Manipulate

Manipulate[expr,{u,u_min,u_max}]

generates a version of expr with controls added to allow interactive manipulation of the value of u.

Manipulate[expr,{u,u_min,u_max,du}]

allows the value of u to vary between u_min and u_max in steps du.

Manipulate[expr,{{u,u_init},u_min,u_max,…}]

takes the initial value of u to be u_init.

Manipulate[expr,{{u,u_init,u_lbl},…}]

labels the controls for u with u_lbl.

Manipulate[expr,{u,{u₁,u₂,…}}]

allows u to take on discrete values u₁,u₂,….

Manipulate[expr,{u,…},{v,…},…]

provides controls to manipulate each of the u,v,….

Manipulate[expr,"c_u"{u,…},"c_v"{v,…},…]

links the controls to the specified controllers on an external device.

Details and Options

The expression expr can be any graphic or other expression. If it is None, only the controls are displayed.
The following forms by default yield particular forms of controls:

	{u,u_min,u_max}	manipulator (slider, animator, etc.)
	{u,u_min,u_max,du}	discrete manipulator with step du
	{u,{x_min,y_min},{x_max,y_max}}	2D slider
	{u,Locator}	a locator in a graphic
	{u,{u₁,u₂,…}}	setter bar for few elements; popup menu for more
	{u,{u₁->lbl₁,u₂->lbl₂,…}}	setter bar or popup menu with labels for elements
	{u,{True, False}}	checkbox
	{u,color}	color slider
	{u}	blank input field
	{u,FormObject[…]}	form with specified fields
	{u,func}	create an arbitrary control from a function
	{{u,u_init},…}	control with initial value u_init
	{{u,u_init,u_lbl},…}	control with label u_lbl
	{{u,…},…,opts}	control with particular options
	Control[…]	general control object
	Delimiter	horizontal delimiter
	string, view, cell expression, etc.	explicit text, view, cell, etc. annotations

Possible annotations given in place of controls include expressions with heads String, Style, Row, Item, Text, ExpressionCell, TextCell as well as views and layout constructs such as TabView, Grid, and Multicolumn.
Inside annotations, controls can be specified using Control.
The label u_lbl for a control can be any expression.
The option setting ControlType->type attempts to use controls of the specified type.
Possible control types include: Animator, Checkbox, CheckboxBar, ColorSetter, ColorSlider, FormControl, InputField, Manipulator, PopupMenu, RadioButton or RadioButtonBar, Setter or SetterBar, Slider, Slider2D, TogglerBar, Trigger, and VerticalSlider. None can also be used.
ControlType options can be given separately for each variable. Options for the controls can also be given within the specification for the variables.
ControlType->Trigger specifies that a particular variable should be controlled by a trigger.
The control specification {u,u_min,u_max,…,Appearance->"Labeled"} yields a slider with the value displayed as a label.
In the form {u,func}, Dynamic[u] is given as the first argument to func.
The form {u,u_min,u_max,func} can also be used; what is evaluated is then func[Dynamic[u],{u_min,u_max}].
The form {u} is equivalent to {u, InputField}. {u, ColorSlider} gives a default color slider as a control.
In the form {u,Locator}, the value of u is a list giving x and y coordinates. The coordinates refer either to the first graphic in expr, or range from 0 to 1 in each direction across expr.
The form {{u,{{x₁,y₁},{x₂,y₂},…}},Locator} sets up a locator for each of the {x_i,y_i}, and makes the value of u be the list of all {x_i,y_i}.
The form {{u,u_init},{x_min,y_min},{x_max,y_max},Locator} specifies a range for the locators.
The option setting LocatorAutoCreate->All specifies that new locators should be added for clicks that do not hit existing locators. Click deletes locators.
{{u,{}},Locator,LocatorAutoCreate->All} starts with no locators, but allows locators to be created.
If a variable u is used more than once, linked controls for it are given.
The option setting ControlPlacement->pos specifies that controls should be placed at position pos relative to expr. Possible settings for pos are Bottom, Left, Right, and Top.
The placement of an annotation can be specified by enclosing it in an Item that has a ControlPlacement option.
The following overall options can be given:

Alignment	Automatic	how to align the output in the display area
AppearanceElements	Automatic	overall control elements to include in the displayed output
AutoAction	False	whether to change controls automatically when the mouse is over them
AutorunSequencing	Automatic	how autorun should use the controls
BaselinePosition	Automatic	alignment relative to surrounding text
BaseStyle	{}	base style specifications for the Manipulate
Bookmarks	{}	bookmark settings
ContentSize	Automatic	the absolute size for the content area
ContinuousAction	Automatic	whether to update continuously when controls are changed
ControllerLinking	Automatic	when to activate links to external controllers
ControllerMethod	Automatic	how external controllers should operate
ControllerPath	Automatic	what external controllers to try to use
ControlPlacement	Automatic	placement of controls
ControlType	Automatic	type of controls to use
Deinitialization	None	an expression to be evaluated if the output from the Manipulate is deleted
Deployed	False	whether to make the displayed output deployed
Evaluator	Automatic	the kernel to use for evaluations
FrameLabel	None	labels for the outer frame
FrameMargins	Automatic	margins inside the overall frame
ImageMargins	0	margins around the whole Manipulate
Initialization	None	an expression to be evaluated when output is first displayed
InterpolationOrder	Automatic	interpolation order for animating transitions between bookmarks
LabelStyle	{}	style specifications for the controls area
LocalizeVariables	True	whether to localize the variables
Paneled	True	whether to put the displayed output in a panel
PreserveImageOptions	True	whether to preserve image size and other options when regenerating graphics
RotateLabel	False	whether to rotate y labels on the frame
SaveDefinitions	False	whether to save all definitions associated with expr
ShrinkingDelay	0	how long to delay before shrinking if the displayed object gets smaller
SynchronousInitialization	True	whether to perform initialization synchronously
SynchronousUpdating	Automatic	whether to update synchronously
TouchscreenAutoZoom	False	whether to zoom to fullscreen when activated on a touchscreen
TouchscreenControlPlacement	Automatic	placement of controls on a touchscreen
TrackedSymbols	Full	symbols whose changes trigger updates in the output
UndoTrackedVariables	None	variables that, when changed, should be tracked by the front end's undo mechanism
UnsavedVariables	None	variables whose values should not be saved

The options ControlPlacement and ControlType can be given separately for each variable, in the form {u,spec,opts}.
Manipulate is a scoping construct that implements lexical scoping.
Manipulate generates a DynamicModule object, with the variables u, v, etc. specified as local.
With the default setting UnsavedVariables->{}, values of the variables u, v, etc. are automatically saved in notebooks, and restored when the notebooks are reopened.
With a setting Initialization:>expr, the expression expr is evaluated when Manipulate is executed, or when the result is first displayed in a particular session.
The setting for AppearanceElements can be a list in any order of the following: "ContentResizeArea", "HideControlsButton", "ManipulateMenu", "SnapshotButton". By default, only "ManipulateMenu" is included.
Clicking the snapshot button creates a cell directly below the Manipulate output, containing input of the form With[{u=u_val,…},expr] specifying current values of all variables.
With the setting ContinuousAction->None, an explicit Update button is displayed, and expr is not reevaluated until this is clicked.
With the default setting TrackedSymbols->Automatic, only symbols that appear explicitly in expr are tracked.
TrackedSymbols->True tracks symbols that appear in the controls of Manipulate.
With TrackedSymbols->All, the output is updated whenever any symbol encountered in its evaluation is changed.
With the default setting ControllerLinking->Automatic, controls in a Manipulate respond to specified controllers on an external device whenever the Manipulate is part of the current selection.
Controllers on an external device such as a gamepad can include joysticks, buttons, etc.
Typical external controller specifications include:

"X" or "X1"	x	primary x value
"Y" or "Y1"	y	primary y value
"Z" or "Z1"	z	primary z value
"XY" or "XY1"	{x,y}	primary 2-axis controller value
"XYZ" or "XYZ1"	{x,y,z}	primary 3-axis controller value
"X2", "Y2", "XY2", etc.	x, y, {x,y}, etc.	values from a secondary controller
"XCyclic", etc.	x, etc.	values taken to be cyclic
"XAbsolute", etc.	x, etc.	values from absolute control positions
"B1", "B2", etc.	b₁, b₂, etc.	toggling button states
"B1Absolute", "B2Absolute", etc.	b₁, b₂, etc.	instantaneous button states

Raw external controller specification names for devices active in a particular Wolfram System session can typically be found using ControllerInformation.
With the default setting ControllerMethod->Automatic, a controller specification such as "X" means that the displacement of a control such as a joystick by default determines the rate of change of the corresponding variable x. If an alternate state is selected, for example by depressing the joystick, then the absolute position of the control directly determines the value of x.
With a controller specification such as "XAbsolute", the absolute position of a control such as a joystick determines the value of the corresponding variable x.
With a controller specification such as "XCyclic", the value of the corresponding variable x is typically taken to wrap around cyclically when the control reaches the end of its range.
A controller specification such as "XYZ" may be associated with multiple controls on an external device, such as axes on two distinct joysticks on a single controller.
On a gamepad or other device with two joysticks, "XY1" typically refers to the left joystick, and "XY2" to the right one. If a gamepad has a "hat" control, this is typically referred to as "XY3".
Button controller specifications such as "B1" toggle between True and False whenever the corresponding button is pressed.
Absolute specifications such as "B1Absolute" yield True while the button is being pressed, and False otherwise.
The settings for BaseStyle and LabelStyle are appended to the default styles typically given by the "Manipulate" and "ManipulateLabel" styles in the current stylesheet.