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LAGRANGIAN MECHANICS PRINCIPLES

Author Horia Orasanu

ABSTRACT

There are many instances in which the basic physics is known (or postulated), and the behavior of a complex system is to be determined. A typical example is that in which there are too many particles for the problem to be tractable in terms of single-particle equations, and too few for a statistical analysis to apply. In such situations, use of a computer may furnish information on enough specific cases for the general behavior of the system to be discernable. If the basis physics is postulated but not known, a computer simulation can relate the theory to observations on complex systems and thus test the theory theoretical and experimental physics have been joined over the last three decades by that of computational physics

. INTRODUCTION

A wide range of numerical techniques are available, from simple searches to sophisticated methods, such as annealing, an algorithm for finding global minima which was inspired by an actual physical process. Some of the techniques are repetitive applications of deterministic equations. Others invoke stochastic processes (using “random” numbers), to focus on the important features.The research of Professor Gatland involves data analysis and the mathematical modeling and simulation of microscopic physical processes. These activities encompass both research and instructionIn this lecture we recall the definitions of autonomous and non autonomous Dynamical Systems as well as their different concepts of attractors. After that we introduce the different notions of robustness of attractors under perturbation (Upper semicontinuity, Lower semicontinuityIn this section, we derive the kinematic model along with the dynamic equations of motion of the snake robot in a Lagrangian framework. Moreover, we use partial feedback linearization to write the model in a simpler form for model-based control design.

In order to perform control design, we need to write the governing equations of the system in an implementable way. This is often done by choosing a local coordinate chart and writing the system equations with respect to (w.r.t.) these coordinates. According to the illustration of the snake robot in Figure Figure1,1, we choose the vector of the generalized coordinates of the N-link snake robot as x = [q1,q2,…,qN−1,θN,px,py]T ∈ ℝN+2, where qi with i∈{1,…,N−1} denotes the ith joint angle, θN denotes the head angle, and the pair (px,py) describes the position of the CM of the robot w.r.t. the global x−y axes. Since the robot is not subject to nonholonomic velocity constraints, the vector of the generalized velocities is defined as x˙=[q˙1,q˙2,…,q˙N−1,θ˙N,p˙x,p˙y]T∈RN+2. Using these coordinates, it is possible to specify the kinematic map of the robot. In this paper, we denote the first N elements of the vector x, i.e. (q1,…,qN−1,θN), as the angular coordinates, and the corresponding dynamics as the angular dynamics of the system.

2. FORMULATION

Let us assume that the motion of complex system entities takes place on fractal curves (continuous, but non-differentiable). A manifold that is compatible with such movement defines a fractal space. The fractal nature of space generates the breaking of differential time reflection invariance. In such a context, the usual definitions of the derivative of a given function with respect to time [6,7],

dFdt=limΔt→0+F(t+Δt)−F(t)Δt=limΔt→0−F(t)−F(t−Δt)Δt

(1)

are equivalent in the differentiable case. The passage from one to the other is performed via Δt → − Δt transformation (time reflection invariance at the infinitesimal level). In the non-differentiable case, (dQ+dt) and (dQ−dt) are defined as explicit functions of t and dt,

dQdt+limΔt→0+Q(t,t+Δt)−Q(t,Δt)Δt

and:

dQdt=limΔt→0−Q(t,Δt)−Q(t,t−Δt)Δt

(2)

The sign (+) corresponds to the forward process, while (−) corresponds to the backward process. Then, in space coordinates dX, we can write [6,7]:

dX±=dx±+dξ±=v±dt+dξ±

(3)

with v± the forward and backward mean speeds,

v+=dx+dt=limΔt→0+⟨X(t+Δt)−X(t)Δt⟩v−=dx−dt=limΔt→0−⟨X(t)+X(t−Δt)Δt⟩

(4)

and dξ± a measure of non-differentiability (a fluctuation induced by the fractal properties of trajectory) having the average:

⟨dξ±⟩=0,

(5)

where the symbol 〈〉 defines the mean value.

While the speed-concept is classically a single concept, if space is a fractal, then we must introduce two speeds (v+ and v−), instead of one. These “two-values” of the speed vector represent a specific consequence of non-differentiability that has no standard counterpart (according to differential physics).

However, we cannot favor v+ as compared to v−. The only solution is to consider both the forward (dt > 0) and backward (dt < 0) processes. Then, it is necessary to introduce the complex speed [6,7]:

Vˆ=v++v−2−iv+−v−2=dx++dx−2dt−idx+−dx−2dt=VD−iVF,VD=v++v−2,VF=v+−v−2

(6)

If VD is differentiable and resolution scale (dt) speed independent, then VF is non-differentiable and resolution scale (dt) speed dependent.

Using the notations dx± = d±x, Equation (6) becomes:

Vˆ=(d++d−2dt−id+−d−2dt)x

(7)

This enables us to define the operator:

dˆdt=d+−d−2dt−id+−d−2dt

(8)

Let us now assume that the fractal curve is immersed in a three-dimensional space and that X of components Xi (i = 1, 2, 3) is the position vector of a point on the curve. Let us also consider a function f(X, t) and the following series expansion up to the second order:

df=f(Xi+dXi,t+dt)−f(Xi,dt)=(∂∂XidXi+∂∂tdt)f(Xi,t)+12(∂∂XidXi+∂∂tdt)2f(Xi,t)

(9)

Using notations, dXi±=d±Xi

, the forward and backward average values of this relation take the form:

⟨d±f⟩=⟨∂f∂tdt⟩+⟨∇f⋅d±X⟩+12⟨∂2f∂t2(dt)2⟩++⟨∂2f∂Xi∂td±Xidt⟩+12⟨∂2f∂Xi∂Xld±Xid±Xl⟩

(10)

We shall stipulate the following: the mean values of function f and its derivatives coincide with themselves, and the differentials d±Xi and dt are independent. Therefore, the averages of their products coincide with the product of averages. Thus, Equation (10) becomes:

d±f=∂f∂tdt+∇f⟨d±X⟩+12∂2f∂t2⟨(dt)2⟩++∂2f∂Xi∂t⟨d±Xidt⟩+12∂2f∂Xi∂Xl⟨d±Xid±Xl⟩

(11)

or more, using Equation (3),

d±f=∂f∂tdt+∇fd±x+12∂2f∂t2(dt)2+∂2f∂Xi∂td±xidt++12∂2f∂Xi∂Xl(d±xid±xl+⟨dξi±dξl±⟩),i,l=1,We note that in Nottale’s works [6,7], the fractal operator (19) for DF = 2 plays the role of the “covariant derivative operator”. We shall call the operator (19) the “generalized covariant derivative operator”.

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here we consider as say prof dr mircea orasanu and prof horia orasanu as followings

VARIATIONS PRINCIPLES IN NONHOLONOMIC PROBLEM

Author Horia Orasanu

ABSTRACT

Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind,[1] which treat constraints explicitly as extra equations, often using Lagrange multipliers;[2][3] or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.[1][4] In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.\1 1 1 INTRODUCTION

No new physics is introduced in Lagrangian mechanics compared to Newtonian mechanics. Newton’s laws can include non-conservative forces like friction; however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system.

o new physics is introduced in Lagrangian mechanics compared to Newtonian mechanics. Newton’s laws can include non-conservative forces like friction; however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system

Complex systems are large interdisciplinary research topics that have been studied by means of a mixed basic theory that mainly derives from physics and computer simulation. Such systems are made of many interacting elementary units that are called “agents”.

The way in which such a system manifests itself cannot be exclusively predicted only by the behavior of individual elements. Its manifestation is also induced by the manner in which the elements relate in order to influence global behavior. The most significant properties of complex systems are emergence, self-organization, adaptability, etc. [1–4].

Examples of complex systems can be found in human societies, brains, the Internet, ecosystems, biological evolution, stock markets, economies and many others [1, 2]. Particularly, polymers are examples of such complex systems. Their forms include a multitude of organizations starting from simple, linear chains of identical structural units and ending with very complex chains consisting of sequences of amino acids that form the building blocks of living fields. One of the most intriguing polymers in nature is DNA, which creates cells by means of a simple, but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems, such as organs, which, in turn, form even more complex systems, such as organisms. The study of complex systems can offer a glimpse into the realistic dynamics of polymers and solve certain difficult problems (protein folding) [1–4].

Correspondingly, theoretical models that describe the dynamics of complex systems are sophisticated [1–4]. However, the situation can be standardized taking into account that the complexity of interaction processes imposes various temporal resolution scales, while pattern evolution implies different freedom degrees [5].

In order to develop new theoretical models, we must admit that complex systems displaying chaotic behavior acquire self-similarity (space-time structures seem to appear) in association with strong fluctuations at all possible space-time scales [1–4]. Then, in the case of temporal scales that are large with respect to the inverse of the highest Lyapunov exponent, the deterministic trajectories are replaced by a collection of potential trajectories, while the concept of definite positions by that of probability density. One of the most interesting examples is the collision process in complex systems, a case in which the dynamics of the particles can be described by non-differentiable curves.

Since non-differentiability appears as the universal property of complex systems, it is necessary to construct a non-differentiable physics. Thus, the complexity of the interaction processes is replaced by non-differentiability; accordingly, it is no longer necessary to use the whole classical “arsenal” of quantities from standard physics (differentiable physics).

This topic was developed within scale relativity theory (SRT) [6,7] and non-standard scale relativity theory (NSSRT) [8–22]. In this case, we assume that the movements of complex system entities take place on continuous, but non-differentiable, curves (fractal curves), so that all physical phenomena involved in the dynamics depend not only on space-time coordinates, but also on space-time scale resolution. From such a perspective, physical quantities describing the dynamics of complex systems may be considered fractal functions [6,7]. Moreover, the entities of the complex system may be reduced to and identified with their own trajectories, so that the complex system will behave as a special fluid lacking interaction (via their geodesics in a non-differentiable (fractal) space). We have called such fluid a “fractal fluid” [8–22].

In the present paper, we shall introduce new concepts, like non-differentiable entropy, informational non-differentiable entropy, informational non-differentiable energy, etc., in the NSSRT approach (the scale relativity theory with an arbitrary constant fractal dimension). Based on a fractal potential, which is the “source” of the non-differentiability of trajectories of the complex system entities, we establish the relationships among non-differentiable entropy. The correlation fractal potential-non-differentiable entropy implies uncertainty relations in the hydrodynamic representation, while the correlation of informational non-differentiable entropy/informational non-differentiable energy implies specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. The constant value of the informational non-differentiable energy made explicit for the harmonic oscillator induces a quantification condition. We note that there exists a large class of complex systems that take smooth trajectories. However, the analysis of the dynamics of these classes is reducible to the above-mentioned statements by neglecting their fractality.

2. Hallmarks of Non-Differentiability

Let us assume that the motion of complex system entities takes place on fractal curves (continuous, but non-differentiable). A manifold that is compatible with such movement defines a fractal space. The fractal nature of space generates the breaking of differential time reflection invariance. In such a context, the usual definitions of the derivative of a given function with respect to time [6,7],

dFdt=limΔt→0+F(t+Δt)−F(t)Δt=limΔt→0−F(t)−F(t−Δt)Δt

(1)

are equivalent in the differentiable case. The passage from one to the other is performed via Δt → − Δt transformation (time reflection invariance at the infinitesimal level). In the non-differentiable case, (dQ+dt) and (dQ−dt) are defined as explicit functions of t and dt,

dQdt+limΔt→0+Q(t,t+Δt)−Q(t,Δt)Δt

and:

dQdt=limΔt→0−Q(t,Δt)−Q(t,t−Δt)Δt

(2)

The sign (+) corresponds to the forward process, while (−) corresponds to the backward process. Then, in space coordinates dX, we can write [6,7]:

dX±=dx±+dξ±=v±dt+dξ±

(3)

with v± the forward and backward mean speeds,

v+=dx+dt=limΔt→0+⟨X(t+Δt)−X(t)Δt⟩v−=dx−dt=limΔt→0−⟨X(t)+X(t−Δt)Δt⟩

(4)

and dξ± a measure of non-differentiability (a fluctuation induced by the fractal properties of trajectory) having the average:

⟨dξ±⟩=0,

(5)

where the symbol 〈〉 defines the mean value.

While the speed-concept is classically a single concept, if space is a fractal, then we must introduce two speeds (v+ and v−), instead of one. These “two-values” of the speed vector represent a specific consequence of non-differentiability that has no standard counterpart (according to differential physics).

However, we cannot favor v+ as compared to v−. The only solution is to consider both the forward (dt > 0) and backward (dt < 0) processes. Then, it is necessary to introduce the complex speed [6,7]:

Vˆ=v++v−2−iv+−v−2=dx++dx−2dt−idx+−dx−2dt=VD−iVF,VD=v++v−2,VF=v+−v−2

(6)

If VD is differentiable and resolution scale (dt) speed independent, then VF is non-differentiable and resolution scale (dt) speed dependent.

Using the notations dx± = d±x, Equation (6) becomes:

Vˆ=(d++d−2dt−id+−d−2dt)x

References

1. Liljebäck P, Stavdahl Ø, Beitnes A (2006) SnakeFighter – development of a water hydraulic fire fighting snake robot. In: Proc. IEEE international conference on control, automation, robotics, and vision ICARCV, Singapore.

2. Wang Z, Appleton E (2003) The concept and research of a pipe crawling rescue robot. Adv Robot 17.4: 339–358.

3. Fjerdingen SA, Liljebäck P, Transeth AA (2009) A snake-like robot for internal inspection of complex pipe structures (PIKo). In: Proc. IEEE/RSJ international conference on intelligent robots and systems, St. Louis, MO, USA.

4. Dacic DB, Nesic D, Teel AR, Wang W. Path following for nonlinear systems with unstable zero dynamics: an averaging solution. IEEE Trans Automatic Control. 2011;56:880–886. doi: 10.1109/TAC.2011.2105130. [Cross Ref]

5. Hirose S. Biologically inspired robots: snake-like locomotors and manipulators. Oxford, England: Oxford University Press; 1993.

6. Matsuno F, Sato H (2005) Trajectory tracking control of snake robots based on dynamic model. In: Proc. IEEE international conference on robotics and automation, 3029–3034. 18-22 April 2005.

7. Date H, Hoshi Y, Sampei M (2000) Locomotion control of a snake-like robot based on dynamic manipulability. In: Proc. IEEE/RSJ international conference on intelligent robots and systems, Takamatsu, Japan.

8. Tanaka M, Matsuno F (2008) Control of 3-dimensional snake robots by using redundancy. In: Proc. IEEE international conference on robotics and automation, 1156–1161, Pasadena, CA.

9. Ma S, Ohmameuda Y, Inoue K, Li B (2003) Control of a 3-dimensional snake-like robot. In: Proc. IEEE international conference on robotics and automation, vol. 2, 2067–2072, Taipei, Taiwan.

10. Tanaka M, Matsuno F (2009) A study on sinus-lifting motion of a snake robot with switching constraints. In: Proc. IEEE international conference on robotics and automation, 2270–2275. 12-17 May 2009.

11. Prautsch P, Mita T, Iwasaki T (2000) Analysis and control of a gait of snake robot. Trans IEE J Ind Appl Soc 120-D: 372–381.

12. McIsaac K, Ostrowski J. Motion planning for anguilliform locomotion. IEEE Trans Robot Automation. 2003;19:637–652. doi: 10.1109/TRA.2003.814495. [Cross Ref]

13. Hicks G, Ito K. A method for determination of optimal gaits with application to a snake-like serial-link structure. IEEE Trans Automatic Control. 2005;50:1291–1306. doi: 10.1109/TAC.2005.854583. [Cross Ref]

14. Ma S, Ohmameuda Y, Inoue K (2004) Dynamic analysis of 3-dimensional snake robots. In: Proc. IEEE/RSJ international conference on intelligent robots and systems, 767–772. 28 Sept.-2 Oct. 2004.

15. Ma S. Analysis of creeping locomotion of a snake-like robot. Adv Robot. 2001;15(2):205–224. doi: 10.1163/15685530152116236. [Cross Ref]

16. Liljebäck P, Pettersen KY, Stavdahl Ø, Gravdahl JT (2013) Snake robots – modelling, mechatronics, and control. Advances in industrial control. Springer.

17. Liljebäck P, Haugstuen IU, Pettersen KY. Path following control of planar snake robots using a cascaded approach. IEEE Trans Control Syst Technol. 2012;20:111–126.

18. Rezapour E, Pettersen KY, Liljebäck P, Gravdahl JT (2013) Path following control of planar snake robots using virtual holonomic constraints. Paper presented at the IEEE international conference on robotics and biomimetics, Shenzhen, China. [PMC free article] [PubMed]

19. Liljebäck P, Pettersen KY, Stavdahl Ø, Gravdahl JT. Controllability and stability analysis of planar snake robot locomotion. IEEE Trans Automatic Control. 2013;56(6):1365–1380. doi: 10.1109/TAC.2010.2088830. [Cross Ref]

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21. Maggiore M, Consolini L. Virtual holonomic constraints for Euler-Lagrange systems. IEEE Trans on Automatic Control. 2013;58(4):1001–1008. doi: 10.1109/TAC.2012.2215538. [Cross Ref]

22. Consolini L, Maggiore M (2010) Control of a bicycle using virtual holonomic constraints. In: Proc. 49th IEEE conference on decision and control, Atlanta, Georgia, USA, December 15-17, 2010.

23. Shiriaev A, Perram JW, Canudas-de-Wit C. Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach. IEEE Trans Automatic Control. 2005;50(8):1164–1176. doi: 10.1109/TAC.2005.852568. [Cross Ref]

24. Freidovich L, Robertsson A, Shiriaev A, Johansson R. Periodic motions of the Pendubot via virtual holonomic constraints: theory and experiments. Automatica. 2008;44(3):785–791. doi: 10.1016/j.automatica.2007.07.011. [Cross Ref]

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26. Bullo F, Lewis A (2005) Geometric control of mechanical systems. Springer.

27. Fossen TI. Marine control systems: guidance, navigation and control of ships, rigs and underwater vehicles. Marine Cybernetics: Trondheim, Norway; 2002.

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here we sure mention some aspects as say prof dr mircea orasanu as prof horia orasanu as followings

DIFFERENTIAL GEOMETRY FOR NONHOLONOMIC CONSTRAINS

Author Horia Orasanu

keywords

differential forms, constraints,nonholonomic optimizations.

ABSTRACT

rovide an introduction to vector bundles (with application to tubular neighborhoods), principal bundles, connections, the general theory of geometric structures (G-structures) and their integrabiliy, mentioning examples such as Riemannian, complex or symplectic structures.

The course will have several parts:

1. One which concentrates on vector-bundles and connections (including parallel transport, curvature and the construction of the first Chern class). Here we will also discuss the tubular neighborhood theorem.

2. One which concentrates on principal bundles and connections, and where we explain that, for principal

-bundles, the resulting theory is equivalent to the one for vector bundles. This part will start with 1-2 lectures about the basic notions/facts from Lie groups that are needed here.

3. While along the way we will mention some examples of geometric structures (such as metrics), in the last parts of the course we will concentrate on a general framework that allows one to treat many geometric structures in an unified way: the framework provided by the theory of

-structures. Here we will present the framework and examples such as: Riemannian metrics, distributions, foliations, symplectic structures, almost complex and complex structures.

4. Finally, in the last two lectures, we will concentrate on the integrability of G-structures and the torsion of G-structures as obstruction to integrability. For instance, for symplectic structures: one talks about almost symplectic structures (non-degenerate two forms) and their integrability is about the form being closed (Darboux theorem); for foliations one talks about sub-bundle of tangent bundle, and their integrability is equivalent to the involutivity (Frobenius theorem); etc etc.

1. INTRODUCTION

Terminology

We’ve got one more section that we need to take care of before we actually start solving partial differential equations. This will be a fairly short section that will cover some of the basic terminology that we’ll need in the next section as we introduce the method of separation of variables.

Let’s start off with the idea of an operator. An operator is really just a function that takes a function as an argument instead of numbers as we’re used to dealing with in functions. You already know of a couple of operators even if you didn’t know that they were operators. Here are some examples of operators.

However, the performance of this technique greatly relies on the setting of penalty factors, which are usually determined by manual trial and error, and the suitable penalty factors are often problem-dependent and difficult to set. In this paper, a differential evolution approach based on a co-evolution mechanism, named CDE, is proposed to solve the constrained problems. First, a special penalty function is designed to handle the constraints. Second, a co-evolution model is presented and differential evolution (DE) is employed to perform evolutionary search in spaces of both solutions and penalty factors. Thus, the solutions and penalty factors evolve interactively and self-adaptively, and both the satisfactory solutions and suitable penalty factors can be obtained simultaneously

An operator is really just a function that takes a function as an argument instead of numbers as we’re used to dealing with in functions. You already know of a couple of operators even if you didn’t know that they were operators. Here are some examples of operators.

Or, if we plug in a function, say , into each of these we get,

These are all fairly simple examples of operators but the derivative and integral are operators. A more complicated operator would be the heat operator. We get the heat operator from a slight rewrite of the heat equation without sources. The heat operator is,

Now, what we really want to define here is not an operator but instead a linear operator. A linear operator is any operator that satisfies,

The heat operator is an example of a linear operator and this is easy enough to show using the basic properties of the partial derivative so let’s do that.

You might want to verify for yourself that the derivative and integral operators we gave above are also linear operators. In fact, in the process of showing that the heat operator is a linear operator we actually showed as well that the first order and second order partial derivative operators are also linear.

The next term we need to define is a linear equation. A linear equation is an equation in the form,

(1)

where L is a linear operator and f is a known function.

Here are some examples of linear partial differential equations.

The first two from this list are of course the heat equation and the wave equation. The third uses the Laplacian and is usually called Laplace’s Equation. We’ll actually be solving the 2-D version of Laplace’s Equation in a few sections. The fourth equation was just made up to give a more complicated example.

Notice as well with the heat equation and the fourth example above that the presence of the and do not prevent these from being linear equations. The main issue that allows these to be linear equations is the fact that the operator in each is linear.

Now just to be complete here are a couple of examples of nonlinear partial differential equations.

We’ll leave it to you to verify that the operators in each of these are not linear however the problem term in the first is the while in the second the product of the two derivatives is the problem term.

Now, if we go back to (1) and suppose that then we arrive at,

(2)

We call this a linear homogeneous equation (recall that L is a linear operator).

Notice that will always be a solution to a linear homogeneous equation (go back to what it means to be linear and use with any two solutions and this is easy to verify). We call the trivial solution. In fact this is also a really nice way of determining if an equation is homogeneous. If L is a linear operator and we plug in into the equation and we get then we will know that the operator is homogeneous.

We can also extend the ideas of linearity and homogeneous to boundary conditions. If we go back to the various boundary conditions we discussed for the heat equation for example we can also classify them as linear and/or homogeneous.

The prescribed temperature boundary conditions,

are linear and will only be homogenous if and .

The prescribed heat flux boundary conditions,

are linear and will again only be homogeneous if and .

Next, the boundary conditions from Newton’s law of cooling,

are again linear and will only be homogenous if and .

The final set of boundary conditions that we looked at were the periodic boundary conditions,

These are all fairly simple examples of operators but the derivative and integral are operators. A more complicated operator would be the heat operator. We get the heat operator from a slight rewrite of the heat equation without sources. The heat operator is,

ese are all fairly simple examples of operators but the derivative and integral are operators. A more complicated operator would be the heat operator. We get the heat operator from a slight rewrite of the heat equation without sources. The heat operator is,

Now, what we really want to define here is not an operator but instead a linear operator. A linear operator is any operator that satisfies,

and these are both linear and homogeneous.

The final topic in this section is not really terminology but is a restatement of a fact that we’ve seen several times in these notes already.

Principle of Superposition

If and are solutions to a linear homogeneous equation then so is for any values of and .

Now, as stated earlier we’ve seen this several times this semester but we didn’t really do much with it. However this is going to be a key idea when we actually get around to solving partial differential equations. Without this fact we would not be able to solve all but the most basic of partial differential equations.

Or, if we plug in a function, say , into each of these we get,

These are all fairly simple examples of operators but the derivative and integral are operators. A more complicated operator would be the heat operator. We get the heat operator from a slight rewrite of the heat equation without sources. The heat operator is,

Now, what we really want to define here is not an operator but instead a linear operator. A linear operator is any operator that satisfies,

The heat operator is an example of a linear operator and this is easy enough to show using the basic properties of the partial derivative so let’s do that.

You might want to verify for yourself that the derivative and integral operators we gave above are also linear operators. In fact, in the process of showing that the heat operator is a linear operator we actually showed as well that the first order and second order partial derivative operators are also linear.

The next term we need to define is a linear equation. A linear equation is an equation in the form,

(1)

where L is a linear operator and f is a known function.

Here are some examples of linear partial differential equations.

The first two from this list are of course the heat equation and the wave equation. The third uses the Laplacian and is usually called Laplace’s Equation. We’ll actually be solving the 2-D version of Laplace’s Equation in a few sections. The fourth equation was just made up to give a more complicated example.

Notice as well with the heat equation and the fourth example above that the presence of the and do not prevent these from being linear equations. The main issue that allows these to be linear equations is the fact that the operator in each is linear.

Now just to be complete here are a couple of examples of nonlinear partial differential equations.

We’ll leave it to you to verify that the operators in each of these are not linear however the problem term in the first is the while in the second the product of the two derivatives is the problem term.

Now, if we go back to (1) and suppose that then we arrive at,

(2)

We call this a linear homogeneous equation (recall that L is a linear operator).

Notice that will always be a solution to a linear homogeneous equation (go back to what it means to be linear and use with any two solutions and this is easy to verify). We call the trivial solution. In fact this is also a really nice way of determining if an equation is homogeneous. If L is a linear operator and we plug in into the equation and we get then we will know that the operator is homogeneous.

We can also extend the ideas of linearity and homogeneous to boundary conditions. If we go back to the various boundary conditions we discussed for the heat equation for example we can also classify them as linear and/or homogeneous.

The prescribed temperature boundary conditions,

are linear and will only be homogenous if and .

The prescribed heat flux boundary conditions,

are linear and will again only be homogeneous if and .

Next, the boundary conditions from Newton’s law of cooling,

are again linear and will only be homogenous if and .

The final set of boundary conditions that we looked at were the periodic boundary conditions,

Exercise Sheet 1 (Julius Ross), Examples Class 1 Notes. Last updated 31 May 2017.

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Exercise Sheet 3 (Julius Ross), Examples Class 3 Notes. Last updated 6 December 2017.

Exercise Sheet 4 (Julius Ross), Examples Class 4 Notes. Last updated 31 May 2017.

Exam from 2009 (Julius Ross), Notes on Revision Examples Class. Last updated 26 May 2017.

and these are both linear and homogeneous.

The final topic in this section is not really terminology but is a restatement of a fact that we’ve seen several times in these notes already.

Principle of Superposition

If and are solutions to a linear homogeneous equation then so is for any values of and .

Now, as stated earlier we’ve seen this several times this semester but we didn’t really do much with it. However this is going to be a key idea when we actually get around to solving partial differential equations. Without this fact we would not be able to solve all but the most basic of partial differential equations.

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here sure we see as say prof dr mircea orasanu and prof horia orasanu as followings

DIFFERENTIAL OPERATOR USING LAPLACE OPERATOR FOR CONSTRAINTS NONHOLONOMIC

Author Horia Orasanu

ABSTRACT

We interrupt our quest to find new recipes for solving differential equations for a moment – let us introduce “differential operators”. To know how to use them will become very handy as soon as you hit the homework assignments in the textbook: Quite some of the problems are written in term of these operators. As you will see soon, this is nothing but an alternative way of writing our differential equations.

1 INTRODUCTION

Before we get to them let us clarify the term “operator” in general:

An operator is a transformation that transforms a function into another function. Operators and corresponding techniques are called operational methods”

This sounds like it comes right out of law school … but if you think about it: it makes sense:

If we do mathematical operations on a function the result most likely will be a function again.

An operator is an abbreviation in order to describe a mathematical procedure.

So, if for example I say my procedure is:

Take the square of a function,

Then, differentiate it,

Then, take the absolute value of the result

And, finally, take the natural logarithm of this,

I cannot write this as one “formula” for an arbitrary function. But I can define an operator that exactly does follow this procedure step by step and simply call it, let’s say “O”, for example.

It usually looks like some capital letter is multiplied to an expression.

As I just said, “O” for example.

However, if this capital letter has been introduced as an operator instead of as a variable, it means you have to apply it to the expression it is “multiplied to”

Yes, and applying our procedure to an arbitrary function f is the written as .

Or, short: . And, since O is an operator, this is not “O multiplied with f” but “O applied to f”. The multiplication sign is used to indicate that the application of O to a function follows the same laws like multiplication:

Anyway, let’s apply our hypothetical operator O to a simple function :

Then,

– we get this just by following the procedure prescribed by the definition of O

So far the principle of operators. In the following we will focus on a very simple one:

The Differential Operator “D”:

Differential Operator:

Applying D to a function y(x) means nothing else but differentiating the function.

This easy! And applying D twice to a function means to differentiate the function twice, or, to get the second derivative of the function:

Consequently:

etc.

That’s right, stands for applying D twice

while , etc.

The differential operator opens us an alternative way to formulate Differential Equations:

D is used in the textbook to express D. E.’s:

e.g.

means: or

e.g.

means: or

etc.

At this state of the lecture the only ‘advantage’ of using the differential operator D is that it’s knowledge allows you to practice on a larger variety of problems from the textbook

(Oh my!). However, later on we will see other, more complicated mathematical operators, such as the(Nabla), or the (Laplace) operators.

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