MaplePrimes - Questions, Posts, help, musings, answers ... all things Maple and math

Ask a Question

Create a Post

Featured Post

572

Suggestions to improve...

MapleSim is a mature product. The rich component library leaves little room for improvement for wide range of applications. It is understandable that latest product releases focused on specialized toolboxes and performance improvements.

Here is what I think could be beneficial for many users, which is not related to performance, but can help improve the user experience:

Crossing connection lines: A view option to render a crossing connection line with an arc at the crossing point of two connection lines of the same type. Right-click on a connection line might be enough.

-> To avoid misinterpretations of routing in crowded model diagrams
Parameters: An option to populate or reset changes in the parameter pane to the parameter default settings view and vice versa

-> When testing or optimizing a model, sometimes changed parameters should become default values or be reset. Doing this manually is error prone and takes time.
Component flip: Selecting more than one component including connection lines and applying a flip to all of them

-> When building a model, it can happen that a nicely laid out arrangement of components needs to be mirrored or rotated in its entirety. Doing this component by component and connection by connection is a lot of work that can be saved by this option.
Initial conditions: An option in the view menu to highlight components where ICs have been changed from ignore to treat as guess or to strictly enforce

-> ICs are set for many purposes. In addition to defining ICs needed for simulations, this can include extracting equations, testing different model states, or temporarily "immobilizing" a model during assembly, to name a few. Undoing a tentative change can easily be forgotten. Combining existing models that work on their own into a new more complex model often results in an overly constrained model that either cannot be assembled or does not simulate.
In debugging a model, ICs of all components must be individually inspected. This takes time. A quick overview that shows components where ICs are not set to ignore would be very helpful in debugging models.
Undo Create subsystem: A reverse operation that inserts the contents of a subsystem into a parent subsystem.

-> With the evolution of a model it is sometimes useful to exclude or include existing components from or to a subsystem. For this purpose, an undo create subsystem operation should preserve existing connections. A time saver.
Subsystem ports: An option to align a subsystem port to the drawing grid to remove “micro” steps in connection lines

-> For perfectionists who do not have the time to learn (and remember) how to fine-tune at the component level
A history or log function of user actions changing a model, its parametrization or internal state.

-> Often a model does not simulate at all or as desired after modifications. Restoring to a configuration that worked by undo only goes back to the last simulation performed. In such a situation, only reloading the latest file version and redoing modifications may restore the desired model, parametrization, or state. This takes time and migth be unsuccessful. A record of user actions would be a great help.
History or log information in file format could also help MapleSim support to reproduce an error.

For me personally, reducing errors (4. > 7. > 2.) would improve the use experience much more than layouting aids (3. > 1., 5., 6.).

June 19

9 4

Read Full Post

Featured Post

ecterrab

11271

The Electromagnetic Field...

This is an interesting exercise, the computation of the Liénard–Wiechert potentials describing the classical electromagnetic field of a moving electric point charge, a problem of a 3^rd year undergrad course in Electrodynamics. The calculation is nontrivial and is performed below using the Physics package, following the presentation in [1] (Landau & Lifshitz "The classical theory of fields"). I have not seen this calculation performed on a computer algebra worksheet before. Thus, this also showcases the more advanced level of symbolic problems that can currently be tackled on a Maple worksheet. At the end, the corresponding document is linked and with it the computation below can be reproduced. There is also a link to a corresponding PDF file with all the sections open.

Moving charges:
The retarded and Liénard-Wiechert potentials, and the fields and

Freddy Baudine⁽¹⁾, Edgardo S. Cheb-Terrab⁽²⁾

(1) Retired, passionate about Mathematics and Physics

(2) Physics, Differential Equations and Mathematical Functions, Maplesoft

Generally speaking, determining the electric and magnetic fields of a distribution of charges involves determining the potentials and , followed by determining the fields and from

In turn, the formulation of the equations for and is simple: they follow from the 4D second pair of Maxwell equations, in tensor notation

where is the electromagnetic field tensor and is the 4D current. After imposing the Lorentz condition

, i.e.

we get

which in 3D form results in

"(∇)^2A-1/(c^2) (((&PartialD;)^2)/(&PartialD;t^2)( A))=-(4 Pi)/c j"

Laplacian(`ϕ`)-(diff(`ϕ`, t, t))/c^2 = -4*Pi*rho/c

where is the current and is the charge density.

Following the presentation shown in [1] (Landau and Lifshitz, "The classical theory of fields", sec. 62 and 63), below we solve these equations for and resulting in the so-called retarded potentials, then recompute these fields as produced by a charge moving along a given trajectory - the so-called Liénard-Wiechert potentials - finally computing an explicit form for the corresponding and .

While the computation of the generic retarded potentials is, in principle, simple, obtaining their form for a charge moving along a given trajectory , and from there the form of the fields and shown in Landau's book, involves nontrivial algebraic manipulations. The presentation below thus also shows a technique to map onto the computer the manipulations typically done with paper and pencil for these problems. To reproduce the contents below, the Maplesoft Physics Updates v.1252 or newer is required.

with(Physics); Setup(coordinates = Cartesian); with(Vectors)

(1)

The retarded potentials and

The equations which determine the scalar and vector potentials of an arbitrary electromagnetic field are input as

(2)

%Laplacian(varphi(X))-(diff(diff(varphi(X), t), t))/c^2 = -4*Pi*rho(X)

(3)

%Laplacian(A_(X))-(diff(diff(A_(X), t), t))/c^2 = -4*Pi*j_(X)

(4)

The solutions to these inhomogeneous equations are computed as the sum of the solutions for the equations without right-hand side plus a particular solution to the equation with right-hand side.

	Computing the solution to the equations for and

The Liénard-Wiechert potentials of a charge moving along

From (13), the potential at the point is determined by the charge , i.e. by the position of the charge e at the earlier time

The quantityis the 3D distance from the position of the charge at the time to the 3D point of observation. In the previous section, the charge was located at the origin and at rest, so , the radial coordinate. If the charge is moving, say on a path , we have

From (13)≡ `ϕ`(r, t) = de(t-r/c)/r and the definition of above, the potential of a moving charge can be written as

When the charge is at rest, in the Lorentz gauge we are working, the vector potential is . When the charge is moving, the form of can be found searching for a solution to "(∇)^2A-1/(c^2) (((&PartialD;)^2)/(&PartialD;t^2)( A))=-(4 Pi)/c j" that gives when . Following [1], this solution can be written as

"A( )^(alpha)=(e u( )^(alpha))/(`R__beta` u^(beta))"

where is the four velocity of the charge, .

Without showing the intermediate steps, [1] presents the three dimensional vectorial form of these potentials and as

`ϕ` = e/(R-`#mover(mi("v"),mo("→"))`/c.`#mover(mi("R"),mo("→"))`) , `#mover(mi("A"),mo("→"))` = e*`#mover(mi("v"),mo("→"))`/(c*(R-`#mover(mi("v"),mo("→"))`/c.`#mover(mi("R"),mo("→"))`))

	Computing the vectorial form of the Liénard-Wiechert potentials

The electric and magnetic fields and of a charge moving along

The electric and magnetic fields at a point are calculated from the potentials and through the formulas

where, for the case of a charge moving on a path , these potentials were calculated in the previous section as (24) and (18)

`ϕ`(x, y, z, t) = e/(LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)-`#mover(mi("R"),mo("→"))`.(`#mover(mi("v"),mo("→"))`/c))

`#mover(mi("A"),mo("→"))`(x, y, z, t) = e*`#mover(mi("v"),mo("→"))`/(c*(LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)-`#mover(mi("R"),mo("→"))`.(`#mover(mi("v"),mo("→"))`/c)))

These two expressions, however, depend on the time only through the retarded time . This dependence is within and through the velocity of the charge . So, before performing the differentiations, this dependence on must be taken into account.

(29)

(30)

The Electric field `#mover(mi("E"),mo("→"))` = -(diff(`#mover(mi("A"),mo("→"))`, t))/c-%Gradient(`ϕ`)

Computation of

	Computation of

Collecting the results of the two previous subsections, we have for the electric field

E_(X) = -(diff(A_(X), t))/c-%Gradient(varphi(X))

(60)

(61)

The book, presents this result as equation (63.8):

`#mover(mi("E"),mo("→"))` = e*(1-v^2/c^2)*(`#mover(mi("R"),mo("→"))`-`#mover(mi("v"),mo("→"))`*R/c)/(R-(`#mover(mi("v"),mo("→"))`.`#mover(mi("R"),mo("→"))`)/c)^3+`&x`(e*`#mover(mi("R"),mo("→"))`/c(R-(`#mover(mi("v"),mo("→"))`.`#mover(mi("R"),mo("→"))`)/c)^6, `&x`(`#mover(mi("R"),mo("→"))`-`#mover(mi("v"),mo("→"))`*R/c, `#mover(mi("a"),mo("→"))`))

where and . To rewrite (61) as in the above, introduce the two triple vector products

(62)

$simplify(E_(X) = -e*(Physics[Vectors][Norm](R_)^2*a_*c-v_*Physics[Vectors][Norm](v_)^2*Physics[Vectors][Norm](R_)-Physics[Vectors][Norm](R_)*Physics[Vectors][`.`](R_, v_)*a_+v_*Physics[Vectors][`.`](R_, a_)*Physics[Vectors][Norm](R_)+c*v_*Physics[Vectors][`.`](R_, v_))/(c*(1-Physics[Vectors][`.`](R_, v_)/(Physics[Vectors][Norm](R_)*c))*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2*Physics[Vectors][Norm](R_))+c*e*(-Physics[Vectors][Norm](v_)^2*R_-Physics[Vectors][Norm](R_)*c*v_+R_*c^2+Physics[Vectors][`.`](R_, a_)*R_+Physics[Vectors][`.`](R_, v_)*v_)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3, {v_*Physics[Vectors][`.`](R_, a_)-Physics[Vectors][`.`](R_, v_)*a_ = Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))})$

E_(X) = e*(-Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))+R_*c*Physics:-Vectors:-`.`(R_, a_)-Physics:-Vectors:-Norm(R_)^2*a_*c+(-c^2*v_+v_*Physics:-Vectors:-Norm(v_)^2)*Physics:-Vectors:-Norm(R_)+R_*c^3-R_*c*Physics:-Vectors:-Norm(v_)^2)/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

(63)

(64)

$simplify(E_(X) = e*(-Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))+R_*c*Physics[Vectors][`.`](R_, a_)-Physics[Vectors][Norm](R_)^2*a_*c+(-c^2*v_+v_*Physics[Vectors][Norm](v_)^2)*Physics[Vectors][Norm](R_)+R_*c^3-R_*c*Physics[Vectors][Norm](v_)^2)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3, {Physics[Vectors][`.`](R_, a_)*R_-Physics[Vectors][Norm](R_)^2*a_ = Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_))})$

E_(X) = (c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_))-Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))+(c-Physics:-Vectors:-Norm(v_))*(c+Physics:-Vectors:-Norm(v_))*(R_*c-Physics:-Vectors:-Norm(R_)*v_))*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

(65)

Split now this result into two terms, one of them involving the acceleration . For that purpose first expand the expression without expanding the cross products

(66)

Introduce the notation used in the textbook, and and proceed with the splitting

E_(X) = e*(-R*c^2*v_+R*v^2*v_+R_*c^3-R_*c*v^2)/(c*R-Physics:-Vectors:-`.`(R_, v_))^3-e*(R*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))-c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_)))/(c*R-Physics:-Vectors:-`.`(R_, v_))^3

(67)

Rearrange only the first term using simplify; that can be done in different ways, perhaps the simplest is using subsop

E_(X) = e*(c-v)*(c+v)*(-R*v_+R_*c)/(c*R-Physics:-Vectors:-`.`(R_, v_))^3-e*(R*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))-c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_)))/(c*R-Physics:-Vectors:-`.`(R_, v_))^3

(68)

By eye this result is mathematically equal to equation (63.8) of the textbook, shown here above before (62) .

	Algebraic manipulation rewriting (68) as the textbook equation (63.8)

The magnetic field

The book does not show an explicit form for , it only indicates that it is related to the electric field by the formula

Thus in this section we compute the explicit form of and show that this relationship mentioned in the book holds. To compute we proceed as done in the previous sections, the right-hand side should be taken at the previous (retarded) time . For clarity, turn OFF the compact display of functions.

We need to calculate

(75)

	Deriving the chain rule

So applying to (75) the chain rule derived in the previous subsection we have

(87)

where is taken as a function of only in . Now that the functionality is understood, turning ON the compact display of functions and displaying the fields by their names,

(88)

The value of is computed lines above as (48)

(89)

The expression for with no dependency is computed lines above, as (28),

A_(x, y, z, t__0) = e*v_/((Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_)/c)*c)

(90)

The expressions for and the velocity in terms of with no dependency are

(91)

(92)

[%Gradient(t__0(X)) = (r_(x, y, z)-r__0_(t__0))/(-c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))), A_(x, y, z, t__0) = e*v_(t__0)/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c)]

(93)

Introducing this into "H(X)=`%Curl`(A_(x,y,z,t[`0`]))+(`%Gradient`(t[`0`](X)))*((&PartialD;A)/(&PartialD;`t__0`))" ,

(94)

Before computing the first term , for readability, re-introduce the velocity , the acceleration , then remove the dependency of these functions on , not relevant anymore since there are no more derivatives with respect to . Performing these substitutions in sequence,

(95)

(96)

Activate now the inert curl

(97)

From (34)≡, and reintroducing

(98)

(99)

To conclude, rearrange this expression as done with the one for the electric field at (65), so first expand (99) without expanding the cross products

(100)

Then introduce the notation used in the textbook, and and split into two terms, one of which contains the acceleration

H_(X) = Physics:-Vectors:-`&x`(R_, v_)*e*(-c^2+v^2)/(c*R-Physics:-Vectors:-`.`(R_, v_))^3-e*(Physics:-Vectors:-`&x`(R_, a_)*R*c-Physics:-Vectors:-`&x`(R_, a_)*Physics:-Vectors:-`.`(R_, v_)+Physics:-Vectors:-`.`(R_, a_)*Physics:-Vectors:-`&x`(R_, v_))/(c*R-Physics:-Vectors:-`.`(R_, v_))^3