Equations of Motion. State space is the set of all possible states of a dynamical system; each state of the system corresponds to a unique point in the state space. same geometrical considerations used to update a trajectory should be used when correcting any joint limit violations that may occur. As the basic phase nears completion, the author is interested in extensions to specific applied areas. ’s fastest-growing privately-owned. In particular, if the Hamiltonian preserves energy, then one trajectory cannot visit two parts of phase space with different energy. By tracing the abundance of "invisible" words -- pronouns, articles and other short. fr/APC_CS/fr/calendar-node-field-start-date/month/2020-09 fr. Here d = attractor dimension = 1 n = phase-space dimension = 2. Physics 6010, Fall 2010 Introduction. For example, the state space of an autonomous Hamiltonian system the ﬂow is conﬁned to a curved constant energy hyper-surface. The shape of the trajectory represents the system. How the Hamilton-Jacobi equa-tion foreshadows quantum mechanics. However, most of the work in this paper involves trajectories at a fixed energy that. The period is independent of amplitude Consider the harmonic oscillator with Hamiltonian H(p;q) = 1 2 p2 + q2. However, specially in molecular simulations, where the number of degrees of freedom is exceedingly high, the representation in generalized coordinates is completely unsuitable. NUTLEY, NJ - Mayor/Parks and Recreation Commissioner Mauro G. Upon extraction of a periodic orbit, nearly-periodic orbits are integrated. In this case, the trajectory is an ellipse, as shown in the Figure below. The mission management system may include a flight management unit to define and manage a flight plan and associated flight trajectory, an airport navigation unit to define and manage a ground plan and associated ground trajectory, and a mission management unit to manage a continuity between the ground and flight phases of the mission and. at 1300 North Thornton Ave. Our Hamiltonian is H(x). 1) are rewritten in the 2D phase space form x˙ i = v i(ssp), the divergence of the velocity ﬁeld v vanishes, namely the ﬂow is in-compressible. The free flight phase, discussed in Section 5, covers the tra­. A phase space plot is a plot of its independent variables. Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems Kozlov, V. There exists a function H(x,p) which is constant along a particle’s trajectory in phase space and is time-independent. the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. The phase space plot should look like a triangle (hence the name Tent) bounded between zero and one. the phase-space volume conservation, as we shall see in sect. The rst is naturally associated with con guration space, extended by time, while the latter is the natural description for working in phase space. An Example: The Pendulum Consider a simple pendulum. Change of variable (Hamiltonian trajectory) → Conservation of the phase space volume! By noting: Jarzynski theorem. Tucci announced during the Tuesday, Aug. To state this a little more formally, we take a 2-dimensional cross section of the phase space: Γ, and define the Poincaré map: ρ:Γ → Γ such that ρ(x) = Φ τ (x), where Φ is the trajectory and is the first time point when the trajectory passes through Γ again. Non-Chaotic Attractor Examples: →for a damped undrivenpendulum, attractor is just a point at θ=ω =0 (0D in 2D phase space). ij(t) = ∂2H=∂Xj∂Xi is the 6 £6 Hessian matrix of the Hamiltonian with respect to the coordinates and momenta evaluated along the approximate trajectory. Phase portrait A plot of two or more dynamical variables against each other. We are looking at the site in the context of our academic space study and the entire Grounds Plan. Phase Space Trajectory : For simple harmonic motion, the position and momentum satisfy (1) (2) where C and are constants, is the angular frequency, t is the time, and. In the Hamiltonian approach, we're in phase space with a coordinate system having positions and momenta on an equal footing. Learn more. Space vector trajectory is also plotted and can be checked at different modulation indexes using the parameter defined in model properties. In field theories in Minkowski space-time, for example, considerable effort is devoted to showing that, in spite of the non-covariant nature of the phase space, the. Trajectory-free approximation of phase space structures 687 Section 2 gives the mathematical preliminaries and notation to provide the mathematical context for the divergence rate. The ruling comes three months after UC Regents voted to phase out the use of those scores by 2025. The trajectory coming out of the saddle point down and to the left must continue down and to the left until it ends at the equilibrium point. ˆ(q;p;t) = 1 N XN j=1 (q q j(t)) (p p j(t)) where q j(t) and p j(t) is the phase space location of the jth trajectory at time t. In two-dimensional phase space, The evolution of a volume element dV = dp x dx in phase space is given by Volume in phase space is conserved under Hamiltonian flow, a property known today as Liouville’s theorem. Starting from a suitable zeroth-order Hamiltonian, for which the Wigner density is available (either analytically or numerically), the phase space distribution is propagated in time via classical trajectories, while the perturbation is gradually switched on. general case, however, the phase-mixing occurs in phase-space. The asymptotic trajectory indicator can be applied to higher dimensional systems coupled to Langevin baths as we demonstrate for a 3D model of the isomerization of ketene. Six-time F1 world champion Lewis Hamilton will make his first foray into team ownership with the launch of X44, which will compete in the (…). Quantum hydrodynamics in spin chains with phase space methods. In order to consistently construct an extended Hamiltonian on the symplectic extended phase space, other approaches (e. Sections II-A, II-D, and II-F detail each of these points in turn. Binary outputs, simple calculation, high precision and fast response time were the main advantages of the phase space trajectory autocorrelation. Doing this for many values of $$t$$ will then give us a sketch of what the solution will be doing in the phase plane. We discuss the case in which there is no such a measure to be normalized in Section II. Suppose further that the coordinates are all angles. Thus, the singularities of Sare already ‘unfolded’ in the phase space (x. same throughout the entire trajectory. Let us present the following time dependent Hamiltonian  (1) The propagator corresponds to this system can be written in the phase space as (2) This propagator is not exactly evaluated for any arbitrary time dependent mass or frequency, because that will lead to non-linear differential equations. For low dimensional systems (2-or 3-D), it is easy to visualize the limit sets in the phase space: a fixed. Some 6 million miles from Earth, a NASA spacecraft is hurtling through space at more than 70,000 miles per hour. where i is the imaginary unit, t is time, is the partial derivative with respect to t, is the reduced Planck's constant (Planck's constant divided by 2π), ψ(t) is the wave function, and is the Hamiltonian (a self-adjoint operator acting on the state space). a Hamiltonian system q0= @F @p (8) p0= @F @x (9) Proof. Hamiltonian to the Lagrangian. geometric or topological features of the control space. The free flight phase, discussed in Section 5, covers the tra­. It represents a major station and an important stage in the probe's journey, where the team succeeded in operating the six propulsion devices for the first time, to be used in placing the probe in its direct path toward the Mars. The phase space contraction and the entropy production rates of Hamiltonian systems in an external field, thermostatted to obtain a stationary state are considered. Hi guys, I need your help to know how to draw the phase trajectory of a system in the phase plane. In c, we present the probability distributions at times t = 0, t = 1 and t = 6 in the first configuration, i. In an n dimensional phase space, the generator of the time translation can be written in terms of a Hamiltonian and a set of Poisson brackets for the phase space variables. In particular, if the Hamiltonian is returned to its original form the system will return to its original state, apart from a phase factor. The inside region, where the pendulum oscillates back and forth, corresponds to. This has been quantified in terms of the notion of “singular structures’ in the Lagrangian descriptor plots, which are easy to recognize visually (Mancho, Wiggins et al. 10 ) subject to the 2-point boundary conditions , equations. The independence of path is a result of the fact that the non-dynamical phase is a line integral, and can be made with no reference to time:. The choice for ΔΦ is dictated by the fact that it represents the variation in action when the particle moves from its initial position to its final position. General and Canonical Transformations. In particular, if the Hamiltonian preserves energy, then one trajectory cannot visit two parts of phase space with different energy. When the pendulum moves, it travels along a path in the four dimensional phase space made up of Q1, Q2, L1 and L2: the phase space trajectory. The action as an integral of the canonical 1-form over a path in the extended phase space. Through rewriting the CRTBP into Port-Hamiltonian framework, we are allowed to design the feedback controller through energy-shaping and dissipation injection. ﬁner than ba. Let phase crest 0 pass through the particle at point a at time 0. In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. In the case that the particle both enters and exits the interaction region then at some time tiit will have phase-space coordinates f~iin the interaction region. A PCM heat sink can help to reduce the overall mass and. 1 From Lagrange to Hamilton. The result is a renormalisation of the Hamiltonian βH which leads to an eﬀective Hamiltonian expressed in terms of a ‘coarse-grained’ magnetisation ﬁeld m¯(x)= 1 (ba)d! Cell dym(y), where the integral runs over a cell of size (ba)d centred on x. Hamiltonian Systems We begin with a foundational assumption of Hamiltonian mechanics: for any particle, the set of all its possible positions is an n-dimensional (n 3) smooth manifold Cin R3, called con guration space. The shape of the trajectory represents the system. If not specified, it will be set to step_size x num_steps. 7700 South River Parkway. Quantum hydrodynamics in spin chains with phase space methods. How the Hamilton-Jacobi equa-tion foreshadows quantum mechanics. , along the horizontal axis and on the vertical axis). Doing this for many values of $$t$$ will then give us a sketch of what the solution will be doing in the phase plane. As the basic phase nears completion, the author is interested in extensions to specific applied areas. dt at Specifically, unlike its classical counterpart, in general, f. Phase Space Trajectory. Author(s) Polkovnikov, Anatoli. Phase, Flux, and Charge Qubits A Josephson qubit can be understood as a nonlinear resonator formed from the Josephson inductance and its junction capacitance. The canonical 1-form and the symplectic 2-form on the cotangent bundle. “Safer at Home. Formula One enters next phase of the season with a bang. Critical temperatures k BTc AFM =J¼4:5 at p =0(16)andk BTc SG=J¼1:1atp =0. Exercise 1. This is called a point transformation: we've just moved to a different coordinate system, we're relabeling the points in configuration space (but possibly in a time-dependent way). If x(0) denotes the initial conditions of a trajectory at time t 1 =0, then this trajectory is periodic if it returns to its initial point in phase space after the time t 2 =T (period), i. One of the best known examples is the Berry phase [M. The action as an integral of the canonical 1-form over a path in the extended phase space. For this purpose, the Bohlin mapping is used to transform the Coulomb potential into a harmonic oscillator problem. 4 Particle-in-a-box problem in phase space formalism Writing E n= p2/2m we have p n= k n= n(h/2a)(5. num_steps – The number of discrete steps over which to simulate Hamiltonian dynamics. Hamiltonian mechanics in the “extended” phase space Jurgen Struckmeier¨ j. By the Boltzmann equation, if an ensemble has a property called 'temperature', then regions of phase space with more energy are less populated. Classical Hamiltonian trajectories are initiated at random points in phase space on a fixed energy shell of a model two degrees of freedom potential, consisting of two interacting minima in an otherwise flat energy plane of infinite extent. [email protected] If the ratio 1 / 2 is irrational, the trajectory in the two dimensional space is not closed as is the case in classical mechanics. The Hamiltonian operator is said to be a “generator” of the R symmetry. Beautiful Class A Office Space with Views of Adjacent Forest Preserve and Des Plaines River Front Lobby Exposure. For Hamiltonian systems, the probability measure is given by the phase space volume suitably normalized because the measure-preserving condition is guaranteed by the Liouville theorem. The phase space is the space of the dependent variables (X), it does not include t, while the state space incorporates t. Equations of Motion. , the quench from a ground state of the Hamiltonian in the trivial phase (θ 2 = π. So it comes from thinking of this in phase space. Thus the node of intersection between the particle trajectory and the phase. 3 Introduction This course principally deals with the variational principles of mechanics, particularly the Lagrangian and Hamiltonian descriptions of dynamical systems. So far the applicability and measurements of the Berry phase were mostly limited to. the electron charge. same loop using different velocities, the dynamic phase may change but the geometric phase will not. Classical Hamiltonian trajectories are initiated at random points in phase space on a fixed energy shell of a model two degrees of freedom potential, consisting of two interacting minima in an otherwise flat energy plane of infinite extent. By the Boltzmann equation, if an ensemble has a property called 'temperature', then regions of phase space with more energy are less populated. We apply this method of localized control to a forced pendulum model and to a system describing. H A mathematical function that can be used to generate the equations of motion of a dynamic system, equal for many such systems to the sum of the. If x(0) denotes the initial conditions of a trajectory at time t 1 =0, then this trajectory is periodic if it returns to its initial point in phase space after the time t 2 =T (period), i. The Sun-Saturn-Cassini, or phase, angle at the time was 72 degrees; hence, the partial illumination of Saturn in this portrait. The symplectic invariancerequirements areactuallymore stringent than just the phase space volume conservation, as we. a probability density w(p,q,t) is introduced in the phase space), the relevant phase space ensemble is called a statistical ensemble. This Hamiltonian deﬁnes a. I want to calculate and draw the phase space trajectory of this damped harmonic oscillator: $$\ddot{x}+\gamma\,\dot{x}+\omega^2x=0$$ for the two cases $\gamma=2\omega$ and $\gamma=\omega$. 2004-10-11 00:00:00 We establish the existence of weak limits of solutions (in the class L p, p%thinsp;≥ 1) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. Since the value of f is the. Often one considers a parameterization of this by boundary data or choice of a Cauchy surface. Its destination is Mars — the Red Planet, some 42 million miles away — where it. The phase space contraction and the entropy production rates of Hamiltonian systems in an external field, thermostatted to obtain a stationary state are considered. c x z + ω sz = 0 (1) with κ = 2r0 Nc λ γ3 and defining the linear frequencies { ωx = ω0Qx (1− ξz/βcη) ωs = ω0Qs (2) where C = 2πR is the ring circumference. Phase space volumes during these segments were then preserved such that the dissipation was explicitly retained in the degrees of freedom of the total system. So also are the phase space integral invariants s : j L at (32) The action integral 5. We apply this method of localized control to a forced pendulum model and to a system describing. 14,15,17 The tangent space to the Hamiltonian phase space can then be decom-posed using an orthonormal basis of ﬁnite-time Lyapunov vectors. Trajectory-free approximation of phase space structures 687 Section 2 gives the mathematical preliminaries and notation to provide the mathematical context for the divergence rate. One of the benefits of phase-space dynamics is that every phase-point on the phase-space. The Hamilton action S for any conceivable (real or imagined) trajectory A → B, connecting two given points q A and q B in a given time T, is defined by the integral S = ∫ 0 T L (q, q ̇, t) d t along the space-time trajectory q(t) connecting the specified space-time events, initial event A = (q A, t A = 0) and final event B = (q B, t B = T. The Moon is both seductively close to Earth and cosmically far away: Decades after the end of the space race, it remains extraordinarily expensive and difficult to actually get there. Relevant Sections in Text: x1. The phase space trajectory of a single particle falling freely from height is? Phase space is a plot between momentum and position, and since kinetic energy increases the momentum must increase with position, so option "2" must be correct, but the answer key shows that answer in option "4". phase space, ensembles and partition functions. con guration space is R3 and its phase space is R6 (locations and momenta). Classical Mechanics, Lecture 16: Hamiltonian Mechanics. This follows from the Liouville theorem. The first trajectory operations took place successfully on August 17, which is among a series of trajectory control maneuvers. The phase space of the (n-dimensional) harmonic oscillator, a. as a trajectory through its state-space (or phase-space). The phase space can be used to diagnose the evolution of cyclones that undergo single phase (pure tropical or extratropical cyclone) or multiple phases of existence (extratropical transition, tropical transition). Thus the dimension of the difficult part of the problem, the part that involves the solution of coupled ordinary differential equations, is reduced by two. Binary outputs, simple calculation, high precision and fast response time were the main advantages of the phase space trajectory autocorrelation. This trajectory was established by integrating orbital model equations forward from the TLI state vector. Berry (1984), Proc. This unit features open living space along with 9' ceiling, gourmet kitchen with stainless steel appliance and quartz counters. ij(t) = ∂2H=∂Xj∂Xi is the 6 £6 Hessian matrix of the Hamiltonian with respect to the coordinates and momenta evaluated along the approximate trajectory. d) ) Describe two new variables that will make the phase space trajectory a circle. 2004-10-11 00:00:00 We establish the existence of weak limits of solutions (in the class L p, p%thinsp;≥ 1) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. Machine Intelligence and Data Science Strategist at Booz Allen Hamilton. Phase III: In-person learning Monday to Thursday and virtual classes on Fridays Phase IV: In-person school five days a week Each phase will be no less than four weeks, and families will have two. [email protected] So let's think of the motion of this oscillator in the x, p plane. So far the applicability and measurements of the Berry phase were mostly limited to. We present a method to control transport in Hamiltonian systems. In classical mechanics, the phase space is the space of all possible states of a physical. ] (c) Draw a 2-dimensional phase space with q labeling the horizontal axis and p labeling the vertical axis. The general form of the Hamiltonian for the Josephson eﬁect is HJ = UJ. To operate with limited sensor horizons in unpredictable environments, autonomous robots use a receding-horizon strategy to plan trajectories, wherein they execute a short plan while creating the n. 3 Phase space and intuition for quantum adiabatic invariants PROFESSOR: Let's do it biogeometrically. “Phase 2 is where we get into probably more of the things you read about in science fiction. We say that evolution is governed by a ﬂow in phase space. The city’s development arm, the Renaissance City Development Association, is in the final phase of a land swap that will clear the remaining hurdles for a 203-unit building at Howard and. Attractors in Phase Space The surfaces in phase space which the pendulum follows, after transient motion decays, are called attractors. Interestingly, that the time steps satisfying the stationary condition turn out to be independent of the symplectic algorithms chosen. So far the applicability and measurements of the Berry phase were mostly limited to. , the convective time derivative is zero). In Lecture 1, we plotted the phase portrait for the simple pendulum, which we reproduce in Figure 1 as an illustration of our discussion. This means that the probability distribution on phase space acts like an incompressible ﬂuid. For this procedure to be valid, it has to be shown that extremizing S 3 with respect to all possible phase space trajectories {p i(t),xi(t)} is equivalent to extremizing S 2 with respect to i{x(τ),t(τ)} for τ being an aﬃne parameter. trajectory could be trapped by the potential. The phase space trajectory of an undamped oscillator is shown below right. Hi guys, I need your help to know how to draw the phase trajectory of a system in the phase plane. B is that part of phase space from which all initial conditions lead to A as t ∗ →: A B • A has dimension d 1: dq i /dt = dH/dp i (q is a generalized coordinate) dp i /dt = -dH/dq i (p is a generalized momentum = mv) p and q constitute the phase space for the dynamics; N-dimensional dynamics have a 2N-dimensional phase space; p i and q i (for i = 1 to N) are the phase. However, most of the work in this paper involves trajectories at a fixed energy that. Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations. fcl+papfcl=o. Phase Space Trajectory : For simple harmonic motion, the position and momentum satisfy (1) (2) where C and are constants, is the angular frequency, t is the time, and. 2020168  Rogério Martins. H A mathematical function that can be used to generate the equations of motion of a dynamic system, equal for many such systems to the sum of the. Represent the continuous function ˆ(q;p;t) in phase space by a discrete sampling with N trajectories. 1-3) Review of Lecture 12 relations: Examples of Hamiltonian mechanics in phase plots 1D Pendulum and phase plot (Simulation) 1D-HO phase-space control (Simulation of “Catcher in the Eye”) Exploring phase space and Lagrangian mechanics more deeply. 8-billion budget for the current fiscal year. of a two degree-of-freedom Hamiltonian system. If we denote the system’s Hamiltonian by H: H ( , , ) , m p U r r 1 2 i N i N 2 = + 1 f = /= G (3. 183 A unified framework for the numerical solution of optimal control problems using pseudospectral methods. 8 Chapter 2. nonlinearity is crucial because the system has many energy levels, but the operating space of the qubit must. Just like a direction field, a phase portrait can be a tool to predict the behaviors of a system’s solutions. Author(s) Polkovnikov, Anatoli. We say that evolution is governed by a ﬂow in phase space. The full four-body phase space is given by Z d 4 = Z Y4 i=1 d3p i (2ˇ)32E i (2ˇ)4. However, there is one idea, not mentioned in the book, that is very useful to sketching and analyzing phase planes, namely nullclines. The second part of the paper, starting with section 5, deals with the classical limit. Then, at each grid point x = (α, β), we can calculate the solution trajectory’s instantaneous direction of motion at that point by using the given. Comments are turned off. In other words a hamiltonian must exists for these new variables. 2 Isotropic oscillator & Kepler problem in phase space formalism andthat A 1,A 2 =2ωA 3 A 2,A 3 =2ωA 1 A 3,A 1 =2ωA 2 which is to say: L ≡ 1 2ω A possesses Poisson bracket properties that mimic. This is called a point transformation: we've just moved to a different coordinate system, we're relabeling the points in configuration space (but possibly in a time-dependent way). 3) Classically, such a conﬁned free particle traces a rectangular trajectory in phase space, of phase area n=2p n·a = nh as required by the “Planck quantization condition. where the phase-space density r@x, tD is the classical analog of the quantum mechanical density operator. Non-Chaotic Attractor Examples: →for a damped undrivenpendulum, attractor is just a point at θ=ω =0 (0D in 2D phase space). The journey just got a bit easier, however, thanks to a freshly published NASA invention. The extratropical transition of tropical cyclones (TCs) is examined here as one trajectory through the phase space. These theoretical results suggest that the initial rotational excitation significantly enhances reactivity with an amount comparable to the effect of initial. I'm really stuck with this and have no idea what to do I've found stuff on google about it but am struggling to follow. Then the distribution can’t spread out. Show that a system x0= F(x) is at the same time a Hamiltonian system and a gradient system i the Hamiltonian His a harmonic function. This can be used to extract the geometric Aharonov-Anandan phase from any closed-trajectory evolution without requiring knowledge or control of the Hamiltonian. phase space under Hamiltonian evolution, then its shape can change but its volume remains the same. The import of the equations is to fill the phase space with the little arrows shown in figure. The rst is naturally associated with con guration space, extended by time, while the latter is the natural description for working in phase space. The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics and may become advantageous for quantum technology. 5 we’ll introduce the concept of phase space and then derive Liouville’s theorem, which has countless applications in statistical mechanics, chaos, and other ﬂelds. calculated along the arc of trajectory Γ joining the initial point to the final point in spacetime; is the momentum vector and is the Hamiltonian function. Metric and curvature in gravitational phase space 3 structure. In addition, we present Gaussian-weighted quasi-classical trajectory and phase space theory calculations of the integral cross section and thermal rate constant for the title reaction. This gives us a point in the $${x_1}\,{x_2}$$ or phase plane that we can plot. The Hamiltonian function corresponding to the ith particle is 2 2 const. For two DOF Hamiltonian systems, the stable and unstable manifolds of the orbit discussed provide. Interestingly, that the time steps satisfying the stationary condition turn out to be independent of the symplectic algorithms chosen. 1) can be rewritten as h= 1 2 zz¯ (2. An isolated system with a given amount of energy is confined to that energy surface in phase space. (b) Use this generalized momentum and derive the Hamiltonian for the system. Con guration space. This model simulates a simple technique to implement space vector modulation. These are called Hamilton’s equations. For each region of phase space there is a local formulation of the system so that the evolution of the system is described by a time-independent Hamiltonian that depends only on the momenta. Phase Space. 2 billion in prime contracts, and ManTech International, No. where i is the imaginary unit, t is time, is the partial derivative with respect to t, is the reduced Planck's constant (Planck's constant divided by 2π), ψ(t) is the wave function, and is the Hamiltonian (a self-adjoint operator acting on the state space). Proposition 1. phase space, ensembles and partition functions; 16. 1) – the Lorenz equations – and the corresponding sketch of Fig. In classical mechanics, the phase space is the space of all possible states of a physical. The post translunar injection (TLI) phase, discussed in Section 4, covers the portion of flight from the translunar injection to CSM separation (10,962. Model potentials studied consist of a single bistable mode plus transverse harmonic modes. In phase space reconstruction, the time series is rearranged into a phase space based on time delay embed-ding. Classical Hamiltonian trajectories are initiated at random points in phase space on a fixed energy shell of a model two degrees of freedom potential, consisting of two interacting minima in an otherwise flat energy plane of infinite extent. https://www. Section 5 contains concluding remarks. We say that evolution is governed by a ﬂow in phase space. FightAIDS@Home - Phase II Introduction. geometric or topological features of the control space. "For the first time, we will evaluate the impact and severity of the hearing loss and tinnitus on the patients' physical, emotional and social functioning. The phase space of the (n-dimensional) harmonic oscillator, a. a Hamiltonian system q0= @F @p (8) p0= @F @x (9) Proof. Some 6 million miles from Earth, a NASA spacecraft is hurtling through space at more than 70,000 miles per hour. Also of interest is the trajectory in phase space (~x(t);p~(t)). Steps to Solve Questions of Phase Space Trajectory (in Hindi) 8m 47s. Quantum hydrodynamics in spin chains with phase space methods. The Hamiltonian governing this motion will have three degrees of freedom (six coordinates in phase space) and will look like. The Hamiltonian function corresponding to the ith particle is 2 2 const. 10) j=1 Recall that for Hamiltonian systems, canonical transformations do not change volume ele-ments. In the Hamiltonian approach, we're in phase space with a coordinate system having positions and momenta on an equal footing. Learn more. Momentum and coordinate graph. All orbits in phase space (i. The city’s development arm, the Renaissance City Development Association, is in the final phase of a land swap that will clear the remaining hurdles for a 203-unit building at Howard and. TikTokers are celebrating their heritage by taking part in the #FashionEdit challenge – and challenging the app’s algorithm in the process In late July, Milan Mathew uploaded a video of herself to TikTok. The second part of the paper, starting with section 5, deals with the classical limit. The energy conservation condition restricts the motion on a (6N-1)-dimensional surface in phase-space, known as the constant-energy hypersurface or simply the constant-energy surface, see Fig. All orbits in phase space (i. Crest 0 now crosses the x axis ~that is, the particle trajectory! at point d. The non-archetypal emergence trajectories remained fixed in a region of phase space characterized by a relatively high conductivity and low connection strength throughout emergence. Date Issued 2020-05-15. In practice, di•erent chaotic trajectories may have di•erent k’s. We apply this method of localized control to a forced pendulum model and to a system describing. In one-dimension the phase space is constructed with a pair of canonically conjugate variables, position x and momentum p, and the general dynamics of a function F (x,p;t)can be written in the form of a continuity equation ∂F ∂t +∇·J = σ, (1) where ∇= ∂ ∂x, ∂ ∂p. 10) j=1 Recall that for Hamiltonian systems, canonical transformations do not change volume ele-ments. 0 in terms of the other variables yields the Hamiltonian on our reduced (6-dimensional) phase space. In case num_steps is not specified, it will be set to $$2\pi$$. ﬁner than ba. Hamiltonian. Phase space In physics, phase space is a concept which uniﬁes classical (Hamiltonian) mechanics and quantum mechanics; in mathematics, phase space is a concept which uniﬁes symplectic geometry with harmonic analysis and PDE. trary phase space trajectory segment i, its conjugate seg-ment i' '. Time-dependent canonical transformations. Immunomedics, Inc. num_steps – The number of discrete steps over which to simulate Hamiltonian dynamics. The goal of this note is to present Lagrangian and Hamiltonian dynamics, and the connection between them. As each member of the ensemble moves through phase space along a trajectory specified by Hamilton's equations of motion, the phase space density evolves in time. The phase space is (R 6) n, since each particle is specified by six coordinates: three position coordinates and three momentum coordinates. is invariant under a canonical transformation. We also notice that the Hamiltonian in Eq. A Phase Space Current use of the term phase space in nonlinear science is conveniently exempliﬁed by (2. Non-Chaotic Attractor Examples: →for a damped undrivenpendulum, attractor is just a point at θ=ω =0 (0D in 2D phase space). “Safer at Home. The idea is that the influence of phase space structures on trajectories will result in differences in arclength of nearby trajectories near a phase space structure. Show that the geometric phase doesn’t depend on time, but only on the path in parameter space, hence the name "geometric". The three panels animate synchronously: (1) the motion of the particle in the potential; (2) the phase space trajectory; and (3) the time series of. 3 Introduction This course principally deals with the variational principles of mechanics, particularly the Lagrangian and Hamiltonian descriptions of dynamical systems. This 2-dimensional space is called phase space. 119, 7017 (2003). Some fundamental issues regarding time translation and phase. Since 2017, when SecureLink first partnered with Vista, the company has nearly tripled its revenue, earning it a place on the Inc. Show that the geometric phase doesn’t depend on time, but only on the path in parameter space, hence the name "geometric". the value of n. So it comes from thinking of this in phase space. In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism. These theoretical results suggest that the initial rotational excitation significantly enhances reactivity with an amount comparable to the effect of initial. In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. Machine Intelligence and Data Science Strategist at Booz Allen Hamilton. de GSI Accelerator Seminar Darmstadt, 03 July 2003 Extended phase space – p. This is a three-dimensional phase space because there are three ﬁrst order ODEs involving three dependent variables (x, y and z) governing. [Hint: Perhaps you should check that these solutions are consistent with those in part (a). nonlinearity is crucial because the system has many energy levels, but the operating space of the qubit must. The analysis of phase-space trajectories is central to. Metric and curvature in gravitational phase space 3 structure. Tucci announced during the Tuesday, Aug. phase space argument page 4 of 5 4 ()3/2 3 N 2 N N B VmE h Ω= Thus, as advertised above, the volume of a shell of constant energy in phase space is virtually the same as the volume of the entire space for EE< 0. Just like a direction field, a phase portrait can be a tool to predict the behaviors of a system’s solutions. “Safer at Home. The recently introduced statistical theory of non-Hamiltonian systems was applied to develop a procedure for constructing an appropriate generalization of Hamiltonian phase space analysis. Liouville's theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i. Evolution of Phase Space Probabilities¶. Suppose further that the coordinates are all angles. Hamiltonian Partial Differential Equations Many of the principal equations of mathematical physics can be formulated as dynamical systems in an infinite dimensional phase space, very often in the form of a Hamiltonian system with infinity mant degrees of freedom. The Hamiltonian operator is said to be a “generator” of the R symmetry. In c, we present the probability distributions at times t = 0, t = 1 and t = 6 in the first configuration, i. attempts allowed: 1. This 6N-dimensional space is called the "phase space" of the system, and the Hamiltonian defines a 'trajectory' that a system takes through this phase space over time. We discuss the case in which there is no such a measure to be normalized in Section II. This is very useful if one considers a production of two particles, each of which subsequently decays into two-body state. Momentum and coordinate graph. 1) does not explicitly depend on S, and therefore the equation (1. 01-3362 GRC. calculated along the arc of trajectory Γ joining the initial point to the final point in spacetime; is the momentum vector and is the Hamiltonian function. Show that a system x0= F(x) is at the same time a Hamiltonian system and a gradient system i the Hamiltonian His a harmonic function. Points within the complex correspond to the neutral atom; if a trajectory maps outside the complex, it will subsequently. "The goal is to follow this cohort for many decades to characterize the longitudinal trajectory of toxicities related to platinum-based chemotherapy," she said. Tempe, AZ 85284. Hamilton’s principal function and extended phase space. First, the basic principle of phase space and its parameter selection scheme are analyzed. Section 3 shows the derivation of the trajectory divergence rate and discusses its properties. Some fundamental issues regarding time translation and phase. The energy conservation condition restricts the motion on a (6N-1)-dimensional surface in phase-space, known as the constant-energy hypersurface or simply the constant-energy surface, see Fig. Rescale: Due to the change in resolution, the coarse-grained “picture. How the Hamilton-Jacobi equa-tion foreshadows quantum mechanics. The symplectic invariancerequirements areactuallymore stringent than just the phase space volume conservation, as we. A tangle, as in Fig. The first trajectory operations took place successfully on August 17, which is among a series of trajectory control maneuvers. The three panels animate synchronously: (1) the motion of the particle in the potential; (2) the phase space trajectory; and (3) the time series of. Hamilton’s equations involve partial. This plot corresponds to setting4 =p = 0 in the VR Hamiltonian (6), and plotting k versus x. Also includes a 10x 10 overhead door. Starting from a suitable zeroth-order Hamiltonian, for which the Wigner density is available (either analytically or numerically), the phase space distribution is propagated in time via classical trajectories, while the perturbation is gradually switched on. INTEGRAL INVARIANTS, LIOUVILLE'S THEOREM at Hl(Qk¤Pk¤t)=H(qk¤pk»I)+ 49G For all forms the new Hamiltonian is gpk QP]: G (p,P,r= Q =·——; Q = 4 ) k k (306) GG QG 8Qk gpk G (Q,DJ)¢ P = ——— ; q = 3 k. We also brieﬂy discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase-space Schr¨odinger equation. In particular, if the Hamiltonian preserves energy, then one trajectory cannot visit two parts of phase space with different energy. For low dimensional systems (2-or 3-D), it is easy to visualize the limit sets in the phase space: a fixed. formed Hamiltonian the Hamilton equations of motion for X and , and give the solutions for X(t) and (t) in terms of the initial values X0 and 0. The mission management system may include a flight management unit to define and manage a flight plan and associated flight trajectory, an airport navigation unit to define and manage a ground plan and associated ground trajectory, and a mission management unit to manage a continuity between the ground and flight phases of the mission and. See full list on scholarpedia. Show that a system x0= F(x) is at the same time a Hamiltonian system and a gradient system i the Hamiltonian His a harmonic function. These can in fact be interpreted as a single system in a -dimensional phase space, in which each phase vector is. Hamilton's Equations (N Dimensions) Generalize the above ideas to dimensions N > 1: dq i /dt = dH/dp i (q is a generalized coordinate) dp i /dt = -dH/dq i (p is a generalized momentum = mv) p and q constitute the phase space for the dynamics; N-dimensional dynamics have a 2N-dimensional phase space; p i and q i (for i = 1 to N) are the phase. Answered: Sundas Mustafa on 13 Mar 2015. 2 billion in prime contracts, both described how. An equation of the form (4) is called a Hamiltonian system. Interestingly, that the time steps satisfying the stationary condition turn out to be independent of the symplectic algorithms chosen. While only approximate, this is a powerful result because it provides a general (linear) solution to the dynamical equations in the vicinity of any nominal trajectory. The next phase would push ahead to in-space manufacturing and the mining of valuable metals. To state this a little more formally, we take a 2-dimensional cross section of the phase space: Γ, and define the Poincaré map: ρ:Γ → Γ such that ρ(x) = Φ τ (x), where Φ is the trajectory and is the first time point when the trajectory passes through Γ again. This follows from the Liouville theorem. Metric and curvature in gravitational phase space 3 structure. Kris Kolluri, CEO of Cooper’s Ferry Partnership, a proponent of various development projects in the city, said, “The trajectory of Camden’s renaissance continues to be strong. If each point in the phase space is considered as a random quantity with a particular probability ascribed to every possible state (i. I swept it under the carpet and then ignored it. When the pendulum moves, it travels along a path in the four dimensional phase space made up of Q1, Q2, L1 and L2: the phase space trajectory. Mask mandates and other prevention methods remain in effect and are even more important to contain the virus, Cooper said. This model simulates a simple technique to implement space vector modulation. See full list on scholarpedia. The analysis of phase-space trajectories is central to. "For the first time, we will evaluate the impact and severity of the hearing loss and tinnitus on the patients' physical, emotional and social functioning. 7700 South River Parkway. 3) Classically, such a conﬁned free particle traces a rectangular trajectory in phase space, of phase area n=2p n·a = nh as required by the “Planck quantization condition. The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics and may become advantageous for quantum technology. Note that the Hamiltonian H in (1. It’s a move members of. Physics 6010, Fall 2010 Introduction. Points within the complex correspond to the neutral atom; if a trajectory maps outside the complex, it will subsequently. In particular, if the Hamiltonian is returned to its original form the system will return to its original state, apart from a phase factor. Coca-Cola Amatil Limited (OTCPK:CCLAF) Q2 2020 Earnings Conference Call August 19, 2020 8:00 PM ET Company Participants Ana Metelo - Group Head, Investor Relations Alison Watkins - Group Managing. 394 Chapter 14 Hamilton’s equations and phase space A typical example is the predator-prey system of equations x˙1 = ax1 −bx1x2, x˙2 = bx1x2 −cx2, which govern the population density x1(t) of the prey and the population density x2(t) of. We demonstrate that this method has sufficient accuracy to reproduce phase space structures such as turnstiles for a 1D model of the isomerization of ketene in an external field. The Sun-Saturn-Cassini, or phase, angle at the time was 72 degrees; hence, the partial illumination of Saturn in this portrait. This model simulates a simple technique to implement space vector modulation. The resulting equation can then be plotted in momentum-space to provide a visual representation of the system’s behavior. In particular, if the Hamiltonian preserves energy, then one trajectory cannot visit two parts of phase space with different energy. points in phase space shown in Figure 2(a). c x z + ω sz = 0 (1) with κ = 2r0 Nc λ γ3 and defining the linear frequencies { ωx = ω0Qx (1− ξz/βcη) ωs = ω0Qs (2) where C = 2πR is the ring circumference. , along the horizontal axis and on the vertical axis). The area of the ellipse is equal to the product of the energy and the cycle duration of the oscillator, so that in case of energy loss because of damping the ellipse converts to a logarithmic spiral. individual trajectories and Hamilton’s equations. The non-archetypal emergence trajectories remained fixed in a region of phase space characterized by a relatively high conductivity and low connection strength throughout emergence. This is a three-dimensional phase space because there are three ﬁrst order ODEs involving three dependent variables (x, y and z) governing. Phase space is a mathematical concept whose graph is the plot of the momentum of a particle versus the position of particle in the physical space. Equations will not display properly in Safari-please use another browser. The phase space of the (n-dimensional) harmonic oscillator, a. Let us present the following time dependent Hamiltonian  (1) The propagator corresponds to this system can be written in the phase space as (2) This propagator is not exactly evaluated for any arbitrary time dependent mass or frequency, because that will lead to non-linear differential equations. 1 From Lagrange to Hamilton. As we saw in Chapter 2, the Lagrangian formulation of the. This is known as the. Polking of Rice University. 1 Phase space Phase space is the abstract space of which Ndimensions correspond to the Ngeneralized coordinates and another Ntheir corresponding conjugate momenta. We also brieﬂy discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase-space Schr¨odinger equation. Phase Space. An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian—there are no other functionally independent conserved quantities on the phase space. When the pendulum moves, it travels along a path in the four dimensional phase space made up of Q1, Q2, L1 and L2: the phase space trajectory. Hamiltonian to the Lagrangian. Do not just write it down. The separatrix, plotted in red, is given by , where is the Hamiltonian of the system. It suffices to exhibit a trial loop with energy larger than the linear phonon energy at the same pseudoaction to prove the existence of a Discrete Breather with a frequency above the linear. For a much more sophisticated phase plane plotter, see the MATLAB plotter written by John C. Binary outputs, simple calculation, high precision and fast response time were the main advantages of the phase space trajectory autocorrelation. This gives us a point in the $${x_1}\,{x_2}$$ or phase plane that we can plot. The Moon is both seductively close to Earth and cosmically far away: Decades after the end of the space race, it remains extraordinarily expensive and difficult to actually get there. c) Sketch a 2-d phase space plot that shows a trajectory for the system, appropriate for the Hamiltonian variables. Poincare, Lagrange, Hamiltonian, and Jacobi mechanics (Unit 1 Ch. 5 starting this Friday, September 4th at 5pm. The phase φ of the oscillator The phase angle φ can be determined from eq. A PCM heat sink can help to reduce the overall mass and. (NASDAQ:IMMU) Q2 2020 Results Conference Call August 05, 2020 05:00 PM ET Company Participants Chau Cheng - Senior Director-Investor Relations Dr. ’Phase space’ is a special but important case, a state space with symplectic structure, preserved by the ﬂow. of a two degree-of-freedom Hamiltonian system. The second part of the paper, starting with section 5, deals with the classical limit. The phase space is therefore 2n-dimensional. in the phase space formulation of quantum mechanics, one can propose two more ways for generating trajectory-based dynamics which conserves the canonical distribution in the quantum phase space [i. In this case, the trajectory is an ellipse, as shown in the Figure below. The software that Hamilton and her team developed for Apollo 11 was used in all subsequent Apollo missions, and it was later adapted for NASA's space shuttle and Skylab missions. One of the best known examples is the Berry phase [M. The asymptotic trajectory indicator can be applied to higher dimensional systems coupled to Langevin baths as we demonstrate for a 3D model of the isomerization of ketene. Hamiltonian mechanics was originally founded on variational principles and to some concrete degree the phase space of a system with n degrees of freedom can be modeled as either R 2n or a subset of it; but the modern theory manages to start from symplectic structure on manifolds, develop vector fields on cotangent bundles and have Hamiltonian. Hamilton's Equations (N Dimensions) Generalize the above ideas to dimensions N > 1: dq i /dt = dH/dp i (q is a generalized coordinate) dp i /dt = -dH/dq i (p is a generalized momentum = mv) p and q constitute the phase space for the dynamics; N-dimensional dynamics have a 2N-dimensional phase space; p i and q i (for i = 1 to N) are the phase. The systems are a massive particle in a one. 12, Unit 2 Ch. A system evolves merely by following the direction indicated by the arrows. Phase portrait A plot of two or more dynamical variables against each other. Even though the example is in. The appendix summarises some useful properties of canonical transforma- tions and Lagrange brackets in phase space, and the general mathematical techniques. NASA Technical Reports Server (NTRS) Chronis, Themis G. Some Over-the-Rhine residents are pushing back on the design of FC Cincinnati's future West End Stadium, namely its light-up exterior… Construction of FC Cincinnati’s \$250 million soccer. The ruling comes three months after UC Regents voted to phase out the use of those scores by 2025. Proposition 1. Thus, the time evolution or trajectory of a system as specified by Hamilton's equations of motion, can be expressed by giving the phase space vector, $$x$$ as a function of time. The area of the ellipse is equal to the product of the energy and the cycle duration of the oscillator, so that in case of energy loss because of damping the ellipse converts to a logarithmic spiral. Phase III: In-person learning Monday to Thursday and virtual classes on Fridays Phase IV: In-person school five days a week Each phase will be no less than four weeks, and families will have two. Throughout ChaosBook we reserve the term ‘phase space’ to Hamiltonian ﬂows. the Bloch Hamiltonian only "knows" about the 'hopping matrix elements" between orbitals, but not how the orbitals are embedded in space the Berry curvature in 1<-space of {at. In an n dimensional phase space, the generator of the time translation can be written in terms of a Hamiltonian and a set of Poisson brackets for the phase space variables. Phase space is a mathematical concept whose graph is the plot of the momentum of a particle versus the position of particle in the physical space. In two-dimensional phase space, The evolution of a volume element dV = dp x dx in phase space is given by Volume in phase space is conserved under Hamiltonian flow, a property known today as Liouville’s theorem. Section 5 contains concluding remarks. de GSI Accelerator Seminar Darmstadt, 03 July 2003 Extended phase space – p. All orbits in phase space (i. Show the direction of travel around the trajectory. Such a set of phase points is called a phase space ensemble. Previous Year Questions (in Hindi) 8m 21s. For two DOF Hamiltonian systems, the stable and unstable manifolds of the orbit discussed provide. Model potentials studied consist of a single bistable mode plus transverse harmonic modes. Then, at each grid point x = (α, β), we can calculate the solution trajectory’s instantaneous direction of motion at that point by using the given. For this procedure to be valid, it has to be shown that extremizing S 3 with respect to all possible phase space trajectories {p i(t),xi(t)} is equivalent to extremizing S 2 with respect to i{x(τ),t(τ)} for τ being an aﬃne parameter. c) Sketch a 2-d phase space plot that shows a trajectory for the system, appropriate for the Hamiltonian variables. istence of a well-defined function H, on the reduced phase space P,, which we call the reduced Hamiltonian associated with H and #. Learn more from Team Tami. HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS 227 p 0 + ∆p 0 z 0 + ∆z 0 p 0 Area at t = 0 Area at t = t 1 ∆p. • Rotations in space. fr/APC_CS/fr/calendar-node-field-start-date/month/2020-09 fr. Note that the Hamiltonian H in (1. POSTULATES 1. Show that the geometric phase doesn’t depend on time, but only on the path in parameter space, hence the name "geometric". Jarzynski theorem: steered dynamics. The phase space can be used to diagnose the evolution of cyclones that undergo single phase (pure tropical or extratropical cyclone) or multiple phases of existence (extratropical transition, tropical transition). at 1300 North Thornton Ave. Doing this for many values of $$t$$ will then give us a sketch of what the solution will be doing in the phase plane. A one-dimensional mass-spring system traces out an ellipse in phase-space. Corresponding to any region of phase space, B o, there exists a corresponding region in which the. This 6N-dimensional space is called the "phase space" of the system, and the Hamiltonian defines a 'trajectory' that a system takes through this phase space over time. fr/APC_CS/en/calendar-node-field-start-date/month/2020-09 en. A quick guide to sketching phase planes Section 6. By manifold we mean a smooth di erentiable d-dimensional space which looks like Rd only locally. To do so, we draw a grid on the phase plane. Velocity Phase Space. (i) We have dH dt = r pH dp dt + r q dq dt = r pHr q+ r qHr p= 0 (10) (ii) This is obtained very similarly. 12, Unit 2 Ch. The phase space is the space of the dependent variables (X), it does not include t, while the state space incorporates t. The phase space of the (n-dimensional) harmonic oscillator, a. If interactions between these particles can be ignored, we can consider the N particles as being within some volume in the 6-dimensional phase space (q i,p i), i =1,2,3, and Liouville’stheoremfor thelatter phase space impliesthat the 6-. In phase space, η represents the position or phase point, and the flow is expressed through Hamilton’s equations. Recall the basic setup for an autonomous system of two DEs: dx dt = f(x,y) dy dt = g(x,y). same throughout the entire trajectory. The two most important features regarding the experimental detection of a quantum global phase are [i] that it is a statistical object, and [ii] it can be measured only relatively. o, and δx is the computed linear deviation from the nominal trajectory. Wyatt, Evolution of classical and quantum phase space distributions: A new trajectory approach for phase-space dynamics, J. This model simulates a simple technique to implement space vector modulation. de GSI Accelerator Seminar Darmstadt, 03 July 2003 Extended phase space – p. is invariant under a canonical transformation. Arsenal manager Joe Montemurro says his side can do ‘even better’ following the opening Women’s Super League 6-1 victory over Reading on Sunday afternoon. 2 Isotropic oscillator & Kepler problem in phase space formalism andthat A 1,A 2 =2ωA 3 A 2,A 3 =2ωA 1 A 3,A 1 =2ωA 2 which is to say: L ≡ 1 2ω A possesses Poisson bracket properties that mimic. 5(17), where k B is the Boltzmann constant, indicate the locations of PM-AFM and PM-SG thermal phase transitions, respectively, at G =0. POSTULATES 1. This Hamiltonian deﬁnes a. On Tuesday, Governor Roy Cooper announced that North Carolina will take a modest step forward move into Phase 2. How to add trajectory into a phase plane? Follow 98 views (last 30 days) Alex on 24 Feb 2014. Hamilton’s equations on a symplectic manifold. For a one-dimensional oscillator (with a two-dimensional phase space), the two LCEs therefore must be , so the motion is quasiperiodic and cannot be chaotic. Such a set of phase points is called a phase space ensemble. If not specified, it will be set to step_size x num_steps. The rst is naturally associated with con guration space, extended by time, while the latter is the natural description for working in phase space. The result is a renormalisation of the Hamiltonian βH which leads to an eﬀective Hamiltonian expressed in terms of a ‘coarse-grained’ magnetisation ﬁeld m¯(x)= 1 (ba)d! Cell dym(y), where the integral runs over a cell of size (ba)d centred on x. , along the horizontal axis and on the vertical axis). 5 starting this Friday, September 4th at 5pm. Details of the calculation:. phase space, ensembles and partition functions. Finally, in Section 15. hamiltonian dynamics (level 1) Jump to News forum EXAM SCHEME repository GRE PHYSICS CSIR NET PHYSICAL SCIENCE SYLLABUS TRAJECTORY MEDIA PLAYER CSIR NET VIDEO LECTURES - MODULAR 1-VECTOR CALCULUS-LEVEL-1 2-VECTOR CALCULUS-LEVEL-2 3-DIFFERENTIAL EQUATIONS-LEVEL-1 4-DIFFERENTIAL EQUATIONS-LEVEL-2 5-LINEAR ALGEBRA-LEVEL-1 6-LINEAR ALGEBRA-LEVEL. We refer to the one in the spirit of the phase space continuity equation as “equi-librium continuity dynamics” (ECD), and the other.