}

{\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb{R}

che mappa su $\mathbb {R}$ $\mathbb {R}$ $\mathbb{R}$ (e non su $\mathbb {C}$ $\mathbb {C}$ ${\mathbb {C}}$ per ragioni fisiche). Affinché l'azione sia locale è necessario imporre ulteriori restrizioni sul funzionale: se $\phi \in {\mathcal {C}}$ $\phi \in {\mathcal {C}}$ $\phi \in {\mathcal {C}}$ si assume che $S(\phi )$ $S(\phi )$ $S(\phi )$ sia l' integrale su $M$ ${\displaystyle M}$ $M$ della lagrangiana ${\mathcal {\mathcal {L}}}(\phi ,\partial \phi ,\partial \partial \phi ,...,x)$ ${\mathcal {\mathcal {L}}}(\phi ,\partial \phi ,\partial \partial \phi ,...,x)$ ${\mathcal {{\mathcal L}}}(\phi ,\partial \phi ,\partial \partial \phi ,...,x)$ , che è funzione di $\phi$ $\phi$ $\phi$ , delle sue derivate e della posizione. Esplicitamente, l'azione è definita nel seguente modo:

{\mathcal {S}}[\phi ]\equiv \int _{M}{\mathcal {\mathcal {L}}}(\phi (x),\partial \phi (x),\partial \partial \phi (x),...,x)d^{n}x\qquad \forall \phi \in {\mathcal {C}}

{\mathcal {S}}[\phi ]\equiv \int _{M}{\mathcal {\mathcal {L}}}(\phi (x),\partial \phi (x),\partial \partial \phi (x),...,x)d^{n}x\qquad \forall \phi \in {\mathcal {C}}

{\mathcal S}[\phi ]\equiv \int _{M}{\mathcal {{\mathcal L}}}(\phi (x),\partial \phi (x),\partial \partial \phi (x),...,x)d^{n}x\qquad \forall \phi \in {\mathcal {C}}

La maggior parte delle volte si assume che la lagrangiana dipenda soltanto dal valore del campo e dalla sua derivata prima, sebbene questo non sia vero in generale.

Se $M$ ${\displaystyle M}$ $M$ è compatto , le condizioni al contorno si ottengono specificando i valori di $\phi$ $\phi$ $\phi$ sulla frontiera . In caso contrario si possono fornire opportuni limiti per $\phi$ $\phi$ $\phi$ quando $x$ ${\displaystyle x}$ $x$ tende all' infinito . Questo rende possibile ottenere l'insieme delle funzioni $\phi$ $\phi$ $\phi$ tali che tutte le derivate funzionali di $S$ ${\displaystyle S}$ $S$ su $\phi$ $\phi$ $\phi$ sono nulle e $\phi$ $\phi$ $\phi$ soddisfa le condizioni al contorno date. Tale insieme è determinato, considerando le condizioni al contorno, dalle soluzioni on shell delle equazioni di Eulero-Lagrange:

{\frac {\delta {\mathcal {S}}}{\delta \varphi }}=-\partial _{\mu }\left({\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}\right)+{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial \phi }}=0

{\frac {\delta {\mathcal {S}}}{\delta \varphi }}=-\partial _{\mu }\left({\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}\right)+{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial \phi }}=0

{\frac {\delta {\mathcal {S}}}{\delta \varphi }}=-\partial _{\mu }\left({\frac {\partial {\mathcal {{\mathcal L}}}}{\partial (\partial _{\mu }\phi )}}\right)+{\frac {\partial {\mathcal {{\mathcal L}}}}{\partial \phi }}=0

Il membro sinistro è la derivata funzionale dell'azione rispetto a $\phi$ $\phi$ $\phi$ . In meccanica classica la lagrangiana è data dalla differenza tra l' energia cinetica $T$ ${\displaystyle T}$ $T$ e l' energia potenziale $U$ ${\displaystyle U}$ $U$ .

Si consideri una trasformazione infinitesima su ${\mathcal {C}}$ ${\mathcal {C}}$ ${\mathcal {C}}$ generata da un funzionale $Q$ ${\displaystyle Q}$ $Q$ tale che:

Q\left[\int _{N}{\mathcal {\mathcal {L}}}\,\mathrm {d} ^{n}x\right]\approx \int _{\partial N}f^{\mu }[\phi (x),\partial \phi ,\partial \partial \phi ,\ldots ]\mathrm {d} s_{\mu }

Q\left[\int _{N}{\mathcal {\mathcal {L}}}\,\mathrm {d} ^{n}x\right]\approx \int _{\partial N}f^{\mu }[\phi (x),\partial \phi ,\partial \partial \phi ,\ldots ]\mathrm {d} s_{\mu }

Q\left[\int _{N}{\mathcal {{\mathcal L}}}\,{\mathrm {d}}^{n}x\right]\approx \int _{{\partial N}}f^{\mu }[\phi (x),\partial \phi ,\partial \partial \phi ,\ldots ]{\mathrm {d}}s_{{\mu }}

per ogni sottovarietà $N$ ${\displaystyle N}$ $N$ . In modo equivalente:

Q[{\mathcal {\mathcal {L}}}(x)]\approx \partial _{\mu }f^{\mu }(x)\quad \forall x

Q[{\mathcal {\mathcal {L}}}(x)]\approx \partial _{\mu }f^{\mu }(x)\quad \forall x

Q[{\mathcal {{\mathcal L}}}(x)]\approx \partial _{\mu }f^{\mu }(x)\quad \forall x

dove:

{\mathcal {\mathcal {L}}}(x)={\mathcal {\mathcal {L}}}[\phi (x),\partial _{\mu }\phi (x),x]

{\mathcal {\mathcal {L}}}(x)={\mathcal {\mathcal {L}}}[\phi (x),\partial _{\mu }\phi (x),x]

{\mathcal {{\mathcal L}}}(x)={\mathcal {{\mathcal L}}}[\phi (x),\partial _{\mu }\phi (x),x]

Se questo vale on shell e off shell allora $Q$ ${\displaystyle Q}$ $Q$ genera una simmetria off shell . Se invece vale solo on shell, allora $Q$ ${\displaystyle Q}$ $Q$ genera una simmetria on shell . Il funzionale $Q$ ${\displaystyle Q}$ $Q$ è un generatore un gruppo di simmetria di Lie a un parametro.

Per il teorema di Eulero–Lagrange per ogni $N$ ${\displaystyle N}$ $N$ si ha, on shell:

Q\left[\int _{N}{\mathcal {\mathcal {L}}}\,\mathrm {d} ^{n}x\right]=\int _{N}\left[{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial \phi }}-\partial _{\mu }{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}\right]Q[\phi ]\,\mathrm {d} ^{n}x+\int _{\partial N}{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]\,\mathrm {d} s_{\mu }\approx \int _{\partial N}f^{\mu }\,\mathrm {d} s_{\mu }

Q\left[\int _{N}{\mathcal {\mathcal {L}}}\,\mathrm {d} ^{n}x\right]=\int _{N}\left[{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial \phi }}-\partial _{\mu }{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}\right]Q[\phi ]\,\mathrm {d} ^{n}x+\int _{\partial N}{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]\,\mathrm {d} s_{\mu }\approx \int _{\partial N}f^{\mu }\,\mathrm {d} s_{\mu }

Q\left[\int _{N}{\mathcal {\mathcal {L}}}\,\mathrm {d} ^{n}x\right]=\int _{N}\left[{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial \phi }}-\partial _{\mu }{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}\right]Q[\phi ]\,\mathrm {d} ^{n}x+\int _{\partial N}{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]\,\mathrm {d} s_{\mu }\approx \int _{\partial N}f^{\mu }\,\mathrm {d} s_{\mu }

Dato che questo vale per ogni $N$ ${\displaystyle N}$ $N$ vale la relazione:

\partial _{\mu }\left[{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }\right]\approx 0

\partial _{\mu }\left[{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }\right]\approx 0

\partial _{\mu }\left[{\frac {\partial {\mathcal {{\mathcal L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }\right]\approx 0

che è l' equazione di continuità per la corrente di Noether $J^{\mu }$ $J^{\mu }$ $J^\mu$ associata alla simmetria, definita da: ^[6]

J^{\mu }\,=\,{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }

J^{\mu }\,=\,{\frac {\partial {\mathcal {\mathcal {L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }

J^{\mu }\,=\,{\frac {\partial {\mathcal {{\mathcal L}}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }

Se si integra la corrente di Noether su una sezione di tipo tempo si ottiene una quantità conservata detta carica di Noether .

Teoria quantistica dei campi ^[7]

Nel formalismo della seconda quantizzazione è possibile scrivere il teorema di Noether come relazione tra funzioni di correlazione . Siano $O_{1}...O_{n}$ $O_{1}...O_{n}$ $O_{1}...O_{n}$ n operatori generici. La funzione di correlazione è per definizione:

$<O_{1}...O_{n}>={\frac {1}{\mathcal {Z}}}\int D\phi e^{-{\mathcal {S}}}O_{1}...O_{n}$ $<O_{1}...O_{n}>={\frac {1}{\mathcal {Z}}}\int D\phi e^{-{\mathcal {S}}}O_{1}...O_{n}$ $<O_{1}...O_{n}>={\frac {1}{\mathcal {Z}}}\int D\phi e^{-{\mathcal {S}}}O_{1}...O_{n}$

con ${\mathcal {S}}$ ${\mathcal {S}}$ ${\mathcal S}$ azione, ${\mathcal {Z}}$ ${\mathcal {Z}}$ ${\mathcal {Z}}$ funzione di partizione e $D\phi$ $D\phi$ $D\phi$ la misura su tutti i campi fondamentali presenti nell'azione. Considero una generica trasformazione nei campi fondamentali $\phi \longrightarrow \phi '$ $\phi \longrightarrow \phi '$ $\phi \longrightarrow \phi '$ tale che

${\mathcal {S}}(\phi )={\mathcal {S}}'(\phi ')+\delta {\mathcal {S}}(\phi ')$ ${\mathcal {S}}(\phi )={\mathcal {S}}'(\phi ')+\delta {\mathcal {S}}(\phi ')$ ${\mathcal {S}}(\phi )={\mathcal {S}}'(\phi ')+\delta {\mathcal {S}}(\phi ')$

$O_{i}(\phi )=O'_{i}(\phi ')+\delta O_{i}(\phi ')$ $O_{i}(\phi )=O'_{i}(\phi ')+\delta O_{i}(\phi ')$ $O_{i}(\phi )=O'_{i}(\phi ')+\delta O_{i}(\phi ')$

Sarà quindi valida la seguente relazione:

$\int D\phi e^{-{\mathcal {S}}(\phi )}O_{1}...O_{n}=\int D\phi 'e^{-{\mathcal {S}}'(\phi ')-\delta {\mathcal {S}}(\phi ')}(O_{1}+\delta O_{1})...(O_{n}+\delta O_{n})$ $\int D\phi e^{-{\mathcal {S}}(\phi )}O_{1}...O_{n}=\int D\phi 'e^{-{\mathcal {S}}'(\phi ')-\delta {\mathcal {S}}(\phi ')}(O_{1}+\delta O_{1})...(O_{n}+\delta O_{n})$ $\int D\phi e^{-{\mathcal {S}}(\phi )}O_{1}...O_{n}=\int D\phi 'e^{-{\mathcal {S}}'(\phi ')-\delta {\mathcal {S}}(\phi ')}(O_{1}+\delta O_{1})...(O_{n}+\delta O_{n})$

Espandendo al primo ordine $e^{-\delta {\mathcal {S}}(\phi ')}\approx 1-\delta {\mathcal {S}}(\phi ')$ $e^{-\delta {\mathcal {S}}(\phi ')}\approx 1-\delta {\mathcal {S}}(\phi ')$ $e^{-\delta {\mathcal {S}}(\phi ')}\approx 1-\delta {\mathcal {S}}(\phi ')$ . Nella relazione precedente i termini di ordine 0 si elidono, al primo ordine è quindi verificata la seguente relazione:

$\int D\phi e^{-{\mathcal {S}}(\phi )}\delta {\mathcal {S}}(\phi )O_{1}...O_{n}=\int D\phi e^{-{\mathcal {S}}(\phi )}\sum _{i=1}^{n}O_{1}...\delta O_{i}...O_{n}$ $\int D\phi e^{-{\mathcal {S}}(\phi )}\delta {\mathcal {S}}(\phi )O_{1}...O_{n}=\int D\phi e^{-{\mathcal {S}}(\phi )}\sum _{i=1}^{n}O_{1}...\delta O_{i}...O_{n}$ $\int D\phi e^{-{\mathcal {S}}(\phi )}\delta {\mathcal {S}}(\phi )O_{1}...O_{n}=\int D\phi e^{-{\mathcal {S}}(\phi )}\sum _{i=1}^{n}O_{1}...\delta O_{i}...O_{n}$

in cui la sommatoria nel termine di destra indica la somma su tutti i possibili prodotti degli operatori in cui compare una volta sola un $\delta O$ $\delta O$ $\delta O$ . Nel caso di un solo operatore si ha:

$<\delta {\mathcal {S}}\ O>=<\delta O>\qquad \qquad \qquad \qquad \qquad (1)$ $<\delta {\mathcal {S}}\ O>=<\delta O>\qquad \qquad \qquad \qquad \qquad (1)$ $<\delta {\mathcal {S}}\ O>=<\delta O>\qquad \qquad \qquad \qquad \qquad (1)$

Considero ora una trasformazione che soddisfi le ipotesi del teorema di Noether (simmetria continua dell'azione) che posso quindi scrivere come:

$\phi '=\phi +i\epsilon \chi$ $\phi '=\phi +i\epsilon \chi$ $\phi '=\phi +i\epsilon \chi$

con $\epsilon$ $\epsilon$ $\epsilon$ parametro globale piccolo e $\chi$ $\chi$ $\chi$ generica funzione dei campi fondamentali e delle $x$ ${\displaystyle x}$ $x$ . Localizzo $\epsilon$ $\epsilon$ $\epsilon$ , rompendo la simmetria dell'azione altrimenti valida, ed espando in serie al primo ordine. La differenza nell'azione è quindi scrivibile come la somma di due termini, uno proporzionale a $\epsilon$ $\epsilon$ $\epsilon$ che sarà nullo poiché l'azione è invariante per la trasformazione globale ed uno proporzionale a $\partial _{\mu }\epsilon (x)$ $\partial _{\mu }\epsilon (x)$ $\partial _{\mu }\epsilon (x)$ che scrivo come:

$\delta {\mathcal {S}}=\int d^{4}x{\frac {\partial {\mathcal {L}}}{\partial \phi }}\delta \phi =i\int {\mathcal {d}}^{4}xJ_{\mu }(x)\partial _{\mu }\epsilon (x)=-i\int d^{4}x\epsilon (x)\partial _{\mu }J_{\mu }(x)$ $\delta {\mathcal {S}}=\int d^{4}x{\frac {\partial {\mathcal {L}}}{\partial \phi }}\delta \phi =i\int {\mathcal {d}}^{4}xJ_{\mu }(x)\partial _{\mu }\epsilon (x)=-i\int d^{4}x\epsilon (x)\partial _{\mu }J_{\mu }(x)$ $\delta {\mathcal {S}}=\int d^{4}x{\frac {\partial {\mathcal {L}}}{\partial \phi }}\delta \phi =i\int {\mathcal {d}}^{4}xJ_{\mu }(x)\partial _{\mu }\epsilon (x)=-i\int d^{4}x\epsilon (x)\partial _{\mu }J_{\mu }(x)$

per l'ultimo passaggio si è integrato per parti. Analogamente si vede che

$\delta O=i\int d^{4}x\epsilon (x){\frac {\delta O}{\delta \phi (x)}}\chi (x)$ $\delta O=i\int d^{4}x\epsilon (x){\frac {\delta O}{\delta \phi (x)}}\chi (x)$ $\delta O=i\int d^{4}x\epsilon (x){\frac {\delta O}{\delta \phi (x)}}\chi (x)$

Da $(1)$ ${\displaystyle (1)}$ $(1)$ segue che:

$\int d^{4}z\epsilon (z)<\partial _{\mu }J_{\mu }(z)O(y)>=\int d^{4}z\epsilon (z)<{\frac {\delta O(y)}{\delta \phi (z)}}\chi (z)>$ $\int d^{4}z\epsilon (z)<\partial _{\mu }J_{\mu }(z)O(y)>=\int d^{4}z\epsilon (z)<{\frac {\delta O(y)}{\delta \phi (z)}}\chi (z)>$ $\int d^{4}z\epsilon (z)<\partial _{\mu }J_{\mu }(z)O(y)>=\int d^{4}z\epsilon (z)<{\frac {\delta O(y)}{\delta \phi (z)}}\chi (z)>$

Localizzo $\epsilon (z)$ $\epsilon (z)$ $\epsilon (z)$ imponendo la condizione $\epsilon (z)=\epsilon \delta (x-z)$ $\epsilon (z)=\epsilon \delta (xz)$ $\epsilon (z)=\epsilon \delta (x-z)$ . Dalla definizione della delta di Dirac :

$<\partial _{\mu }J_{\mu }(x)O(y)>=<{\frac {\delta O(y)}{\delta \phi (x)}}\chi (x)>$ $<\partial _{\mu }J_{\mu }(x)O(y)>=<{\frac {\delta O(y)}{\delta \phi (x)}}\chi (x)>$ $<\partial _{\mu }J_{\mu }(x)O(y)>=<{\frac {\delta O(y)}{\delta \phi (x)}}\chi (x)>$

Questa condizione estende il risultato del teorema di Noether rendendolo valido anche a livello quantistico. Nel caso $O(y)$ ${\displaystyle O(y)}$ $O(y)$ si una stringa di operatori locali definiti lontani da x si ottiene

$<\partial _{\mu }J_{\mu }(x)O(y)>=0\qquad x\neq y$ $<\partial _{\mu }J_{\mu }(x)O(y)>=0\qquad x\neq y$ $<\partial _{\mu }J_{\mu }(x)O(y)>=0\qquad x\neq y$

che rappresenta l'analogo della conservazione della corrente in teoria dei campi classica.

Integrando sul volume

$\int d^{3}x<\partial _{\mu }J_{\mu }(x)O(y)>=\int d^{3}x<\partial _{0}J_{0}(x)O(y)>+\int d^{3}x<\nabla \cdot {\mathbf {J}}(x)O(y)>=0$ $\int d^{3}x<\partial _{\mu }J_{\mu }(x)O(y)>=\int d^{3}x<\partial _{0}J_{0}(x)O(y)>+\int d^{3}x<\nabla \cdot {\mathbf {J}}(x)O(y)>=0$ $\int d^{3}x<\partial _{\mu }J_{\mu }(x)O(y)>=\int d^{3}x<\partial _{0}J_{0}(x)O(y)>+\int d^{3}x<\nabla \cdot {\mathbf {J}}(x)O(y)>=0$

Per il teorema della divergenza in una teoria di campo a volume infinito il secondo termine è nullo. Sia

${\bar {J_{0}}}(x_{0})\equiv \int d^{3}xJ_{0}(x)$ ${\bar {J_{0}}}(x_{0})\equiv \int d^{3}xJ_{0}(x)$ ${\bar {J_{0}}}(x_{0})\equiv \int d^{3}xJ_{0}(x)$

Si è quindi dimostrato che

$<\partial _{0}{\bar {J}}_{0}(x_{0})O(y)>=0$ $<\partial _{0}{\bar {J}}_{0}(x_{0})O(y)>=0$ $<\partial _{0}{\bar {J}}_{0}(x_{0})O(y)>=0$

${\bar {J}}_{0}(x_{0})$ ${\bar {J}}_{0}(x_{0})$ ${\bar {J}}_{0}(x_{0})$ è quindi una carica conservata.

Esempio

Supponiamo di trattare un sistema bidimensionale, e di considerare una trasformazione di coordinate ${\vec {x}}=(x,y)\rightarrow {\vec {f}}$ ${\vec {x}}=(x,y)\rightarrow {\vec {f}}$ ${\vec {x}}=(x,y)\rightarrow {\vec {f}}$ così definita:

f_{1}=x+s\qquad f_{2}=y

f_{1}=x+s\qquad f_{2}=y

f_{{1}}=x+s\qquad f_{{2}}=y

Secondo il teorema, si ha che:

{\frac {\partial f_{1}}{\partial s}}=1\qquad {\frac {\partial f_{2}}{\partial s}}=0

{\frac {\partial f_{1}}{\partial s}}=1\qquad {\frac {\partial f_{2}}{\partial s}}=0

{\frac {\partial f_{1}}{\partial s}}=1\qquad {\frac {\partial f_{2}}{\partial s}}=0

Quindi, automaticamente si conserverà la quantità:

p_{1}=\sum _{i=1}^{n}{\frac {\partial f_{i}}{\partial s}}(t,0)\,p_{i}=\mathrm {costante}

p_{1}=\sum _{i=1}^{n}{\frac {\partial f_{i}}{\partial s}}(t,0)\,p_{i}=\mathrm {costante}

p_{1}=\sum _{i=1}^{n}{\frac {\partial f_{i}}{\partial s}}(t,0)\,p_{i}=\mathrm {costante}

Questo significa che per un sistema che ha un'invarianza per traslazioni nella direzione $x$ ${\displaystyle x}$ $x$ , si conserverà il momento lineare (quantità di moto) in quella direzione.

Note

^ Yvette Kosmann-Schwarzbach - The Noether Theorems
^ E. Noether, Invariante Variationsprobleme . Göttingen 1918, pp. 235-257. Traduzione di MA Tavel in Transport Theory and Statistical Mechanics (1971), pp. 183-207
^ Thompson, WJ, Angular Momentum: an illustrated guide to rotational symmetries for physical systems , vol. 1, Wiley, 1994, p. 5, ISBN 0-471-55264-X .
^ Alberto Nicolis - The Noether theorem ( PDF ), su phys.columbia.edu . URL consultato il 19 settembre 2015 (archiviato dall' url originale il 13 maggio 2015) .
^ www-physics.ucsd.edu - Noether's Theorem
^ Michael E. Peskin, Daniel V. Schroeder, An Introduction to Quantum Field Theory , Basic Books, 1995, p. 18, ISBN 0-201-50397-2 .
^ Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139644167 .

Bibliografia

( EN ) The heritage of Emmy Noether in algebra, geometry and physics . Bar Ilan University, Tel Aviv (Israel), 2-3 dicembre 1996
( EN ) Herbert Goldstein, Classical Mechanics , 2nd, Reading MA, Addison-Wesley, 1980, pp. 588–596, ISBN 0-201-02918-9 .
( EN ) Yvette Kosmann-Schwarzbach , The Noether theorems: Invariance and conservation laws in the twentieth century , Sources and Studies in the History of Mathematics and Physical Sciences, Springer-Verlag , 2011, ISBN 978-0-387-87868-3 .
( EN ) Cornelius Lanczos, The Variational Principles of Mechanics , 4th, New York, Dover Publications, 1970, pp. 401–5, ISBN 0-486-65067-7 .
( EN ) Dwight E. Neuenschwander, Emmy Noether's Wonderful Theorem , Johns Hopkins University Press, 2010, ISBN 978-0-8018-9694-1 .
( EN ) Hanca, J.; Tulejab, S.; Hancova, M., Symmetries and conservation laws: Consequences of Noether's theorem , in American Journal of Physics , vol. 72, n. 4, 2004, pp. 428–35, Bibcode : 2004AmJPh..72..428H , DOI : 10.1119/1.1591764 .
( EN ) Merced Montesinos e Ernesto Flores, Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem , in Revista Mexicana de Física , vol. 52, 2006, p. 29, Bibcode : 2006RMxF...52...29M , arXiv : hep-th/0602190 .
( EN ) Sardanashvily, G., Gauge conservation laws in a general setting. Superpotential , in International Journal of Geometric Methods in Modern Physics , vol. 6, n. 06, 2009, p. 1047, Bibcode : 2009arXiv0906.1732S , DOI : 10.1142/S0219887809003862 , arXiv : 0906.1732 .

Voci correlate

Collegamenti esterni

( EN ) DV Alekseevskii, Noether theorem , in Encyclopaedia of Mathematics , Springer e European Mathematical Society, 2002.
Noether's Theorem at MathPages.

Portale Matematica

Portale Meccanica

[1] Yvette Kosmann-Schwarzbach - The Noether Theorems

[2] E. Noether, Invariante Variationsprobleme . Göttingen 1918, pp. 235-257. Traduzione di MA Tavel in Transport Theory and Statistical Mechanics (1971), pp. 183-207

[3] Thompson, WJ, Angular Momentum: an illustrated guide to rotational symmetries for physical systems , vol. 1, Wiley, 1994, p. 5, ISBN 0-471-55264-X .

[4] Alberto Nicolis - The Noether theorem ( PDF ), su phys.columbia.edu . URL consultato il 19 settembre 2015 (archiviato dall' url originale il 13 maggio 2015) .

[5] www-physics.ucsd.edu - Noether's Theorem

[Peskin-6] Michael E. Peskin, Daniel V. Schroeder, An Introduction to Quantum Field Theory , Basic Books, 1995, p. 18, ISBN 0-201-50397-2 .

[7] Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139644167 .

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