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As was described in, for example, this post, one can formulate Noether's Theorem also in Hamiltonian Mechanics. Symmetries are then represented by vector fields generated by observables whose Poisson brackets vanish with the Hamiltonian of the system.

My question is: How do you describe transformations of time in this formalism? Flows in the phase space only represent active transformations of the phase space, but they don't change anything about the time variable. It is not possible to derive, for example, conservation of energy as the consequence of a symmetry regarding the system's time-shifts.

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Quantumwhisp
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2 Answers2

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Yes, it is possible, same as for Lagrangian action. Taking the Hamiltonian action as $$ S_H\{q, p\} = \int_{t_i}^{t_f}{dt\; \left[p\dot{q} - H(q, p, t) \right]} $$ consider an infinitesimal time shift $t' = t +\delta t$, with $q'(t') = q(t) +\delta q$, $p'(t') = p(t) +\delta p$, and assume invariance under time translations: $$ S_H\{q, p\} = \int_{t_i}^{t_f}{dt\; \left[p\dot{q} - H(q, p, t) \right]} = \int_{t'_i}^{t'_f}{dt\; \left[p'\dot{q'} - H(q', p', t) \right]} $$ The 2nd equality above gives $$ \int_{t_i}^{t_f}{dt\; \left[p\dot{q} - H(q, p, t) \right]} = \int_{t_i + \delta t_i}^{t_f + \delta t_f}{dt\; \left[p\dot{q} - H(q, p, t) + p\delta \dot{q} + \dot{q}\delta p - \frac{\partial H}{\partial q}(q, p, t)\delta q - \frac{\partial H}{\partial p}(q, p, t)\delta p \right]} $$ or after slight rearrangement, $$ \left[p\dot{q} - H \right]\Big|_{t_f}\delta t_f - \left[p\dot{q} - H \right]\Big|_{t_i}\delta t_i + \int_{t_i + \delta t_i}^{t_f + \delta t_f}{dt\;\left[p\delta \dot{q} + \dot{q}\delta p + \dot{p}\delta q - \dot{q}\delta p \right]} = \\ = \int_{t_i}^{t_f}{dt\; \frac{d}{dt}\left[ p\dot{q}\delta t - H(q,p,t)\delta t + p\delta q\right]} = \int_{t_i}^{t_f}{dt\; \frac{d}{dt}\left[ 2 p\delta q - H(q,p,t)\delta t \right]} = 0 $$ where use is made of $\dot{q}\delta t = \delta q$ and the Hamiltonian EOMs, $ \dot{q}=\frac{\partial H}{\partial p}$, $\dot{p}=-\frac{\partial H}{\partial q}$. For a time translation that leaves end points invariant such that $\delta q\Big|_{t_i} = \delta q\Big|_{t_f} = 0$, the last equality above reduces to $$ \frac{dH}{dt}(q, p, t) = 0 $$ or $$ H(q, p, t) = const. $$

udrv
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  • Is there any way to describe this time-shift, or mixed transformations in space and time, in the same way it is done in the linked article? That means, using oberservables that create a flow in some manyfold, which represents the transformation that leave the action invariant? – Quantumwhisp Dec 27 '15 at 13:43
  • Re: the original article, Qmechanic answered your question separately already. I would add perhaps that if you consider the field theory version with invariance under space-time translations, then you'd end up with conservation of the stress-energy tensor, $\partial^\mu T_\mu^\nu =0$, subsuming energy-momentum conservation. This is probably the "flow-charge" form you are looking for. – udrv Dec 28 '15 at 03:40
  • I am sorry, but what you describe as flow here is some combined 4-vector-field (or lorentztensor), whose time and space components satisfy this continuity equation. What I mean when I say flow is some vectorfield (or a flow that is the intral-curves of this vectorfield) that describes the symmetry-transformations in noethers theorem, as it is done in the article I linked. The article does this only for "vertical" transformations, as I am looking for some kind of flow for "horizontal transformations", if I want to use the language of Qmechanic. – Quantumwhisp Dec 28 '15 at 13:58
  • What I describe is the conservation law for the "current density" corresponding to / expressing Noether's theorem for space-time translations, only in this case the "current" is tensorial, not vectorial. It is far from "some (arbitrary) combined 4-vector field". Please see at least the basics here: https://en.wikipedia.org/wiki/Noether%27s_theorem#Field_theory_version. Check paragraph above eq. for $T_\mu^\nu$. – udrv Dec 28 '15 at 20:40
  • No offense, I know what this current density is and that it is the expression of conservation in field theories, and that it originates from the space/time-translation invariance of the lagrangian. It is just not what I ment when I posted my question. That's what I was trying to say when I said for now, it is just a lorentztensor whose divergence vanishes, as opposed to a Vektorfield that generates transformations, as in the linked article. – Quantumwhisp Dec 29 '15 at 08:56
  • Edit: I reviewed the article I linked, and it is probably not that clear what I mean from that article. What I want to say ist that in hamilton formalism, you can describe symmetry-transformations by using a flow on the phase-space, and that any observable $F$ creates such a flow. That way for example angular-momentum creates roations arround the origin of the coordinate system. I have seen this only for transformations of the phase-space, and I want to see if there is a way to describe transformations of time or mixed transformations of time and phase-space in the same way. – Quantumwhisp Dec 29 '15 at 09:08
  • Do you mean "Hamiltonian flows" of canonical transformations generated by symmetry generators? For instance, as here: http://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf, secs. 4.4.1-4.4.2, around eqs.(4.112-4.114)? In that case I think the generator is the Hamiltonian itself and the flow just the evolution it generates. Otherwise you may want to ask Qmechanic about this. – udrv Dec 29 '15 at 12:38
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Comments to the question (v2):

  1. It should first of all be mentioned that the Hamiltonian generates time-evolution, although I'm sure OP already knows this.

  2. The Hamiltonian Lagrangian $$ \tag{1} L_H(q,\dot{q},p,t) ~:=~\sum_{i=1}^n p_i \dot{q}^i - H(q,p,t)$$
    may be viewed as a first-order Lagrangian system $L_H(z,\dot{z},t)$ in twice as many variable $$\tag{2} (z^1,\ldots,z^{2n}) ~=~ (q^1, \ldots, q^n;p_1,\ldots, p_n).$$

  3. How to apply Noether's theorem for energy conservation in a Lagrangian system is e.g. discussed in my Phys.SE answer here. None of the suggested infinitesimal transformations in my Phys.SE answer are of the form of a vertical vector field $$\tag{3} \delta z^I~=~\varepsilon V^I, \qquad \delta t~=~0, $$
    in such a way that the vector field components $V^I$ do not depend on $\dot{z}$, which seems to be at the core of OP's question.

  4. The corresponding Lagrangian energy function $$\tag{4} h_H(z,\dot{z},t)~:=~ \sum_{I=1}^{2n}\dot{z}^I\frac{\partial L_H(z,\dot{z},t)}{\partial \dot{z}^I} - L_H(z,\dot{z},t)~=~H(z,t) $$ is unsurprisingly the Hamiltonian itself. It will play the role of Noether charge for time translations.

  5. Energy is conserved on-shell if the Hamiltonian Lagrangian $L_H$ (or equivalently, the Hamiltonian $H$), has no explicit time dependence.

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