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Maple provides a state-of-the-art environment for algebraic computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible. The theme of the Physics project for Maple 2023 has been the use of the package in Education, new commands, the consolidation of the functionality introduced in previous releases, also related to the StandardModel package, and several enhancements necessary for producing the new Courseware-Support, Mechanics material.
As part of its commitment to providing the best possible computational environment in Physics, Maplesoft launched a Maple Physics: Research and Development website in 2014, which enabled users to download research versions of the package, ask questions, and provide feedback. The results from this accelerated exchange have been incorporated into the Physics package in Maple 2023. The presentation below illustrates both the novelties and the kind of mathematical formulations that can now be performed.
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Courseware support: Mechanics
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The Physics package, with its various subpackages, is now a resourceful, mature project. For Maple 2023, part of the focus has been in its use
in Education in Physics, starting with Mechanics. The resulting material is now part of Maple's help system and can be opened to interact with it as a Maple document. The presentation is organized as a hyperlinked syllabus covering typical topics appearing in Mechanics undergrad courses of the first years. For the Maple 2023 release, Part I covers:
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New command Lagrange Equations
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There is now a new Physics command, LagrangeEquations, taking advantage of the functional differentiation capabilities. LagrangeEquations receives an expression representing a Lagrangian and returns a sequence of Lagrange equations with as many equations as coordinates are indicated. The formula in the traditional case where the Lagrangian depends on 1st order derivatives of the coordinates and there is only one parameter, , is
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where formally represents the derivative with respect to the coordinates of the particle, equal to the Gradient when working in Cartesian coordinates; represents the equivalent operation, replacing each coordinate by the corresponding generalized velocity and represents the total derivative with respect to , the parameter parametrizing the coordinates. In more general cases the number of parameters can be many. For example, in electrodynamics, the "coordinate" is a tensor field , there are then four coordinates, one for each of the values of the index , and there are four parameters . LagrangeEquations can handle tensors and vectors of the Physics package as well as derivatives using vectorial differential operators (see d_ and Nabla), works by performing functional differentiation (see Fundiff), and handles 1st, and higher order derivatives of the coordinates in the Lagrangian automatically.
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Examples
The Lagrangian of a one-dimensional oscillator - small oscillations
The corresponding Lagrange equation gives Newton's second law, a 2nd order linear ODE for
The Lagrangian of a pendulum of mass and length where the suspension point moves uniformly over a vertical circumference centered at the origin, with a constant frequency
The Lagrange equations
The Maxwell equations can be derived as Lagrange equations as follows. For simplicity, consider Maxwell equations in vacuum. Define first a tensor representing the 4D electromagnetic field potential
The electromagnetic field tensor
The Lagrangian
Maxwell equations in 4D tensorial notation
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New command Substitute
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The command for doing syntactical exact-match substitutions is subs. Frequently, however, what we intend to accomplish with a substitution is more of a mathematical substitution. For different kinds of mathematical substitutions Maple has the commands algsubs, Physics:-SubstituteTensor, Physics:-SubstituteTensorIndices, Physics:-Library:-SubstituteOperator and Physics:-Library:-SubstituteMatrix, resulting in a myriad of commands, useful, however difficult to remember. A new single command, Physics:-Substitute, unifies all of those so that one does not need to remember which one to use.
Examples
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Substitutions of sub-products and sub-sums, possibly involving noncommutative objects
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Substitution of matrices and vectors taking mathematical equality into account
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Set up some non-commutative operands for substitution
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Some typical examples not handled by subs, eval or algsubs
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Substituting tensorial sub-expressions
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Substitutions of tensor indices
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New command StandardModel:-Lagrangian
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One of the distinctive aspects of the Standard Model is the complexity of its Lagrangian. In this context, the new StandardModel:-Lagrangian returns the Lagrangian of the model after symmetry breaking, optionally restricted to only the interaction terms, or only one of its QED, QCD or electroweak sectors, or only one of the different sub-terms involved in the electroweak part; all of that with the covariant derivatives, and sums over leptons and quarks optionally expanded.
The algebraic expressions returned by Lagrangian are fully computable, so you can use them as starting point to construct other Lagrangians (add or subtract terms), or the Action and related field equations (see d_, D_ for covariant derivatives, diff and Fundiff for functional differentiation), or to compute scattering amplitudes (see FeynmanDiagrams and FeynmanIntegral).
Examples
The massless fields of the model are the electromagnetic and gluon fields and the three neutrinos
The Leptons and Quarks of the model are
The Gauge fields, and their related field strengths displayed with Open Face type fonts
To represent the interaction Lagrangians for the QCD and electroweak sectors as sums over leptons and quarks, all of them fermions, it is useful to introduce four anticommutative prefixes, used below as summation indices in the formulas
For readability, omit from the display of formulas the functionality of all the fields entering the Standard Model (see CompactDisplay) and use the lowercase i instead of the uppercase I to represent the imaginary unit
The Lagrangian of the whole Standard Model after symmetry breaking, in its most compact form:
In the output above we see, among other things, the Dirac matrix, and the Cabibbo - Kobayashi - Maskawa matrix , and the tensor indices of different kinds all explicit. See StandardModel for the notational conventions used, which are standard in the literature but for a few things, like a sign in the definition of , that depend on the reference.
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The Quantum Electrodynamics (QED) Lagrangian
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The simplest sector of this Lagrangian (8) is the QED one
The applied form can be obtained using the Library command ApplyProductsOfDifferentialOperators over the output (9) or passing the optional argument applied
Only the interaction part of this Lagrangian is relevant when computing scattering amplitudes. To get that part, you can either expand the covariant derivative operator or or pass the optional keyword expanded, in which case also the trace of gets expanded
then discard the non-interaction terms
or simpler: pass the keyword interaction
All the algebraic expressions returned by Lagrangian are fully computable in that further calculations can proceed starting from them. For example (see FeynmanDiagrams), this is the self-energy of the electron
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The Quantum Chromodynamics (QCD) Lagrangian
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Next in complexity is the QCD Lagrangian
For computing scattering amplitudes, only the interaction part of this Lagrangian is relevant
The amplitude at tree level for the process with two incoming and two outgoing Up quarks (particle and antiparticle) exchanging a gluon
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The Electro-Weak Lagrangian
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The electroweak sector of the Standard Model Lagrangian is significantly more complicated
To decipher this result it is useful to see the structure of physically recognizable terms. For that purpose, you can use the keyword showterms
In this result we see a sum of , and after 'where' there is a list of equations with the formulas represented by each . Take from the above, for instance, only the charged current term that involves interaction between the leptons and the corresponding neutrinos: you can do that with the mouse, copy and paste, or using the option of Lagrangian
The same term in expanded form, useful for computing scattering amplitudes
A process at tree level with a positron and electronic neutrino incoming and the antiparticle of the muon and a muon neutrino outgoing after exchanging a W boson
The term of the electroweak Lagrangian contains the interaction between the Higgs and the Z and W bosons
The probability density at one loop for a process with three Higgs incoming and outgoing: to omit the large - two pages - algebraic result containing Feynman integrals, end the input line with ":", or remove that ending to see and manipulate the integrals. Note this result involves Feynman diagrams with two, three, four, five and six vertices. New in Maple 2023 are the diagrams with more than four vertices.
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New Physics:-Library commands: FactorSum and RemoveCommonFactors
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It is sometimes the case that one needs to factor out something without changing or complicating the algebraic structure of each factor. For this purpose, there is a new Physics:-Library command, FactorSum.
Examples
We want to extract the factors . The standard approach is to use factor or simplify
Both performed the operation requested, but, in addition, performed additional undesired operations or normalizations. An alternative to accomplish this without changing the algebraic structure of each factor, is
The relevance of this is in that more often than not, textbooks write this expression as in (57), not (56), not (55), and when studying a book or trying to follow its computations getting exact match to what we see on paper, is of help in a way that the other commands are not.
Similarly, when we have equations with both sides having a common factor and therefore we want to just cancel out that factor without changing anything else, the routine to use is the new . For example, in
we see, on the left-hand and right-hand sides, the common factors . One simple way of performing this cancellation of common factors on the left and right hand sides, sufficiently identical to what we do with paper and pencil, is
Note that, as shown above, you don't need to load the Physics to use its Library commands
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New coordinates labels
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Notation is one of the most important things to communicate with others in science. In the context of the Physics package, strong emphasis is put on using textbook notation as much as possible regarding input and output. Still, for historical reasons, when using the Physics package, the labels used to refer to a coordinate system had been a single Capital Letter, as in X, Y, ...It was not possible to use, e.g. , or .
That has changed in Maple 2023: any symbol can be used as a coordinate system label.
Examples
A lowercase letter can be used as a coordinate's label
In these cases where the coordinate's system label is not a single Capital Letter, say X, the default coordinates automatically associated to the label, instead of , are now literal subscripts , that is: symbols with a root, in this example x, followed by the double underscore __, then a number. The automatic mapping is in these cases , also using literal subscripts x__0 and x__4.
Hence
Everything else works as with the traditional Capital Letter coordinate-system labels. So is a tensor because it is the 4-vector related to the system of coordinates
Consequently, also receives the automatic reformatting of repeated indices making one of then contravariant
A typical textbook case example: the label has a prime using the new feature of Maple 2023 that allows for primed variables. For example,
Note that the coordinates automatically associated to also have a prime; that is also as in textbooks. One can as well use and at the same time to describe transformations of coordinates between two reference systems
(in the above, the previous label X' is automatically removed to reuse the coordinates with the new label x' )
The case of a coordinate system where the label has a prime is also special. For example, suppose you set a coordinate system without a prime, as in
If you now set the label , because the label is already set, the coordinates associated to are those of with a prime
Finally, it is now also possible to use literalsubscripts with any label. For example, for a Capital Letter label, say instead of , you can set Y with literal subscripts:
Summarizing:
1. The label can now be any symbol; particularly useful is the case of a lowercase letter as a label: we can now represent position vectors in tensorial notation using a more natural textbook-notation (lowercase).
2. We can now use the traditional x and x', or X and X' notations, at the same time, as frequently seen in textbooks when presenting transformations of coordinates
3. The coordinates automatically associated to these new coordinate labels use literal subscripted letters as in , a textbook-like typesetting, instead of the old, not so good looking,
4. It is now possible (optional) to use literal subscripts also with Capital Letter coordinate-system labels
5. Coordinate-system labels with a prime are treated in a special way when the same label without a prime already exists, by just concatenating a prime to the coordinates of the label without prime.
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