# Kinematics (Docstrings)¶

## Point¶

class sympy.physics.vector.point.Point(name)[source]

This object represents a point in a dynamic system.

It stores the: position, velocity, and acceleration of a point. The position is a vector defined as the vector distance from a parent point to this point.

Parameters

name : string

The display name of the Point

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Point('P')
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> O.set_vel(N, u1 * N.x + u2 * N.y + u3 * N.z)
>>> O.acc(N)
u1'*N.x + u2'*N.y + u3'*N.z


symbols() can be used to create multiple Points in a single step, for example:

>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> from sympy import symbols
>>> N = ReferenceFrame('N')
>>> u1, u2 = dynamicsymbols('u1 u2')
>>> A, B = symbols('A B', cls=Point)
>>> type(A)
<class 'sympy.physics.vector.point.Point'>
>>> A.set_vel(N, u1 * N.x + u2 * N.y)
>>> B.set_vel(N, u2 * N.x + u1 * N.y)
>>> A.acc(N) - B.acc(N)
(u1' - u2')*N.x + (-u1' + u2')*N.y

a1pt_theory(otherpoint, outframe, interframe)[source]

Sets the acceleration of this point with the 1-point theory.

The 1-point theory for point acceleration looks like this:

^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP) + 2 ^N omega^B x ^B v^P

where O is a point fixed in B, P is a point moving in B, and B is rotating in frame N.

Parameters

otherpoint : Point

The first point of the 1-point theory (O)

outframe : ReferenceFrame

The frame we want this point’s acceleration defined in (N)

fixedframe : ReferenceFrame

The intermediate frame in this calculation (B)

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import Vector, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.a1pt_theory(O, N, B)
(-25*q + q'')*B.x + q2''*B.y - 10*q'*B.z

a2pt_theory(otherpoint, outframe, fixedframe)[source]

Sets the acceleration of this point with the 2-point theory.

The 2-point theory for point acceleration looks like this:

^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)

where O and P are both points fixed in frame B, which is rotating in frame N.

Parameters

otherpoint : Point

The first point of the 2-point theory (O)

outframe : ReferenceFrame

The frame we want this point’s acceleration defined in (N)

fixedframe : ReferenceFrame

The frame in which both points are fixed (B)

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.a2pt_theory(O, N, B)
- 10*q'**2*B.x + 10*q''*B.y

acc(frame)[source]

The acceleration Vector of this Point in a ReferenceFrame.

Parameters

frame : ReferenceFrame

The frame in which the returned acceleration vector will be defined in

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x

locatenew(name, value)[source]

Creates a new point with a position defined from this point.

Parameters

name : str

The name for the new point

value : Vector

The position of the new point relative to this point

Examples

>>> from sympy.physics.vector import ReferenceFrame, Point
>>> N = ReferenceFrame('N')
>>> P1 = Point('P1')
>>> P2 = P1.locatenew('P2', 10 * N.x)

partial_velocity(frame, *gen_speeds)[source]

Returns the partial velocities of the linear velocity vector of this point in the given frame with respect to one or more provided generalized speeds.

Parameters

frame : ReferenceFrame

The frame with which the velocity is defined in.

gen_speeds : functions of time

The generalized speeds.

Returns

partial_velocities : tuple of Vector

The partial velocity vectors corresponding to the provided generalized speeds.

Examples

>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> p = Point('p')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> p.set_vel(N, u1 * N.x + u2 * A.y)
>>> p.partial_velocity(N, u1)
N.x
>>> p.partial_velocity(N, u1, u2)
(N.x, A.y)

pos_from(otherpoint)[source]

Returns a Vector distance between this Point and the other Point.

Parameters

otherpoint : Point

The otherpoint we are locating this one relative to

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x

set_acc(frame, value)[source]

Used to set the acceleration of this Point in a ReferenceFrame.

Parameters

frame : ReferenceFrame

The frame in which this point’s acceleration is defined

value : Vector

The vector value of this point’s acceleration in the frame

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x

set_pos(otherpoint, value)[source]

Used to set the position of this point w.r.t. another point.

Parameters

otherpoint : Point

The other point which this point’s location is defined relative to

value : Vector

The vector which defines the location of this point

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x

set_vel(frame, value)[source]

Sets the velocity Vector of this Point in a ReferenceFrame.

Parameters

frame : ReferenceFrame

The frame in which this point’s velocity is defined

value : Vector

The vector value of this point’s velocity in the frame

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x

v1pt_theory(otherpoint, outframe, interframe)[source]

Sets the velocity of this point with the 1-point theory.

The 1-point theory for point velocity looks like this:

^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP

where O is a point fixed in B, P is a point moving in B, and B is rotating in frame N.

Parameters

otherpoint : Point

The first point of the 2-point theory (O)

outframe : ReferenceFrame

The frame we want this point’s velocity defined in (N)

interframe : ReferenceFrame

The intermediate frame in this calculation (B)

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import Vector, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.v1pt_theory(O, N, B)
q'*B.x + q2'*B.y - 5*q*B.z

v2pt_theory(otherpoint, outframe, fixedframe)[source]

Sets the velocity of this point with the 2-point theory.

The 2-point theory for point velocity looks like this:

^N v^P = ^N v^O + ^N omega^B x r^OP

where O and P are both points fixed in frame B, which is rotating in frame N.

Parameters

otherpoint : Point

The first point of the 2-point theory (O)

outframe : ReferenceFrame

The frame we want this point’s velocity defined in (N)

fixedframe : ReferenceFrame

The frame in which both points are fixed (B)

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.v2pt_theory(O, N, B)
5*N.x + 10*q'*B.y

vel(frame)[source]

The velocity Vector of this Point in the ReferenceFrame.

Parameters

frame : ReferenceFrame

The frame in which the returned velocity vector will be defined in

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x


## kinematic_equations¶

sympy.physics.vector.functions.kinematic_equations(speeds, coords, rot_type, rot_order='')[source]

Gives equations relating the qdot’s to u’s for a rotation type.

Supply rotation type and order as in orient. Speeds are assumed to be body-fixed; if we are defining the orientation of B in A using by rot_type, the angular velocity of B in A is assumed to be in the form: speed[0]*B.x + speed[1]*B.y + speed[2]*B.z

Parameters

speeds : list of length 3

The body fixed angular velocity measure numbers.

coords : list of length 3 or 4

The coordinates used to define the orientation of the two frames.

rot_type : str

The type of rotation used to create the equations. Body, Space, or Quaternion only

rot_order : str or int

If applicable, the order of a series of rotations.

Examples

>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
...     order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']

sympy.physics.vector.functions.partial_velocity(vel_vecs, gen_speeds, frame)[source]

Returns a list of partial velocities with respect to the provided generalized speeds in the given reference frame for each of the supplied velocity vectors.

The output is a list of lists. The outer list has a number of elements equal to the number of supplied velocity vectors. The inner lists are, for each velocity vector, the partial derivatives of that velocity vector with respect to the generalized speeds supplied.

Parameters

vel_vecs : iterable

An iterable of velocity vectors (angular or linear).

gen_speeds : iterable

An iterable of generalized speeds.

frame : ReferenceFrame

The reference frame that the partial derivatives are going to be taken in.

Examples

>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import partial_velocity
>>> u = dynamicsymbols('u')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> vel_vecs = [P.vel(N)]
>>> gen_speeds = [u]
>>> partial_velocity(vel_vecs, gen_speeds, N)
[[N.x]]

sympy.physics.vector.functions.get_motion_params(frame, **kwargs)[source]

Returns the three motion parameters - (acceleration, velocity, and position) as vectorial functions of time in the given frame.

If a higher order differential function is provided, the lower order functions are used as boundary conditions. For example, given the acceleration, the velocity and position parameters are taken as boundary conditions.

The values of time at which the boundary conditions are specified are taken from timevalue1(for position boundary condition) and timevalue2(for velocity boundary condition).

If any of the boundary conditions are not provided, they are taken to be zero by default (zero vectors, in case of vectorial inputs). If the boundary conditions are also functions of time, they are converted to constants by substituting the time values in the dynamicsymbols._t time Symbol.

This function can also be used for calculating rotational motion parameters. Have a look at the Parameters and Examples for more clarity.

Parameters

frame : ReferenceFrame

The frame to express the motion parameters in

acceleration : Vector

Acceleration of the object/frame as a function of time

velocity : Vector

Velocity as function of time or as boundary condition of velocity at time = timevalue1

position : Vector

Velocity as function of time or as boundary condition of velocity at time = timevalue1

timevalue1 : sympyfiable

Value of time for position boundary condition

timevalue2 : sympyfiable

Value of time for velocity boundary condition

Examples

>>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols
>>> from sympy import symbols
>>> R = ReferenceFrame('R')
>>> v1, v2, v3 = dynamicsymbols('v1 v2 v3')
>>> v = v1*R.x + v2*R.y + v3*R.z
>>> get_motion_params(R, position = v)
(v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z)
>>> a, b, c = symbols('a b c')
>>> v = a*R.x + b*R.y + c*R.z
>>> get_motion_params(R, velocity = v)
(0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z)
>>> parameters = get_motion_params(R, acceleration = v)
>>> parameters[1]
a*t*R.x + b*t*R.y + c*t*R.z
>>> parameters[2]
a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z