User:EditingPencil/sandbox/noether

Consider the transformation where the change of coordinates also depends on the generalized velocities.$$\backslash begin\{align\}$$

q^r\to q^r+\delta q^r\\

\delta q^r=\epsilon\phi^r(q,\dot{q},t)\\

\end{align}If the above is a dynamical symmetry, then the lagrangian changes by:$$\backslash delta\; L=\backslash epsilon\backslash frac\; d\; \{dt\}F(q,\backslash dot\; q,t)$$and the new Lagrangian is said to be dynamically equivalent to the old Lagrangian as it ensures the resultant equations of motion being the same.

Using the change in Lagrangian property of a dynamical symmetry:$$\backslash frac\; d\{dt\}F=\backslash frac\{\backslash partial\; F\}\{\backslash partial\; q^r\}\backslash dot\; q^r+\backslash frac\{\backslash partial\; F\}\{\backslash partial\; \backslash dot\; q^r\}\backslash ddot\; q^r+\backslash frac\{\backslash partial\; F\}\{\backslash partial\; t\}=\backslash frac\{\backslash delta\; L\}\{\backslash epsilon\}=\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q^r\}\backslash phi^r+\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q^r\}\backslash frac\{\backslash partial\; \backslash phi^r\}\{\backslash partial\; t\}\backslash right)+p\_s\backslash frac\{\backslash partial\; \backslash phi^s\}\{\backslash partial\; q^r\}\backslash dot\; q^r+p\_s\backslash frac\{\backslash partial\; \backslash phi^s\}\{\backslash partial\; \backslash dot\; q^r\}\backslash ddot\; q^r$$Since the $\backslash ddot\; q$ terms appear only once in either side, it's coefficients must be equal for this to be true, giving the relation:$$p\_s\backslash frac\{\backslash partial\; \backslash phi^s\}\{\backslash partial\; \backslash dot\; q^r\}=\backslash frac\{\backslash partial\; F\}\{\backslash partial\; \backslash dot\; q^r\}$$The above relation is useful in dervations involving infinitesimal generators of transformations.

Consider the symmetry transformations of fields in classical field theory as:

$\backslash begin\{align\}$

x\rightarrow&\, x'=x+\delta x(x)\\

\phi(x)\rightarrow&\, \phi'(x')=\phi(x)+\delta\phi(x)

\end{align}

Since the parameters can change, that is $x\backslash rightarrow\; x\text{'}=x+\backslash delta\; x(x)$, it is no longer sufficient to consider only change in Lagrangians independently as the volume element also contributes changes, nonetheless the same can be shown using change in action. The change in volume element is given by

$\backslash det(\backslash frac\{\backslash partial\; x\text{'}\}\{\backslash partial\; x\})=|(1+\backslash partial\_0\backslash delta\; x^0)(1+\backslash partial\_1\backslash delta\; x^1)\backslash cdots(1+\backslash partial\_\{n-1\}\; \backslash delta\; x^\{n-1\})|+\backslash mathcal\; O(\backslash delta\; x^2)=1+\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu\; +\backslash mathcal\; O(\backslash delta\; x^2)$

The terms arising in change in action is collected into effective change in Lagrangian as $\backslash delta\; \backslash int\backslash mathcal\; L\; \backslash ,\; d^n\; x=\backslash int\; \backslash mathcal\; L\backslash ,\backslash delta(d^nx)+\backslash int\; \backslash delta\backslash mathcal\; L\backslash ,d^nx=:\backslash int\backslash delta\; \backslash mathcal\; L\_\backslash text\{eff\}\backslash ,d^nx$ which gives $\backslash delta\; \backslash mathcal\; L\_\backslash text\{eff\}=\backslash mathcal\; L\backslash ,\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu+\backslash delta\backslash mathcal\; L$. Hence the modified conditions for dynamical symmetry are given as $\backslash delta\; L\_\backslash text\{eff\}=L\backslash ,\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu+\backslash delta\; L\; =\; \backslash partial\_\backslash mu\; F^\backslash mu$.

In Hamiltonian formalism, to avoid excess momentum terms in the Hamiltonian, one can make the switch from the covariant De-Donder Weyl formalism to a more traditional formalism where only time evolution is considered. This also has the benifit of providing the Hamiltonian as a generator of time translation. This is done using $\backslash mathcal\; L[\backslash phi,\backslash partial\backslash phi,x]\backslash rightarrow\; L[\backslash phi,\backslash dot\backslash phi,t]\backslash equiv\backslash int\backslash ,\backslash mathcal\; L[\backslash phi,\backslash dot\backslash phi,\backslash partial\_i\backslash phi[\backslash phi],\backslash mathbf\; x,t]\backslash ,d^3\backslash mathbf\; x$. Since both formalisms relate to minimization of the same action, they are said to be equivalent. The equations of motion are changed as follows:

$\backslash frac\{\backslash partial\; \backslash mathcal\; L\}\{\backslash partial\; \backslash phi\}=\backslash partial\_\backslash mu\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\_\backslash mu\backslash phi\}\backslash rightarrow\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}=\backslash frac\; d\{dt\}\backslash frac\{\backslash partial\; \backslash mathcal\; L\}\{\backslash partial\; \backslash dot\backslash phi\}$

In this formalism, the boundary conditions $\backslash phi\_i(x)=\backslash phi(x,t\_i)\backslash rightarrow\; \backslash phi\_f(x)=\backslash phi(x,t\_f)$ are given and the solution of the equations of motion relating $\backslash phi(x,t)$ are such that they satisfy the given boundary condition on $t=t\_i,t=t\_f$ and for all $x$. The relating Poisson brackets are also hence, defined in terms of the coordinates, $\backslash phi$ and respective conjugate momentum $\backslash pi=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\backslash phi\}$.{{Cite book |last=Williams |first=Anthony G. |title=Quantum field theory: classical mechanics to gauge field theories |date=2022 |publisher=Cambridge University Press |isbn=978-1-108-47090-2 |location=Cambridge, United Kingdom ; New York, NY}}

Corresponding $\backslash partial\_\backslash mu\; F^\backslash mu[\backslash phi,\backslash partial\backslash phi,x]\backslash rightarrow\; \backslash int\backslash ,\backslash partial\_\backslash mu\; F^\backslash mu[\backslash phi,\backslash dot\backslash phi,\backslash partial\_i\backslash phi[\backslash phi],\backslash mathbf\; x,t]\backslash ,d^3\backslash mathbf\; x=\backslash partial\_0\backslash int\; F^0\backslash ,d^3\backslash mathbf\; x\backslash equiv\; \backslash partial\_0\; F$ given that the vector component approaches zero on the boundaries. If $\backslash delta\; \backslash mathcal\; L\_\backslash text\{eff\}=0\backslash rightarrow\; \backslash partial\_0\; F=0$, hence in such conditions $F$ corresponds to conserved charge in the De-Donder Weyl formalism.

$\backslash delta\; L\_\backslash text\{eff\}=\backslash int(\backslash mathcal\; L\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu+\backslash delta\; \backslash mathcal\; L)\backslash ,d^3x=\; \backslash int\; \backslash mathcal\; L\backslash partial\_0\backslash delta\; x^0\backslash ,d^3x+\; \backslash int\; \backslash delta(\backslash mathcal\; L\backslash ,d^3x)\; =\backslash int\; \backslash mathcal\; L\backslash partial\_0\backslash delta\; x^0\backslash ,d^3x+\backslash delta\; L=\backslash partial\_0\; F$

If Euler Lagrange relation is satisfied for the provided Lagrangian, the invariants of motion can be derived as:$$\backslash delta\; L-\backslash epsilon\backslash frac\; d\; \{dt\}F(q,\backslash dot\; q,t)=\; \backslash epsilon\backslash phi\backslash cancelto\{=0\}\{\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; q\}-\backslash frac\{d\}\{dt\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q\}\backslash right)L\}+\backslash epsilon\backslash frac\{d\}\{dt\}\backslash left(\backslash phi\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q\}L-\; F\backslash right)=\backslash epsilon\backslash frac\{d\}\{dt\}\backslash left(\backslash phi\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q\}L-\; F\backslash right)=0$$Hence $\backslash left(\backslash phi\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q\}L-F\backslash right)=p\backslash phi-F$ is a constant of motion. Since Euler Lagrange equation is satisfied, the derived Noether invariant also generates the same symmetry transformation.

Update this with noether theorem from "Lagrangian Interaction An Introduction to Relativistic Symmetry in Electrodynamics and Gravitation - Doughty" including change in time.

Consider the symmetry transformations of fields in classical field theory as:

$\backslash begin\{align\}$

x\rightarrow&\, x'=x+\delta x(x)\\

\phi(x)\rightarrow&\, \phi'(x')=\phi(x)+\delta\phi(x)

\end{align}

The total variation of fields is defined as $\backslash Delta\backslash phi\backslash equiv\; \backslash phi\text{'}(x)-\backslash phi(x)=-\backslash partial\_\backslash mu\backslash phi(x)\backslash delta\; x^\backslash mu+\backslash delta\backslash phi(x)$ upto first order. Similarly it also follows that $\backslash Delta\backslash partial\backslash phi=\backslash partial\backslash Delta\backslash phi$.

$\backslash delta\; L=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}\backslash Delta\backslash phi(x)+\backslash frac\{\backslash partial\; L\}\{\backslash partial\backslash partial\backslash phi\}\backslash Delta\backslash partial\backslash phi+\backslash partial\_\backslash mu\; L\backslash delta\; x^\backslash mu=\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\backslash phi\}\backslash right)\backslash Delta\backslash phi(x)+\backslash partial\_\backslash mu(\backslash pi^\backslash mu\; \backslash Delta\backslash phi+L\backslash delta\; x^\backslash mu)-L\backslash partial\_\backslash mu\backslash delta\; x^\backslash mu$

Using equations of motion i.e. in the on-shell condition and combining terms in the symmetry condition $\backslash delta\; L\_\backslash text\{eff\}=L\backslash ,\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu+\backslash delta\; L\; =\; \backslash partial\_\backslash mu\; F^\backslash mu$, the conservation of current is derived as:

$\backslash partial\_\backslash mu(\backslash pi^\backslash mu\; \backslash Delta\backslash phi+L\backslash delta\; x^\backslash mu-F^\backslash mu)=\backslash partial\_\backslash mu\; j^\backslash mu=0$

Using $\backslash partial\_\backslash mu\; L=\backslash frac\{\backslash partial\; L\}\{\backslash partial\backslash phi\}\backslash partial\_\backslash mu\backslash phi+\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\_\backslash nu\backslash phi\}\backslash partial\_\backslash mu\backslash partial\_\backslash nu\backslash phi+\backslash partial\_\backslash mu^\backslash text\{ex\}L$ the effective change can be expressed as:

$\backslash delta\; L\_\backslash text\{eff\}=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}\backslash Delta\backslash phi(x)+\backslash frac\{\backslash partial\; L\}\{\backslash partial\backslash partial\backslash phi\}\backslash Delta\backslash partial\backslash phi+\backslash partial\_\backslash mu\; (L\backslash delta\; x^\backslash mu)=\backslash frac\{\backslash partial\; L\}\{\backslash partial\backslash phi\}\backslash delta\; \backslash phi+\backslash pi^\backslash nu(-\backslash partial\_\backslash mu\backslash phi\; \backslash partial\_\backslash nu\backslash delta\; x^\backslash mu\; +\backslash partial\_\backslash nu\backslash delta\; \backslash phi)+\backslash partial\_\backslash mu^\backslash text\{ex\}L\backslash delta\; x^\backslash mu\; +L\backslash partial\_\backslash mu\backslash delta\; x^\backslash mu$

which can be used to check for symmetry of transformation i.e. if it can be expressed as a divergence of some vector function $\backslash delta\; L\_\backslash text\{eff\}=\backslash partial\_\backslash mu\; F^\backslash mu$.

If the spatial components of conserved current vanish at the boundary, $\backslash int\backslash partial\_\backslash mu\; j^\backslash mu\backslash ,d^3x=0=\backslash int\; \backslash partial\_0\; j^0\backslash ,d^3x=\backslash partial\_0(\backslash int\; j^0\backslash ,d^3x)$is corresponding conserved charge.

$G=\backslash int\; j^0\backslash ,d^3x=\backslash int\; \backslash pi^0\backslash delta\; \backslash phi\backslash ,d^3x-\backslash int\; \backslash pi^0\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu\backslash ,d^3x+\backslash int\; \backslash mathcal\; L\backslash delta\; x^0\backslash ,d^3x-F$

$\backslash partial\_0\; G=\backslash int\; \backslash partial\_0\backslash pi^0\backslash delta\; \backslash phi\backslash ,d^3x+\backslash int\; \backslash pi^0\backslash partial\_0\backslash delta\; \backslash phi\backslash ,d^3x-\backslash int\; \backslash partial\_0\backslash pi^0\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu\backslash ,d^3x-\backslash int\; \backslash pi^0\backslash partial\_0\backslash partial\_\backslash mu\; \backslash delta\; x^\backslash mu\backslash ,d^3x+\backslash int\backslash partial\_0\; \backslash mathcal\; L\backslash delta\; x^0\backslash ,d^3x-\backslash delta\; L$

The symmetry condition for both classical Lagrangian and field theory Lagrangians can be expressed without making mention of the function, $F$ as follows:

Whether it is possible to simplify the following off-shell quantities as divergence of some vector function:

1. For classical field theories: $\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\_\backslash mu\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\_\backslash mu\; \backslash phi\}\; \backslash right)\backslash Delta\backslash phi\; \backslash ,\backslash equiv\; \backslash ,\; \backslash partial\_\backslash mu\; j^\backslash mu$
2. For classical Lagrangian mechanics: $\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q\}-\backslash frac\; d\{dt\}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q\}\; \backslash right)\backslash delta\; q\; \backslash ,\; \backslash equiv\; \backslash ,\; \backslash frac\; d\{dt\}Q$

The above equations are known as strong equations (also refered to as kinematic or improper) which unlike weak equations (also refered to as dynamical or proper), are relations that are simplified without invoking the equations of motion which are Euler Lagrange equations. According to convention related to these relations, equivalence symbol is used for strong relations and "is equal to" symbol for weak relations. Only the symmetry condition for variations $\backslash delta\; q$ or $\backslash Delta\; \backslash phi$, which is expressed as $\backslash delta\; L\_\backslash text\{eff\}=\backslash partial\_\backslash mu\; F^\backslash mu$ or the invariance of action, is used to derive the strong relations. Since Euler Lagrange equations are satisfied on-shell, the corresponding $\backslash frac\; d\; \{dt\}\; Q$ and $\backslash partial\_\backslash mu\; j^\backslash mu$ also vanish on-shell. The weak form of the above relations is hence derived by combining with on-shell EOM relations, resulting in Noether's first theorem conserved current relations: $\backslash partial\_\backslash mu\; j^\backslash mu=0,\backslash quad\; \backslash frac\{dQ\}\{dt\}=0$.

The same relations occur with additional terms in the case of local symmetries, which is a class of symmetries with infinite generators parametrized in a certain manner by arbitrary functions instead of arbitrary constants, leading to Noether's second theorem. In noether's second theorem, local symmetries are involved where $\backslash delta\; x=0,\backslash Delta\backslash phi=\backslash Delta\backslash phi[f,\backslash partial\_\backslash mu\; f,\backslash partial\_\backslash mu\backslash partial\_\backslash nu\; f,\backslash cdots;x]$ which is a symmetry of the system for any arbitrary function $f$. Hence, the derived strong relations are of the following form:

$\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\_\backslash mu\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\_\backslash mu\; \backslash phi\}\; \backslash right)\backslash frac\{\backslash delta\; \backslash Delta\backslash phi\}\{\backslash delta\; f\}\; \backslash ,\backslash equiv\; \backslash ,\backslash partial\_\backslash mu\backslash left(\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\_\backslash nu\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\_\backslash nu\; \backslash phi\}\; \backslash right)\backslash frac\{\backslash delta\; \backslash Delta\backslash phi\}\{\backslash delta\backslash partial\_\backslash mu\; f\}\backslash right)-\backslash partial\_\backslash mu\backslash partial\_\backslash nu\backslash left(\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\; \backslash phi\}\; \backslash right)\backslash frac\{\backslash delta\; \backslash Delta\backslash phi\}\{\backslash delta\backslash partial\_\backslash mu\; \backslash partial\_\backslash nu\; f\}\backslash right)+\backslash cdots$

The strong form of Noether's first theorem can be combined with that of the second theorem to get another strong relation, particularly since $f(x)=k$ should have its associated Noether's first theorem strong relations with $\backslash Delta\backslash phi=k\backslash frac\{\backslash delta\backslash Delta\backslash phi\}\{\backslash delta\; f\}$ where the constant can be absorbed into the conserved current.{{Citation |last=Brading |first=Katherine |title=Symmetries and Noether's theorems |date=2003-12-04 |work=Symmetries in Physics |pages=89–109 |url=https://doi.org/10.1017/cbo9780511535369.006 |access-date=2025-06-17 |publisher=Cambridge University Press |isbn=978-0-521-82137-7 |last2=Brown |first2=Harvey R.}} https://arxiv.org/pdf/hep-th/0009058

$\backslash partial\_\backslash mu\backslash left(j^\backslash mu-\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\_\backslash nu\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\_\backslash nu\; \backslash phi\}\; \backslash right)\backslash frac\{\backslash delta\; \backslash Delta\backslash phi\}\{\backslash delta\backslash partial\_\backslash mu\; f\}+\backslash partial\_\backslash nu\backslash left(\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash phi\}-\backslash partial\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash partial\; \backslash phi\}\; \backslash right)\backslash frac\{\backslash delta\; \backslash Delta\backslash phi\}\{\backslash delta\backslash partial\_\backslash mu\; \backslash partial\_\backslash nu\; f\}\backslash right)+\backslash cdots\backslash right)\backslash ,\backslash equiv\; \backslash ,\; 0$

which is a strong relation, showing that the divergence should vanish identically for the function.

Add also quasi-invariance of Lagrangian, consequences with singular Lagrangians, etc from Rothe n Rothe.

Using the calculated relations, the following Poisson brackets can be computed as:$$\backslash \{q^r,\backslash epsilon\; (p\_s\backslash phi^s-F)\backslash \}=\backslash delta\; q^r,\backslash quad\; \backslash \{p\_r,\backslash epsilon(p\_s\backslash phi^s-F)\backslash \}=\backslash delta\; p\_r+\backslash epsilon\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q^s\}-\backslash frac\{d\}\{dt\}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q^s\}\backslash right)\backslash frac\{\backslash partial\; \backslash phi^s\}\{\backslash partial\; \backslash dot\; q^r\}$$Hence, the term $p\backslash phi-F$ generates the canonical dynamical symmetry transformation if either the Euler Lagrange relation is satisfied, or if $\backslash frac\{\backslash partial\; \backslash phi\_s\}\{\backslash partial\; \backslash dot\; q^r\}=0\backslash ,\backslash forall\; s,r$ which is a infinitesimal point transformation. Note that in the point transformation condition, the quantity generates the transformation regardless of if the Euler Lagrange equations are satisfied and since they do not depend on the dynamics of the problem are said to be a purely kinematic relation.{{Cite journal |last=Mallesh |first=K. S. |last2=Chaturvedi |first2=Subhash |last3=Balakrishnan |first3=V. |last4=Simon |first4=R. |last5=Mukunda |first5=N. |date=2011-02-01 |title=Symmetries and conservation laws in classical and quantum mechanics |url=https://link.springer.com/article/10.1007/s12045-011-0020-5 |journal=Resonance |language=en |volume=16 |issue=2 |pages=129–151 |doi=10.1007/s12045-011-0020-5 |issn=0973-712X}}

Similar results are obtained in classical field theory, for example, in a Lorentz invariant Lagrangian density where corresponding conserved charges, momentum density $P^\backslash mu$ generates translation of fields and $M^\{\backslash mu\backslash nu\}$of Lorentz invariance generates Lorentz transformation of fields.{{Cite book |last=Greiner |first=Walter |url=http://link.springer.com/10.1007/978-3-642-61485-9 |title=Field Quantization |last2=Reinhardt |first2=Joachim |date=1996 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-78048-9 |location=Berlin, Heidelberg |pages=49-54 |language=en |doi=10.1007/978-3-642-61485-9}}

class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof

The change in generalized velocity and momentum term can be derived as: $$\backslash begin\{align\}$$ p=\frac{\partial L}{\partial \dot q}, \quad& \dot q=\frac {dq}{dt}\\ \delta p_r=\frac{\partial^2L}{\partial q^s\partial\dot q^r}\delta q^s+\frac{\partial^2 L}{\partial \dot q^s\partial \dot q^r}\delta \dot q^s,\quad&\delta \dot q^r=\epsilon \frac{\partial \phi^r}{\partial q^s} \dot q^s+\epsilon \frac{\partial \phi^r}{\partial \dot q^s}\ddot q^s+\epsilon\frac{\partial\phi^r}{\partial t} \\ \end{align} Firstly, the change in momentum can be expressed in a more useful form as follows:$$\backslash delta\; p\_r=\backslash frac\{\backslash partial^2L\}\{\backslash partial\; q^s\backslash partial\backslash dot\; q^r\}\backslash delta\; q^s+\backslash frac\{\backslash partial^2\; L\}\{\backslash partial\; \backslash dot\; q^s\backslash partial\; \backslash dot\; q^r\}\backslash delta\; \backslash dot\; q^s=\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q^s\}\backslash delta\; q^s+\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q^s\}\backslash delta\; \backslash dot\; q^s\backslash right)-\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q^s\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; q^s)-\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q^s\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\backslash dot\; q^s)=\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; L)-p\_s\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; \backslash dot\; q^s)-\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q^s\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; q^s)$$ Simplifying the required poisson brackets, $\backslash begin\{align\}$ \{q^r,\epsilon (p_s\phi^s-F)\}=\epsilon \left(\phi_r+\frac{\partial \dot q^m}{\partial p_r}\cancelto{=0}{\left(p_s\frac{\partial \phi^s}{\partial \dot q^m}-\frac{\partial F}{\partial \dot q^m}\right)}\right)&=\delta q^r\\ \{p_r,\epsilon(p_s\phi^s-F)\}=\epsilon\left(-p_s\frac{\partial \phi^s}{\partial q^r}+\frac{\partial F}{\partial q^r}+\cancelto{=0}{\left(\frac{\partial F}{\partial \dot q^m}-p_s\frac{\partial \phi^s}{\partial \dot q^m}\right)}\left(\frac{\partial \dot q^m}{\partial q^r}\right)_{q,p,t}\right) &=\epsilon\left(-p_s\frac{\partial \phi^s}{\partial q^r}+\frac{\partial F}{\partial q^r}\right)\\ \end{align} As a preliminary result, for any function of $(q,\backslash dot\; q,t)$, $\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}\backslash frac\{d\}\{dt\}-\backslash frac\{d\}\{dt\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}=\backslash frac\{\backslash partial\}\{\backslash partial\; q^r\}+\backslash frac\{\backslash partial\; \backslash ddot\; q^s\}\{\backslash partial\; \backslash dot\; q^r\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^s\}$ which can be used to calculated the quantity: $\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}\backslash left(\backslash frac\; \{dF\}\{dt\}\backslash right)-p\_s\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}\backslash left(\backslash frac\; \{d\}\{dt\}\backslash phi^s\backslash right)\backslash right)-\backslash dot\; p\_s\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash phi^s)=\backslash frac\{d\}\{dt\}\backslash cancel\{\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}F-p\_s\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}\backslash phi^s\backslash right)\}+\backslash frac\{\backslash partial\; \backslash ddot\; q^s\}\{\backslash partial\; \backslash dot\; q^r\}\backslash cancel\{\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^s\}F-p\_m\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^s\}\backslash phi^m\backslash right)\}-p\_s\backslash frac\{\backslash partial\; \backslash phi^s\}\{\backslash partial\; q^r\}+\backslash frac\{\backslash partial\; F\}\{\backslash partial\; q^r\}=\backslash \{p\_r,(p\backslash phi-F)\backslash \}$ This relation can be restated and combined with the formula for $\backslash delta\; p\_r$to give the required relation for momentum. $\backslash \{\; p\_r,\backslash epsilon(p\_s\backslash phi^s-F)\backslash \}=\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; L)-p\_s\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; \backslash dot\; q^s)\; -\; \backslash dot\; p\_s\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash dot\; q^r\}(\backslash delta\; q^s)=\backslash delta\; p\_r+\backslash epsilon\backslash left(\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q^s\}-\backslash frac\{d\}\{dt\}\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q^s\}\backslash right)\backslash frac\{\backslash partial\; \backslash phi^s\}\{\backslash partial\; \backslash dot\; q^r\}$