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- Symmetries and conservation laws: Consequences of Noether’s theorem
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We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived.

In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, a special case exists in which the action is scale invariant if it satisfies the following two constraints: 1 it must depend upon a scale-free Lagrangian, and 2 the Lagrangian must change under scale in the same way as the inverse time,.

Our contribution lies in the derivation of a generalised Lagrangian, in the form of a power series expansion, that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models—state space models based upon differential equations—that can be used to distinguish scale symmetry from scale freeness in empirical data. We establish face validity with an analysis of simulated data, in which we show how scale symmetry can be identified and how the associated conserved quantities can be estimated in neuronal time series.

Considerations of the way in which a dynamical system changes under transformation of scale offer insight into its operational principles. Scale freeness is a paradigm that has been observed in a variety of physical and biological phenomena and describes a situation in which appropriately scaling the space and time coordinates of any evolution of the system yields another possible evolution. In the brain, scale freeness has drawn considerable attention, as it has been associated with optimal information transmission capabilities.

Scale symmetry describes a special case of scale freeness, in which a system is perfectly unchanged under transformation of scale. Here we show that scale symmetry can be identified, and the related conserved quantities measured, in both simulations and real-world data. Our contribution allows for the first such statistical characterisation of the quantity that is conserved purely by virtue of scale symmetry.

PLoS Comput Biol 16 5 : e This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: E. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist. A symmetry is a transformation to a physical law that leaves its mathematical form invariant [ 1 ]. For instance, the known laws of physics are invariant under translation in space, rotation in space, and translation in time.

In other words: having taken all influencing factors into account, it is impossible for an external observer to determine whether a dynamical system has been shifted to a new location, rotated by a fixed angle, or whether its onset has been shifted in time.

However, the laws are generally not invariant under transformation of scale. Richard Feynman famously described an intuitive example of why this is the case for a scale transformation within a gravitational field. He asked the audience to consider a thought experiment in which an intricate cathedral made of matchsticks was increased in size to the point where it would instead be made of great logs, thus collapsing under its own weight.

The scale dependence of this system is further emphasized by his observation that:. The little cathedral made with matchsticks is attracted to the Earth.

So, to make the comparison I should make the big cathedral attracted to an even bigger Earth. Too bad—a bigger Earth would attract it even more and the sticks would break even more surely. Scale symmetries are therefore not universally applicable in the same way as translation in space, rotation in space, and translation in time. However, there are known constraints see Materials and Methods under which scale symmetries can arise in dynamical systems. In Noether demonstrated that for every continuous symmetry of the action of a dynamical system there exists a corresponding conservation law [ 3 ].

This theorem tells us that it is by virtue of the symmetries of translation in space, rotation in space, and translation in time that the corresponding quantities of linear momentum, angular momentum, and energy are conserved, respectively. It is the purpose of the present work to devise a method for estimating scale symmetries and their associated conserved quantities in empirical time series. We then show that an equation of motion leads to a scale invariant action under the constraints that its Lagrangian: 1 is scale-free, and 2 transforms inversely with time under change of scale.

In the second section, the main contribution of this paper is presented via the derivation of a generalised scale-symmetric Lagrangian, in the form of a power series expansion, which can be used to model time series from any scale-free system that follows the principle of stationary action. In the first section, we demonstrate proof of principle by showing that the generalised Lagrangian can be used to distinguish scale symmetry from scale freeness via simulations of a classical particle.

In the second section, using murine calcium imaging and macaque monkey fMRI datasets, we show that neural systems support a neurobiologically-based quantity that is conserved by virtue of scale symmetry. We use two ground truth datasets in the form of noiseless particle trajectories that are known a priori to be a scale-symmetric, and b scale-free. In Eqs 28 through 30 we show that the scale-symmetric case arises when the particle experiences a force that varies inversely as the cube of its distance from the origin, which in turn is known to result in an logarithmic spiral trajectory [ 4 ] Fig 1A , left.

In the scale-free case we use an inverse square force law which is known to result, for instance in planetary orbits, in an elliptical trajectory [ 5 ] Fig 1B , left. Note that we use a version of 34 in which we accommodate both an x and y coordinate, as shown in the accompanying code, to allow for the particles to trace 2D trajectories see S1 Text.

This means that the conserved quantity in 23 can be related directly to the geometric properties of the logarithmic spiral in the scale-symmetric case e. A In order from left to right: 1 the trajectory of a particle moving under the influence of a force that varies inversely as the cube of position; 2 Approximate lower bound log model evidence given by the free energy F following Bayesian model reduction for scale-symmetric sy.

We use Dynamic Expectation Maximisation DEM [ 6 ] to infer the latent states and estimate the parameters and hyperparameters; i. Using Bayesian model inversion, followed by model reduction, we show that the correct model is identified Fig 1A and 1B centre. We subsequently use the posterior expectations of the parameters for the full scale-free and reduced scale-symmetric models to show that the Noether conserved quantity or Noether charge is constant in time for the scale-symmetric Fig 1A , right , but not for the scale-free case Fig 1B right.

Here, we analyse murine calcium imaging [ 9 ] rest and task and macaque monkey fMRI [ 10 ] rest and anaesthetised datasets, using the same techniques as with the particle simulations described above.

The fMRI datasets are taken from the Nathan Kline Institute Macaque Dataset 1, in which twelve fMRI scans each approximately 10 minutes long are acquired in a single monkey in an awake state and twelve in an anaesthetized state. The macaque monkey was sedated with dexdomitor 0. Pre-processing of the murine calcium imaging [ 11 ] and macaque monkey fMRI [ 12 ] datasets were carried out as described previously.

We show all results obtained for the resting states in Fig 2. K Approximate lower bound log model evidence given by the free energy F following Bayesian model reduction for scale-symmetric sy. We find that there is higher model evidence for scale symmetry, as opposed to scale freeness, in a single region Fig 2C , left. All other regions show either higher model evidence for scale freeness, or else cannot be statistically classified either as scale-symmetric or scale-free—these regions are not coloured and appear white Fig 2C , right.

No region emerges as scale-symmetric in the task state. We show a sample timecourse from the region classified as scale-symmetric, together with the estimated data following model inversion Fig 2D. We then run both the full scale-free and reduced scale-symmetric models forward, with low noise in the absence of external inputs, in order to show the way in which the pure equations of motion evolve in time Fig 2E.

In the macaque monkey fMRI data, we observe higher model evidence for scale symmetry, as opposed to scale freeness, in a single cortical network Fig 2I. We show a sample timecourse, together with the estimated data following model inversion, from this scale-symmetric network Fig 2J.

We also calculate the variational free energy Fig 2K , associated probabilities Fig 2L , and Noether conserved quantity Fig 2M for this scale-symmetric network. No network emerges as scale-symmetric in the anaesthetised state.

Note that, although we focus on neuroimaging data in this study, these tools can be used to distinguish between scale symmetry and scale freeness in any dynamical system with measurable time series without restriction upon data dimensionality—provided that the system: a operates with scale-free dynamics and b follows the principle of stationary action.

In fact, as we show in Eqs 28 to 30 see Materials and Methods , the only way for a classical 1D time-independent Lagrangian to qualify as scale-symmetric is if its potential energy term varies as the inverse square of position. More generally, we show that a scale-free dynamical system that follows the principle of stationary action is only scale-symmetric in the special case that its Lagrangian scales inversely with time see Eq 12 in Materials and Methods.

This restrictive condition may explain why symmetries under change of scale are not usually discussed in the context of dynamical systems. Another reason could be that a symmetry is often defined as being contingent on a Lagrangian remaining invariant, which would only be possible in a scale transformation if the rescaling factors preceding the spatial and temporal variables cancelled each other in every term.

In the case of a non-unity Jacobian, quantities conserved by virtue of scale symmetry only exist if one redefines what is meant by scale symmetry to include a factor that cancels the Jacobian. No other definition leads to a conservation law. In other words, instead of satisfying the sufficient but not necessary condition of an invariant Lagrangian, we allow for the existence of scale symmetry via the sufficient and necessary condition of an invariant action.

To demonstrate the practical applicability of the theoretical results, we derive an expression for a generalised scale-symmetric Lagrangian in the form of a power series expansion see Eq 16 in Materials and Methods and show that this can be used to distinguish scale symmetry from scale freeness in ground truth models of classical particle trajectories.

Assuming that neural systems operate with scale-free dynamics [ 13 — 15 ] and evolve via a stationary action principle [ 16 — 18 ], we therefore establish a link between scaling properties and conservative aspects of neuronal message passing e. When describing angular momentum one can turn to familiar real-world examples involving e. Yet, if asked to provide a similarly intuitive understanding of the quantity conserved by virtue of scale symmetry, we would be hard-pressed.

We build an inference tool that allows for nonlinear effects with complex noise to be identified in the context of a hypothesised scale free dynamical systems architecture. Crucially, this solution to the inverse problem from data to mechanism enables us to test for alternate scaling principles. We hope that our theoretical framework, as well as the data and code we have made publicly available, will allow researchers to apply this methodology across a broader range of datasets, in order to reveal clues as to the biological underpinning of conservation laws arising by virtue of scale symmetries in neural systems.

The action can be evaluated for any trajectory, but trajectories that satisfy the equation of motion and thus might be followed in reality are distinguished because they render the action stationary. That is, a small variation of any trajectory that satisfies the equation of motion leaves the value of the action unchanged to first order [ 21 ]. Almost all of modern physics, including field-theoretic descriptions of electromagnetism, gravity and quantum theory, can be re-cast in terms of the principle of stationary action.

In this work, we consider a Lagrangian with explicit time-dependence to facilitate the analysis of driven systems. The principle of stationary action tells us that the trajectory q t followed by the system from any chosen initial point q i at time t i to any chosen final point q f at time t f renders the action, given by: 1 stationary.

One can then use standard arguments [ 22 ] to show that any trajectory q t for which the action is stationary is a solution of the Euler-Lagrange equation: 3. Scale freeness describes a situation in which different levels of magnification of a dynamical system are indistinguishable to within a multiplicative constant [ 23 ].

More explicitly, using 1 and 7 , we see that the system is scale-free if: 8. This shows that the scaled path described by 5 also renders the action stationary, i. Scale freeness has been observed in a variety of physical and biological settings [ 24 ]. These include neural systems across different species [ 25 , 26 ], in which evidence for scale freeness is identified by signatures of critical neuronal dynamics [ 27 , 28 ], and is considered to offer functional [ 29 ], developmental [ 30 ], as well as evolutionary [ 31 , 32 ] advantages.

However, some studies recognize the lack of sufficient orders of magnitude in spatial and temporal scale within such studies in neuroscience [ 33 ]. Furthermore, there are known limitations inherent in indirectly inferring scale freeness on the basis of proximity to power law behavior in neural cascading events [ 34 ] or in power frequency plots [ 35 ].

We say that a system is scale-symmetric if it is impossible to determine the magnification at which its evolution is observed. In other words, scale symmetry means that a system is perfectly unchanged under transformation of scale, i.

Since the path of integration is arbitrary, it follows that the action is scale-symmetric if and only if the Lagrangian satisfies: 12 where the identity defines the scaled Lagrangian and the equality describes the condition for scale symmetry. We therefore see that scale symmetry can exist in scale-free systems if these can be described by a Lagrangian that scales inversely with time.

Here, we present the main contribution of this paper via the derivation of a generalised scale-symmetric Lagrangian that can be used to identify scale symmetry in time series from any scale-free dynamical system that follows the principle of stationary action. We can write an expression for a Lagrangian as a sum over power terms: 13 where x , y and z are constants and C xyz is an arbitrary expansion coefficient.

This in turn means that we can replace the triple summation in 13 with a double summation: 16 which describes a family of scale-symmetric Lagrangians. This means that one can use an expansion of this expression to the desired number of terms as a forward generative model. This model is capable of creating data and also of applying an inverse procedure fitting to recover key model parameters from arbitrary time series.

This can be done for any dynamical system i. Note that one can in principle restrict the allowed values of the exponents in 16 to be natural numbers in order to obtain an analytic function. However, for the purpose of the work presented here we do not place such a restriction, in order to allow for greater flexibility in subsequent time series analyses.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant and, more generally, homogeneous under the action of a symmetry. View PDF on arXiv.

In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, a special case exists in which the action is scale invariant if it satisfies the following two constraints: 1 it must depend upon a scale-free Lagrangian, and 2 the Lagrangian must change under scale in the same way as the inverse time,. Our contribution lies in the derivation of a generalised Lagrangian, in the form of a power series expansion, that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models—state space models based upon differential equations—that can be used to distinguish scale symmetry from scale freeness in empirical data. We establish face validity with an analysis of simulated data, in which we show how scale symmetry can be identified and how the associated conserved quantities can be estimated in neuronal time series. Considerations of the way in which a dynamical system changes under transformation of scale offer insight into its operational principles. Scale freeness is a paradigm that has been observed in a variety of physical and biological phenomena and describes a situation in which appropriately scaling the space and time coordinates of any evolution of the system yields another possible evolution.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. Mathematical Physics. Authors: Stephen C.

Conservation of Linear Momentum. Homogeneity of space: If each particle of an isolated system is shifted by a constant distance ε, then the system's Lagrangian.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Tuleja and M.

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. Cosserat and F. Cosserat in This theorem only applies to continuous and smooth symmetries over physical space. Noether's theorem is used in theoretical physics and the calculus of variations.

Zheng Xiao, Long Wei. Google Scholar. Article views PDF downloads Cited by 0. Applying the well-known Lie symmetry method, we analysis the symmetry properties of the equation.

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We transform the time fractional SMK equation to nonlinear ordinary differential equation ODE of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in the Erdelyi-Kober EK sense. We solve the reduced fractional ODE using a power series technique.

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The derivation of conservation laws for invariant variational problems is based on Noether's theorem. It is shown that the use of Lie-. Bäcklund transformation.

Ogier D. 11.05.2021 at 13:11PDF | We derive conservation laws from symmetry operations using the principle of least action. These derivations, which are examples of.

Ursulina V. 16.05.2021 at 10:17PDF | Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law.

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