Physics[Parameters] - set symbols to work as constant parameters
Parameters(a, b, ...)
a, b, ...
The Parameters command allows you to define the parameters of a theory in such a way that no functionality can be attached to them. For example, if m is defined as a parameter through Parameters(m), then m⁡t returns m, without any functionality.
To know which names are defined as parameters at some point in a session, call Parameters() without any arguments; the result is a set with the required information.
The cancellation of a parameter definition can be done by unassigning the variable. The unassignment is automatically taken into account in all subsequent calculations, as well as in any Parameters() requests for information.
F ≔ k⁢x
Notice that functionality has been attached to k automatically.
In a model where k is a constant, the above, withⅆkⅆt, is undesired. You would like to be able to define F in the simple way it has been done, and then have F⁡t return a result with k, not k⁡t. This situation is addressed by the Parameters command. For example:
In this way, instead of the undesired result k⁡t, you now have the constant k defined as a parameter, with no functionality attached.
A typical use for the Parameters command is when computing equations of motion departing from a Lagrangian or a Hamiltonian (the Energy). Consider a harmonic oscillator of mass m, and k is a constant parametrizing the restoring force. The Energy (Hamiltonian) in terms of the momentum p and position q is given by:
H ≔ p22⁢m+1⁢k⁢q22
where in the above, p and q represent functions of time, while m and k represent constant parameters. Because m and k have been set by the Parameters command, no functionality is attached to them.
Now you can compute the Hamilton equations directly.
eq1 ≔ ⅆⅆt⁢q⁡t=diff⁡H⁡t,p⁡t
eq2 ≔ ⅆⅆt⁢p⁡t=−diff⁡H⁡t,q⁡t
It is now easy to see that the Energy of this oscillator is a constant; that is, it does not depend on t: differentiate the Energy (the Hamiltonian H), and introduce the equations of motion that were previously derived.
The same computation can be performed without using Parameters. Define H as a mapping, then you must use more complicated syntax to specify the parameters. See the last example in the help page for D for a demonstration of this method.
To query about the objects defined as parameters at any moment, enter the Parameters command with no arguments.
To unset the symbol k as a parameter, it suffices to unassign it.
k ≔ 'k'
Now k is not in the list of parameters, and it depends on t in the function H.
Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics[*], Physics[diff]
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