# Masses, Inertias & Particles, RigidBodys (Docstrings)#

## Particle#

class sympy.physics.mechanics.particle.Particle(name, point, mass)[source]#

A particle.

Parameters:

name : str

Name of particle

point : Point

A physics/mechanics Point which represents the position, velocity, and acceleration of this Particle

mass : sympifyable

A SymPy expression representing the Particle’s mass

Explanation

Particles have a non-zero mass and lack spatial extension; they take up no space.

Values need to be supplied on initialization, but can be changed later.

Examples

```>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> po = Point('po')
>>> m = Symbol('m')
>>> pa = Particle('pa', po, m)
>>> # Or you could change these later
>>> pa.mass = m
>>> pa.point = po
```
angular_momentum(point, frame)[source]#

Angular momentum of the particle about the point.

Parameters:

point : Point

The point about which angular momentum of the particle is desired.

frame : ReferenceFrame

The frame in which angular momentum is desired.

Explanation

The angular momentum H, about some point O of a particle, P, is given by:

H = r x m * v

where r is the position vector from point O to the particle P, m is the mass of the particle, and v is the velocity of the particle in the inertial frame, N.

Examples

```>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v, r = dynamicsymbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> A = O.locatenew('A', r * N.x)
>>> P = Particle('P', A, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.angular_momentum(O, N)
m*r*v*N.z
```
kinetic_energy(frame)[source]#

Kinetic energy of the particle.

Parameters:

frame : ReferenceFrame

The Particle’s velocity is typically defined with respect to an inertial frame but any relevant frame in which the velocity is known can be supplied.

Explanation

The kinetic energy, T, of a particle, P, is given by

‘T = 1/2 m v^2’

where m is the mass of particle P, and v is the velocity of the particle in the supplied ReferenceFrame.

Examples

```>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy import symbols
>>> m, v, r = symbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.kinetic_energy(N)
m*v**2/2
```
linear_momentum(frame)[source]#

Linear momentum of the particle.

Parameters:

frame : ReferenceFrame

The frame in which linear momentum is desired.

Explanation

The linear momentum L, of a particle P, with respect to frame N is given by

L = m * v

where m is the mass of the particle, and v is the velocity of the particle in the frame N.

Examples

```>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v = dynamicsymbols('m v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> A = Particle('A', P, m)
>>> P.set_vel(N, v * N.x)
>>> A.linear_momentum(N)
m*v*N.x
```
property mass#

Mass of the particle.

parallel_axis(point, frame)[source]#

Returns an inertia dyadic of the particle with respect to another point and frame.

Parameters:

point : sympy.physics.vector.Point

The point to express the inertia dyadic about.

frame : sympy.physics.vector.ReferenceFrame

The reference frame used to construct the dyadic.

Returns:

The inertia dyadic of the particle expressed about the provided point and frame.

property point#

Point of the particle.

property potential_energy#

The potential energy of the Particle.

Examples

```>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
>>> P.potential_energy
g*h*m
```

## RigidBody#

class sympy.physics.mechanics.rigidbody.RigidBody(name, masscenter, frame, mass, inertia)[source]#

An idealized rigid body.

Explanation

This is essentially a container which holds the various components which describe a rigid body: a name, mass, center of mass, reference frame, and inertia.

All of these need to be supplied on creation, but can be changed afterwards.

Examples

```>>> from sympy import Symbol
>>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
>>> from sympy.physics.mechanics import outer
>>> m = Symbol('m')
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer (A.x, A.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, A, m, inertia_tuple)
>>> # Or you could change them afterwards
>>> m2 = Symbol('m2')
>>> B.mass = m2
```

Attributes

 name (string) The body’s name. masscenter (Point) The point which represents the center of mass of the rigid body. frame (ReferenceFrame) The ReferenceFrame which the rigid body is fixed in. mass (Sympifyable) The body’s mass. inertia ((Dyadic, Point)) The body’s inertia about a point; stored in a tuple as shown above.
angular_momentum(point, frame)[source]#

Returns the angular momentum of the rigid body about a point in the given frame.

Parameters:

point : Point

The point about which angular momentum is desired.

frame : ReferenceFrame

The frame in which angular momentum is desired.

Explanation

The angular momentum H of a rigid body B about some point O in a frame N is given by:

H = I . w + r x Mv

where I is the central inertia dyadic of B, w is the angular velocity of body B in the frame, N, r is the position vector from point O to the mass center of B, and v is the velocity of the mass center in the frame, N.

Examples

```>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> M, v, r, omega = dynamicsymbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, 1 * N.x)
>>> I = outer(b.x, b.x)
>>> B = RigidBody('B', P, b, M, (I, P))
>>> B.angular_momentum(P, N)
omega*b.x
```
property central_inertia#

The body’s central inertia dyadic.

kinetic_energy(frame)[source]#

Kinetic energy of the rigid body.

Parameters:

frame : ReferenceFrame

The RigidBody’s angular velocity and the velocity of it’s mass center are typically defined with respect to an inertial frame but any relevant frame in which the velocities are known can be supplied.

Explanation

The kinetic energy, T, of a rigid body, B, is given by

‘T = 1/2 (I omega^2 + m v^2)’

where I and m are the central inertia dyadic and mass of rigid body B, respectively, omega is the body’s angular velocity and v is the velocity of the body’s mass center in the supplied ReferenceFrame.

Examples

```>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody
>>> from sympy import symbols
>>> M, v, r, omega = symbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (b.x, b.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, inertia_tuple)
>>> B.kinetic_energy(N)
M*v**2/2 + omega**2/2
```
linear_momentum(frame)[source]#

Linear momentum of the rigid body.

Parameters:

frame : ReferenceFrame

The frame in which linear momentum is desired.

Explanation

The linear momentum L, of a rigid body B, with respect to frame N is given by

L = M * v*

where M is the mass of the rigid body and v* is the velocity of the mass center of B in the frame, N.

Examples

```>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> M, v = dynamicsymbols('M v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (N.x, N.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, N, M, Inertia_tuple)
>>> B.linear_momentum(N)
M*v*N.x
```
parallel_axis(point)[source]#

Returns the inertia dyadic of the body with respect to another point.

Parameters:

point : sympy.physics.vector.Point

The point to express the inertia dyadic about.

Returns:

The inertia dyadic of the rigid body expressed about the provided point.

property potential_energy#

The potential energy of the RigidBody.

Examples

```>>> from sympy.physics.mechanics import RigidBody, Point, outer, ReferenceFrame
>>> from sympy import symbols
>>> M, g, h = symbols('M g h')
>>> b = ReferenceFrame('b')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
>>> B.potential_energy
M*g*h
```

## inertia#

sympy.physics.mechanics.functions.inertia(frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0)[source]#

Simple way to create inertia Dyadic object.

Parameters:

frame : ReferenceFrame

The frame the inertia is defined in

ixx : Sympifyable

the xx element in the inertia dyadic

iyy : Sympifyable

the yy element in the inertia dyadic

izz : Sympifyable

the zz element in the inertia dyadic

ixy : Sympifyable

the xy element in the inertia dyadic

iyz : Sympifyable

the yz element in the inertia dyadic

izx : Sympifyable

the zx element in the inertia dyadic

Explanation

If you do not know what a Dyadic is, just treat this like the inertia tensor. Then, do the easy thing and define it in a body-fixed frame.

Examples

```>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')
>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)
```

## inertia_of_point_mass#

sympy.physics.mechanics.functions.inertia_of_point_mass(mass, pos_vec, frame)[source]#

Inertia dyadic of a point mass relative to point O.

Parameters:

mass : Sympifyable

Mass of the point mass

pos_vec : Vector

Position from point O to point mass

frame : ReferenceFrame

Reference frame to express the dyadic in

Examples

```>>> from sympy import symbols
>>> from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass
>>> N = ReferenceFrame('N')
>>> r, m = symbols('r m')
>>> px = r * N.x
>>> inertia_of_point_mass(m, px, N)
m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)
```

## linear_momentum#

sympy.physics.mechanics.functions.linear_momentum(frame, *body)[source]#

Linear momentum of the system.

Parameters:

frame : ReferenceFrame

The frame in which linear momentum is desired.

body1, body2, body3… : Particle and/or RigidBody

The body (or bodies) whose linear momentum is required.

Explanation

This function returns the linear momentum of a system of Particle’s and/or RigidBody’s. The linear momentum of a system is equal to the vector sum of the linear momentum of its constituents. Consider a system, S, comprised of a rigid body, A, and a particle, P. The linear momentum of the system, L, is equal to the vector sum of the linear momentum of the particle, L1, and the linear momentum of the rigid body, L2, i.e.

L = L1 + L2

Examples

```>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, linear_momentum
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = Point('Ac')
>>> Ac.set_vel(N, 25 * N.y)
>>> I = outer(N.x, N.x)
>>> A = RigidBody('A', Ac, N, 20, (I, Ac))
>>> linear_momentum(N, A, Pa)
10*N.x + 500*N.y
```

## angular_momentum#

sympy.physics.mechanics.functions.angular_momentum(point, frame, *body)[source]#

Angular momentum of a system.

Parameters:

point : Point

The point about which angular momentum of the system is desired.

frame : ReferenceFrame

The frame in which angular momentum is desired.

body1, body2, body3… : Particle and/or RigidBody

The body (or bodies) whose angular momentum is required.

Explanation

This function returns the angular momentum of a system of Particle’s and/or RigidBody’s. The angular momentum of such a system is equal to the vector sum of the angular momentum of its constituents. Consider a system, S, comprised of a rigid body, A, and a particle, P. The angular momentum of the system, H, is equal to the vector sum of the angular momentum of the particle, H1, and the angular momentum of the rigid body, H2, i.e.

H = H1 + H2

Examples

```>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, angular_momentum
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> angular_momentum(O, N, Pa, A)
10*N.z
```

## kinetic_energy#

sympy.physics.mechanics.functions.kinetic_energy(frame, *body)[source]#

Kinetic energy of a multibody system.

Parameters:

frame : ReferenceFrame

The frame in which the velocity or angular velocity of the body is defined.

body1, body2, body3… : Particle and/or RigidBody

The body (or bodies) whose kinetic energy is required.

Explanation

This function returns the kinetic energy of a system of Particle’s and/or RigidBody’s. The kinetic energy of such a system is equal to the sum of the kinetic energies of its constituents. Consider a system, S, comprising a rigid body, A, and a particle, P. The kinetic energy of the system, T, is equal to the vector sum of the kinetic energy of the particle, T1, and the kinetic energy of the rigid body, T2, i.e.

T = T1 + T2

Kinetic energy is a scalar.

Examples

```>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, kinetic_energy
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> kinetic_energy(N, Pa, A)
350
```

## potential_energy#

sympy.physics.mechanics.functions.potential_energy(*body)[source]#

Potential energy of a multibody system.

Parameters:

body1, body2, body3… : Particle and/or RigidBody

The body (or bodies) whose potential energy is required.

Explanation

This function returns the potential energy of a system of Particle’s and/or RigidBody’s. The potential energy of such a system is equal to the sum of the potential energy of its constituents. Consider a system, S, comprising a rigid body, A, and a particle, P. The potential energy of the system, V, is equal to the vector sum of the potential energy of the particle, V1, and the potential energy of the rigid body, V2, i.e.

V = V1 + V2

Potential energy is a scalar.

Examples

```>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, potential_energy
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> Pa = Particle('Pa', P, m)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> a = ReferenceFrame('a')
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> potential_energy(Pa, A)
M*g*h + g*h*m
```

## Lagrangian#

sympy.physics.mechanics.functions.Lagrangian(frame, *body)[source]#

Lagrangian of a multibody system.

Parameters:

frame : ReferenceFrame

The frame in which the velocity or angular velocity of the body is defined to determine the kinetic energy.

body1, body2, body3… : Particle and/or RigidBody

The body (or bodies) whose Lagrangian is required.

Explanation

This function returns the Lagrangian of a system of Particle’s and/or RigidBody’s. The Lagrangian of such a system is equal to the difference between the kinetic energies and potential energies of its constituents. If T and V are the kinetic and potential energies of a system then it’s Lagrangian, L, is defined as

L = T - V

The Lagrangian is a scalar.

Examples

```>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, Lagrangian
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> Lagrangian(N, Pa, A)
-M*g*h - g*h*m + 350
```