Generalized coordinates

From Wikipedia, the free encyclopedia

In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart. The generalized velocities are the time derivatives of the generalized coordinates of the system. 

 

An example of a generalized coordinate is the angle that locates a point moving on a circle. The adjective "generalized" distinguishes these parameters from the traditional use of the term coordinate to refer to Cartesian coordinates: for example, describing the location of the point on the circle using x and y coordinates.

Although there may be many choices for generalized coordinates for a physical system, parameters which are convenient are usually selected for the specification of the configuration of the system and which make the solution of its equations of motion easier. If these parameters are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system.

Constraints and degrees of freedom

Open straight path

 

Open curved path F(x, y) = 0

 

Closed curved path C(x, y) = 0

 

One generalized coordinate (one degree of freedom) on paths in 2D. Only one generalized coordinate is needed to uniquely specify positions on the curve. In these examples, that variable is either arc length s or angle θ. Having both of the Cartesian coordinates (x, y) are unnecessary since either x or y is related to the other by the equations of the curves. They can also be parameterized by s or θ.

 

Open curved path F(x, y) = 0. Multiple intersections of radius with path.

 

Closed curved path C(x, y) = 0. Self-intersection of path.

 

The arc length s along the curve is a legitimate generalized coordinate since the position is uniquely determined, but the angle θ is not since there are multiple positions for a single value of θ.

 

Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of Lagrange's equations of motion. However, it can also occur that a useful set of generalized coordinates may be dependent, which means that they are related by one or more constraint equations.

Holonomic constraints

Closed curved surface S(x, y, z) = 0

 

Two generalized coordinates, two degrees of freedom, on curved surfaces in 3d. Only two numbers (u, v) are needed to specify the points on the curve, one possibility is shown for each case. The full three Cartesian coordinates (x, y, z) are not necessary because any two determines the third according to the equations of the curves.

For a system of N particles in 3D real coordinate space, the position vector of each particle can be written as a 3-tuple in Cartesian coordinates:

${\displaystyle \mathbf {r} _{1}=(x_{1},y_{1},z_{1})\,,\quad \mathbf {r} _{2}=(x_{2},y_{2},z_{2})\,,\ldots \,,\mathbf {r} _{N}=(x_{N},y_{N},z_{N})\,.}$

Any of the position vectors can be denoted r_k where k = 1, 2, ..., N labels the particles. A holonomic constraint is a constraint equation of the form for particle k

${\displaystyle f(\mathbf {r} _{k},t)=0}$

which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time t will appear explicitly in the constraint equations. At any instant of time, when t is a constant, any one coordinate will be determined from the other coordinates, e.g. if x_k and z_k are given, then so is y_k. One constraint equation counts as one constraint. If there are C constraints, each has an equation, so there will be C constraint equations. There is not necessarily one constraint equation for each particle, and if there are no constraints on the system then there are no constraint equations.

So far, the configuration of the system is defined by 3N quantities, but C coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is n = 3N − C. (In D dimensions, the original configuration would need ND coordinates, and the reduction by constraints means n = ND − C). It is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as generalized coordinates in this context, denoted q_j(t). It is convenient to collect them into an n-tuple

${\displaystyle \mathbf {q} (t)=(q_{1}(t),q_{2}(t),\ldots ,q_{n}(t))}$

which is a point in the configuration space of the system. They are all independent of one other, and each is a function of time. Geometrically they can be lengths along straight lines, or arc lengths along curves, or angles; not necessarily Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates equals the number of degrees of freedom, n. A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, or the arc length traversed by a bead along a wire.

If it is possible to find from the constraints as many independent variables as there are degrees of freedom, these can be used as generalized coordinates The position vector r_k of particle k is a function of all the n generalized coordinates and time

${\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} (t),t)\,,}$

and the generalized coordinates can be thought of as parameters associated with the constraint.

The corresponding time derivatives of q are the generalized velocities,

${\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}=({\dot {q}}_{1}(t),{\dot {q}}_{2}(t),\ldots ,{\dot {q}}_{n}(t))}$

(each dot over a quantity indicates one time derivative). The velocity vector v_k is the total derivative of r_k with respect to time

${\displaystyle \mathbf {v} _{k}={\dot {\mathbf {r} }}_{k}={\frac {d\mathbf {r} _{k}}{dt}}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}\,.}$

and so generally depends on the generalized velocities and coordinates. Since we are free to specify the initial values of the generalized coordinates and velocities separately, the generalized coordinates q_j and velocities dq_j/dt can be treated as independent variables.

Non-holonomic constraints

A mechanical system can involve constraints on both the generalized coordinates and their derivatives. Constraints of this type are known as non-holonomic. First-order non-holonomic constraints have the form

${\displaystyle g(\mathbf {q} ,{\dot {\mathbf {q} }},t)=0\,,}$

An example of such a constraint is a rolling wheel or knife-edge that constrains the direction of the velocity vector. Non-holonomic constraints can also involve next-order derivatives such as generalized accelerations.

Physical quantities in generalized coordinates

Kinetic energy

The total kinetic energy of the system is the energy of the system's motion, defined as

${\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}{\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}\,,}$

in which · is the dot product. The kinetic energy is a function only of the velocities v_k, not the coordinates r_k themselves. By contrast an important observation is,

${\displaystyle {\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\right){\dot {q}}_{i}{\dot {q}}_{j}+\sum _{i=1}^{n}\left(2{\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial t}}\right){\dot {q}}_{i}+\left({\frac {\partial \mathbf {r} _{k}}{\partial t}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial t}}\right)\!,}$

which illustrates the kinetic energy is in general a function of the generalized velocities, coordinates, and time if the constraint also varies with time, so T = T(q, dq/dt, t).

In the case the constraint on the particle is time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy has no time-dependence and is a homogeneous function of degree 2 in the generalized velocities,

${\displaystyle {\dot {\mathbf {r} }}_{k}\cdot {\dot {\mathbf {r} }}_{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\right){\dot {q}}_{i}{\dot {q}}_{j}\,.}$

Still for the time-independent case, this expression is equivalent to taking the line element squared of the trajectory for particle k,

${\displaystyle ds_{k}^{2}=d\mathbf {r} _{k}\cdot d\mathbf {r} _{k}=\sum _{i,j=1}^{n}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\right)dq_{i}dq_{j}\,,}$

and dividing by the square differential in time, dt², to obtain the velocity squared of particle k. Thus for time-independent constraints it is sufficient to know the line element to quickly obtain the kinetic energy of particles and hence the Lagrangian.

It is instructive to see the various cases of polar coordinates in 2d and 3d, owing to their frequent appearance. In 2d polar coordinates (r, θ),

${\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\,,}$

in 3d cylindrical coordinates (r, θ, z),

${\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+{\dot {z}}^{2}\,,}$

in 3d spherical coordinates (r, θ, φ),

${\displaystyle \left({\frac {ds}{dt}}\right)^{2}={\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2}\,.}$

Generalized momentum

The generalized momentum "canonically conjugate to" the coordinate q_i is defined by

${\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}$

If the Lagrangian L does not depend on some coordinate q_i, then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a conserved quantity, because the time derivative is zero implying the momentum is a constant of the motion;

${\displaystyle {\dot {p}}_{i}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0\,.}$

Examples

Bead on a wire

Bead constrained to move on a frictionless wire. The wire exerts a reaction force C on the bead to keep it on the wire. The non-constraint force N in this case is gravity. Notice the initial position of the wire can lead to different motions.

For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form f(r) = 0, where the position of the bead can be written r = (x(s), y(s)), in which s is a parameter, the arc length s along the curve from some point on the wire. This is a suitable choice of generalized coordinate for the system. Only one coordinate is needed instead of two, because the position of the bead can be parameterized by one number, s, and the constraint equation connects the two coordinates x and y; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.

Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively

${\displaystyle f(\mathbf {r} ,t)=0\,,\quad \mathbf {r} =(x(s,t),y(s,t))}$

which now both depend on time t due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates and explicitly in the constraint equations.

Simple pendulum

Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f(x, y) = 0, the constraint force C is the tension in the rod. Again the non-constraint force N in this case is gravity.

 

Dynamic model of a simple pendulum.

The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum.

A simple pendulum consists of a mass M hanging from a pivot point so that it is constrained to move on a circle of radius L. The position of the mass is defined by the coordinate vector r=(x, y) measured in the plane of the circle such that y is in the vertical direction. The coordinates x and y are related by the equation of the circle

${\displaystyle f(x,y)=x^{2}+y^{2}-L^{2}=0,}$

that constrains the movement of M. This equation also provides a constraint on the velocity components,

${\displaystyle {\dot {f}}(x,y)=2x{\dot {x}}+2y{\dot {y}}=0.}$

Now introduce the parameter θ, that defines the angular position of M from the vertical direction. It can be used to define the coordinates x and y, such that

${\displaystyle \mathbf {r} =(x,y)=(L\sin \theta ,-L\cos \theta ).}$

The use of θ to define the configuration of this system avoids the constraint provided by the equation of the circle.

Notice that the force of gravity acting on the mass m is formulated in the usual Cartesian coordinates,

${\displaystyle \mathbf {F} =(0,-mg),}$

where g is the acceleration of gravity.

The virtual work of gravity on the mass m as it follows the trajectory r is given by

${\displaystyle \delta W=\mathbf {F} \cdot \delta \mathbf {r} .}$

The variation δr can be computed in terms of the coordinates x and y, or in terms of the parameter θ,

${\displaystyle \delta \mathbf {r} =(\delta x,\delta y)=(L\cos \theta ,L\sin \theta )\delta \theta .}$

Thus, the virtual work is given by

${\displaystyle \delta W=-mg\delta y=-mgL\sin \theta \delta \theta .}$

Notice that the coefficient of δy is the y-component of the applied force. In the same way, the coefficient of δθ is known as the generalized force along generalized coordinate θ, given by

${\displaystyle F_{\theta }=-mgL\sin \theta .}$

To complete the analysis consider the kinetic energy T of the mass, using the velocity,

${\displaystyle \mathbf {v} =({\dot {x}},{\dot {y}})=(L\cos \theta ,L\sin \theta ){\dot {\theta }},}$

so,

${\displaystyle T={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} ={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2})={\frac {1}{2}}mL^{2}{\dot {\theta }}^{2}.}$

D'Alembert's form of the principle of virtual work for the pendulum in terms of the coordinates x and y are given by,

${\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {x}}}}-{\frac {\partial T}{\partial x}}=F_{x}+\lambda {\frac {\partial f}{\partial x}},\quad {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {y}}}}-{\frac {\partial T}{\partial y}}=F_{y}+\lambda {\frac {\partial f}{\partial y}}.}$

This yields the three equations

${\displaystyle m{\ddot {x}}=\lambda (2x),\quad m{\ddot {y}}=-mg+\lambda (2y),\quad x^{2}+y^{2}-L^{2}=0,}$

in the three unknowns, x, y and λ.

Using the parameter θ, those equations take the form

${\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\theta }}}}-{\frac {\partial T}{\partial \theta }}=F_{\theta },}$

which becomes,

${\displaystyle mL^{2}{\ddot {\theta }}=-mgL\sin \theta ,}$

or

${\displaystyle {\ddot {\theta }}+{\frac {g}{L}}\sin \theta =0.}$

This formulation yields one equation because there is a single parameter and no constraint equation.
This shows that the parameter θ is a generalized coordinate that can be used in the same way as the Cartesian coordinates x and y to analyze the pendulum.

Double pendulum

A double pendulum

The benefits of generalized coordinates become apparent with the analysis of a double pendulum. For the two masses m_i, i=1, 2, let r_i=(x_i, y_i), i=1, 2 define their two trajectories. These vectors satisfy the two constraint equations,

${\displaystyle f_{1}(x_{1},y_{1},x_{2},y_{2})=\mathbf {r} _{1}\cdot \mathbf {r} _{1}-L_{1}^{2}=0,\quad f_{2}(x_{1},y_{1},x_{2},y_{2})=(\mathbf {r} _{2}-\mathbf {r} _{1})\cdot (\mathbf {r} _{2}-\mathbf {r} _{1})-L_{2}^{2}=0.}$

The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x_i, y_i i=1, 2 and the two Lagrange multipliers λ_i, i=1, 2 that arise from the two constraint equations.

Now introduce the generalized coordinates θ_i i=1,2 that define the angular position of each mass of the double pendulum from the vertical direction. In this case, we have

${\displaystyle \mathbf {r} _{1}=(L_{1}\sin \theta _{1},-L_{1}\cos \theta _{1}),\quad \mathbf {r} _{2}=(L_{1}\sin \theta _{1},-L_{1}\cos \theta _{1})+(L_{2}\sin \theta _{2},-L_{2}\cos \theta _{2}).}$

The force of gravity acting on the masses is given by,

${\displaystyle \mathbf {F} _{1}=(0,-m_{1}g),\quad \mathbf {F} _{2}=(0,-m_{2}g)}$

where g is the acceleration of gravity. Therefore, the virtual work of gravity on the two masses as they follow the trajectories r_i, i=1,2 is given by

${\displaystyle \delta W=\mathbf {F} _{1}\cdot \delta \mathbf {r} _{1}+\mathbf {F} _{2}\cdot \delta \mathbf {r} _{2}.}$

The variations δr_i i=1, 2 can be computed to be

${\displaystyle \delta \mathbf {r} _{1}=(L_{1}\cos \theta _{1},L_{1}\sin \theta _{1})\delta \theta _{1},\quad \delta \mathbf {r} _{2}=(L_{1}\cos \theta _{1},L_{1}\sin \theta _{1})\delta \theta _{1}+(L_{2}\cos \theta _{2},L_{2}\sin \theta _{2})\delta \theta _{2}}$

Thus, the virtual work is given by

${\displaystyle \delta W=-(m_{1}+m_{2})gL_{1}\sin \theta _{1}\delta \theta _{1}-m_{2}gL_{2}\sin \theta _{2}\delta \theta _{2},}$

and the generalized forces are

${\displaystyle F_{\theta _{1}}=-(m_{1}+m_{2})gL_{1}\sin \theta _{1},\quad F_{\theta _{2}}=-m_{2}gL_{2}\sin \theta _{2}.}$

Compute the kinetic energy of this system to be

${\displaystyle T={\frac {1}{2}}m_{1}\mathbf {v} _{1}\cdot \mathbf {v} _{1}+{\frac {1}{2}}m_{2}\mathbf {v} _{2}\cdot \mathbf {v} _{2}={\frac {1}{2}}(m_{1}+m_{2})L_{1}^{2}{\dot {\theta }}_{1}^{2}+{\frac {1}{2}}m_{2}L_{2}^{2}{\dot {\theta }}_{2}^{2}+m_{2}L_{1}L_{2}\cos(\theta _{2}-\theta _{1}){\dot {\theta }}_{1}{\dot {\theta }}_{2}.}$

Euler–Lagrange equation yield two equations in the unknown generalized coordinates θ_i i=1, 2, given by^[14]

${\displaystyle (m_{1}+m_{2})L_{1}^{2}{\ddot {\theta }}_{1}+m_{2}L_{1}L_{2}{\ddot {\theta }}_{2}\cos(\theta _{2}-\theta _{1})+m_{2}L_{1}L_{2}{\ddot {\theta _{2}}}^{2}\sin(\theta _{1}-\theta _{2})=-(m_{1}+m_{2})gL_{1}\sin \theta _{1},}$

and

${\displaystyle m_{2}L_{2}^{2}{\ddot {\theta }}_{2}+m_{2}L_{1}L_{2}{\ddot {\theta }}_{1}\cos(\theta _{2}-\theta _{1})+m_{2}L_{1}L_{2}{\ddot {\theta _{1}}}^{2}\sin(\theta _{2}-\theta _{1})=-m_{2}gL_{2}\sin \theta _{2}.}$

The use of the generalized coordinates θ_i i=1, 2 provides an alternative to the Cartesian formulation of the dynamics of the double pendulum.

Spherical pendulum

Spherical pendulum: angles and velocities.

For a 3d example, a spherical pendulum with constant length l free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form

${\displaystyle f(\mathbf {r} )=x^{2}+y^{2}+z^{2}-l^{2}=0\,,}$

where the position of the pendulum bob can be written

${\displaystyle \mathbf {r} =(x(\theta ,\phi ),y(\theta ,\phi ),z(\theta ,\phi ))\,,}$

in which (θ, φ) are the spherical polar angles because the bob moves in the surface of a sphere. The position r is measured along the suspension point to the bob, here treated as a point particle. A logical choice of generalized coordinates to describe the motion are the angles (θ, φ). Only two coordinates are needed instead of three, because the position of the bob can be parameterized by two numbers, and the constraint equation connects the three coordinates x, y, z so any one of them is determined from the other two.

Generalized coordinates and virtual work

The principle of virtual work states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is, δW=0 for any variation δr. When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F_i=0.

Let the forces on the system be F_j, j=1, ..., m be applied to points with Cartesian coordinates r_j, j=1,..., m, then the virtual work generated by a virtual displacement from the equilibrium position is given by

${\displaystyle \delta W=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot \delta \mathbf {r} _{j}.}$

where δr_j, j=1, ..., m denote the virtual displacements of each point in the body.

Now assume that each δr_j depends on the generalized coordinates q_i, i=1, ..., n, then

${\displaystyle \delta \mathbf {r} _{j}={\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\delta {q}_{1}+\ldots +{\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\delta {q}_{n},}$

and

${\displaystyle \delta W=\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\right)\delta {q}_{1}+\ldots +\left(\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\right)\delta {q}_{n}.}$

The n terms

${\displaystyle F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{i}}},\quad i=1,\ldots ,n,}$

are the generalized forces acting on the system. Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,

${\displaystyle F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}},\quad i=1,\ldots ,n,}$

where v_j is the velocity of the point of application of the force F_j.

In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is

${\displaystyle \delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.}$

Niacin

From Wikipedia, the free encyclopedia

Niacin

Names
Pronunciation	/ˈnaɪəsɪn/
Preferred IUPAC name Pyridine-3-carboxylic acid
Other names Nicotinic acid (INN) Bionic Vitamin B3 Vitamin PP
Identifiers
CAS Number	59-67-6
3D model (JSmol)	Interactive image
3DMet	B00073
Beilstein Reference	109591
ChEBI	CHEBI:15940
ChEMBL	ChEMBL573
ChemSpider	913
DrugBank	DB00627
ECHA InfoCard	100.000.401
EC Number	200-441-0
Gmelin Reference	3340
IUPHAR/BPS	1588
KEGG	D00049
MeSH	Niacin
PubChem CID	938
RTECS number	QT0525000
UNII	2679MF687A
Properties
Chemical formula	C 6NH 5O 2
Molar mass	123.1094 g mol−1
Appearance	White, translucent crystals
Density	1.473 g cm−3
Melting point	237 °C; 458 °F; 510 K
Solubility in water	18 g L−1
log P	0.219
Acidity (pKa)	2.0, 4.85
Isoelectric point	4.75
Refractive index (nD)	1.4936
Dipole moment	0.1271305813 D
Thermochemistry
Std enthalpy of formation (ΔfHo298)	−344.9 kJ mol−1
Std enthalpy of combustion (ΔcHo298)	−2.73083 MJ mol−1
Pharmacology
ATC code	C04AC01 (WHO) C10AD02
License data	EU EMA: by Nicotinic acid
Routes of administration	Intramuscular, by mouth
Pharmacokinetics:
Biological half-life	20–45 min
Hazards
EU classification (DSD)	Xi
R-phrases	R36/37/38
S-phrases	S26, S36
NFPA 704	1 1 0
Flash point	193 °C (379 °F; 466 K)
Autoignition temperature	365 °C (689 °F; 638 K)
Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa).

Niacin, also known as nicotinic acid, is an organic compound and a form of vitamin B3, an essential human nutrient. It has the formula C
₆H
₅NO
₂ and belongs to the group of the pyridinecarboxylic acid.

Niacin is obtained in the diet from a variety of whole and processed foods, with highest contents in fortified packaged foods, tuna, some vegetable and other animal sources. Some countries require its addition to grains. Medication and supplemental niacin are primarily used to treat high blood cholesterol and pellagra (niacin deficiency). Insufficient niacin in the diet can cause nausea, skin and mouth lesions, anemia, headaches, and tiredness. The lack of niacin may also be observed in pandemic deficiency diseases, which are caused by a lack of five crucial vitamins (niacin, vitamin C, thiamin, vitamin D, and vitamin A) and are usually found in areas of widespread poverty and malnutrition.

This colorless, water-soluble solid is a derivative of pyridine, with a carboxyl group (COOH) at the 3-position. Other forms of vitamin B₃ include the corresponding amide nicotinamide (niacinamide), where the carboxyl group has been replaced by a carboxamide group (CONH
₂), as well as more complex amides and a variety of esters.

Niacin and nicotinamide are both precursors of the coenzymes nicotinamide adenine dinucleotide (NAD) and nicotinamide adenine dinucleotide phosphate (NADP) in vivo. NAD converts to NADP by phosphorylation in the presence of the enzyme NAD+ kinase. NADP and NAD are coenzymes for many dehydrogenases, participating in many hydrogen transfer processes. NAD is important in catabolism of fat, carbohydrate, protein, and alcohol, as well as cell signaling and DNA repair, and NADP mostly in anabolism reactions such as fatty acid and cholesterol synthesis. High energy requirements (brain) or high turnover rate (gut, skin) organs are usually the most susceptible to their deficiency.

Niacin supplementation has not been found useful for decreasing the risk of cardiovascular disease in those already on a statin, but appears to be effective in those not taking a statin. Although niacin and nicotinamide are identical in their vitamin activity, nicotinamide does not have the same pharmacological effects (lipid-modifying effects) as niacin. Nicotinamide does not reduce cholesterol or cause flushing. As the precursor for NAD and NADP, niacin is also involved in DNA repair.

Medical uses

Treatment of deficiency

Niacin and niacinamide are used for prevention and treatment of pellagra.

Abnormal lipids

Niacin has sometimes been used in addition to other lipid-lowering medications. Systematic reviews found no effect of niacin on cardiovascular disease or death, in spite of raising HDL cholesterol, and reported side effects including an increased risk of diabetes.

Contraindications

Niacin is contraindicated with active liver disease, persistent elevated serum transaminases, active peptic ulcer disease, or arterial bleeding.

Side effects

The most common adverse effects are flushing (e.g., warmth, redness, itching or tingling), headache, pain, abdominal pain, diarrhea, dyspepsia, nausea, vomiting, rhinitis, pruritus and rash. These can be minimized by initiating therapy at low dosages, increasing dosage gradually, and avoiding administration on an empty stomach. High doses of niacin often temporarily reduce blood pressure as a result of acute vasodilation. In the longer term, high-dose niacin use may persistently lower blood pressure in individuals with hypertension, but more research is needed to determine the extent of this effect.

Facial flushing

Flushing usually lasts for about 15 to 30 minutes, though it can sometimes last up to two hours. It is sometimes accompanied by a prickly or itching sensation, in particular, in areas covered by clothing. Flushing can be blocked by taking 300 mg of aspirin half an hour before taking niacin, by taking one tablet of ibuprofen per day or by co-administering the prostaglandin receptor antagonist laropiprant. Taking niacin with meals also helps reduce this side effect. Acquired tolerance will also help reduce flushing; after several weeks of a consistent dose, most patients no longer experience flushing. Reduction of flushing focuses on altering or blocking the prostaglandin-mediated pathway. Slow- or "sustained"-release forms of niacin have been developed to lessen these side effects. One study showed the incidence of flushing was significantly lower with a sustained-release formulation, though doses above 2 g per day have been associated with liver damage, in particular, with slow-release formulations.

Prostaglandin (PGD₂) is the primary cause of the flushing reaction, with serotonin appearing to have a secondary role in this reaction. The effect is mediated by prostaglandin E2 and D2 due to GPR109A activation of epidermal Langerhans cells and keratinocytes. Langerhans cells use cyclooxygenase type 1 (COX-1) for PGE2 production and are more responsible for acute flushing, while keratinocytes are COX-2 dependent and are in active continued vasodilation. Flushing was often thought to involve histamine, but histamine has been shown not to be involved in the reaction.

Gastrointestinal and hepatic

Gastrointestinal complaints, such as indigestion, nausea and liver failure, have also been reported. Hepatotoxicity is possibly related to metabolism via amidation resulting in NAD production. The time-release form has a lower therapeutic index for lowering serum lipids relative to this form of toxicity.

Diabetes

The high doses of niacin used to improve the lipid profile have been shown to elevate blood sugar by 5-10%, thereby worsening existing diabetes mellitus. In a meta-analysis of 11 trials with non-diabetic participants, niacin therapy increased the relative risk of new-onset diabetes by 34%.

Other

Side effects of heart arrhythmias have also been reported. Increased prothrombin time and decreased platelet count have been reported; therefore, these should be monitored closely in patients who are also taking anticoagulants.

Particularly the time-release variety, at extremely high doses, can cause acute toxic reactions. Extremely high doses of niacin can also cause niacin maculopathy, a thickening of the macula and retina, which leads to blurred vision and blindness. This maculopathy is reversible after niacin intake ceases.

Pregnancy

Niacin in doses used to lower cholesterol levels has been associated with birth defects in laboratory animals, with possible consequences for infant development in pregnant women.

Deficiency

A man with pellagra, which is caused by a chronic lack of vitamin B₃ in the diet

Between 1906 and 1940 more than 3 million Americans were affected by pellagra, with more than 100,000 deaths. Joseph Goldberger was assigned to study pellagra by the Surgeon General of the United States and produced good results. In the late 1930s, studies by Tom Spies, Marion Blankenhorn, and Clark Cooper established that niacin cured pellagra in humans. The disease was greatly reduced as a result.

At present, niacin deficiency is sometimes seen in developed countries, and it is usually apparent in conditions of poverty, malnutrition, and chronic alcoholism. It also tends to occur in less developed areas where people eat maize (corn) as a staple food, as maize is the only grain low in digestible niacin. A cooking technique called nixtamalization i.e., pretreating with alkali ingredients, increases the bioavailability of niacin during maize meal/flour production. For this reason, people who consume corn as tortillas or hominy are not at risk of niacin deficiency.

Mild niacin deficiency has been shown to slow metabolism, causing decreased tolerance to cold.
Severe deficiency of niacin in the diet causes the disease pellagra, which is characterized by diarrhea, dermatitis, and dementia, as well as Casal's necklace lesions on the lower neck, hyperpigmentation, thickening of the skin, inflammation of the mouth and tongue, digestive disturbances, amnesia, delirium, and eventually death, if left untreated. Common psychiatric symptoms of niacin deficiency include irritability, poor concentration, anxiety, fatigue, restlessness, apathy, and depression. Studies have indicated that, in patients with alcoholic pellagra, niacin deficiency may be an important factor influencing both the onset and severity of this condition. Patients with alcoholism typically experience increased intestinal permeability, leading to negative health outcomes.

Hartnup disease is a hereditary nutritional disorder resulting in niacin deficiency. This condition was first identified in the 1950s by the Hartnup family in London. It is due to a deficit in the intestines and kidneys, making it difficult for the body to break down and absorb dietary tryptophan (an essential amino acid that is utilized to synthesize niacin). The resulting condition is similar to pellagra, including symptoms of red, scaly rash, and sensitivity to sunlight. Oral niacin is given as a treatment for this condition in doses ranging from 40–200 mg, with a good prognosis if identified and treated early. Niacin synthesis is also deficient in carcinoid syndrome, because of metabolic diversion of its precursor tryptophan to form serotonin.

Dietary recommendations

The U.S. National Academy of Medicine (then the Institute of Medicine [IOM]) updated Estimated Average Requirements (EARs) and Recommended Dietary Allowances (RDAs) for B vitamins in 1998.

The European Food Safety Authority (EFSA) refers to the collective set of information as Dietary Reference Values (DRV), with Population Reference Intake (PRI) instead of RDA, and Average Requirement instead of EAR. AI and UL defined the same as in United States. For women (including those pregnant or lactating), men and children the PRI is 1.6 mg niacin per megajoule (MJ) of energy consumed. As the conversion is 1 MJ = 238.8 kcal, an adult consuming 2388 calories should be consuming 16 mg niacin. This is comparable to U.S. RDAs. The niacin UL is set at 10 mg/day, which is much less than the U.S. value. The UL applies to niacin as a supplement consumed as one dose, and in intended to avoid the skin flush reaction. This explains why the PRI can be higher than the UL.

Both the DRI and DRV describe amounts needed as niacin equivalents (NE), calculated as 1 mg NE = 1 mg niacin or 60 mg of the essential amino acid tryptophan. This is because the amino acid is utilized to synthesize the vitamin.

For U.S. food and dietary supplement labeling purposes the amount in a serving is expressed as a percent of Daily Value (%DV). For niacin labeling purposes 100% of the Daily Value was 20 mg, but as of May 27, 2016 it was revised to 16 mg to bring it into agreement with the RDA. A table of the old and new adult Daily Values is provided at Reference Daily Intake. The original deadline to be in compliance was July 28, 2018, but on September 29, 2017 the FDA released a proposed rule that extended the deadline to January 1, 2020 for large companies and January 1, 2021 for small companies.

Food sources

Niacin is found in a variety of whole and processed foods, including fortified packaged foods, meat from various animal sources, seafoods, and spices.

Among whole food sources with the highest niacin content per 100 grams:
Meats

• cooked skipjack tuna, 18.8 mg
• cooked light meat turkey, 11.8 mg
• cooked, lean ground pork, 11.1 mg
• cooked venison, 10.8 mg
• cooked, lean veal, 8.0 mg

Plant foods and spices

• sesame seed flour, 12.5 mg
• ground ginger, 9.6 mg
• dried tarragon, 9.0 mg
• dried, green sweet peppers, 7.4 mg
• grilled portobello mushrooms, 6.2 mg
• roasted sunflower seeds, 4.1 mg
• dehydrated apricots, 3.6 mg
• baked potato, 3.1 mg

Fortified breakfast cereals have among the highest niacin contents (more than 20 mg per 100 grams). Whole grain flours, such as from wheat, rice, barley or corn, and pasta have niacin contents in a range of 3–10 mg per 100 grams.

Pharmacology

Pharmacodynamics

The therapeutic effects of niacin are partly mediated through the activation of G protein-coupled receptors, including niacin receptor 1 (NIACR1) and niacin receptor 2 (NIACR2) which are highly expressed in adipose tissue, spleen, immune cells, and keratinocytes, but not in other expected organs such as liver, kidney, heart or intestine. NIACR1 and NIACR2 inhibit cyclic adenosine monophosphate (cAMP) production and thus fat breakdown in adipose tissue and free fatty acids available for liver to produce triglycerides and very-low-density lipoproteins (VLDL) and consequently low-density lipoprotein (LDL). A decrease in free fatty acids also suppresses liver expression of apolipoprotein C3 and PPARg coactivator-1b, thus increasing VLDL turnover and reducing its production.

The mechanism behind niacin increasing HDL is not totally understood, but seems to occur in various ways. Niacin increases apolipoprotein A1 levels due to anticatabolic effects resulting in higher reverse cholesterol transport. It also inhibits HDL hepatic uptake, down-regulating production of the cholesterol ester transfer protein (CETP) gene. Finally, it stimulates the ABCA1 transporter in monocytes and macrophages and upregulates peroxisome proliferator-activated receptor gamma, resulting in reverse cholesterol transport.

Niacin reduces secondary outcomes associated with atherosclerosis, such as low-density lipoprotein cholesterol (LDL), very low-density lipoprotein cholesterol (VLDL-C), and triglycerides (TG), but increases high-density lipoprotein cholesterol (HDL).^[48] Despite the importance of other cardiovascular risk factors, high HDL was associated with fewer cardiovascular events independent of LDL reduction. Other effects include anti-thrombotic and vascular inflammation, improving endothelial function, and plaque stability. As mediators produced from adipocytes, adipokines, such as tumor necrosis factor (TNF)-a, interleukins and chemokines, have pro-inflammatory effects, while others, such as adiponectin, have anti-inflammatory effects that influence the onset of atherosclerosis. Niacin also appears to upregulate brain-derived neurotrophic factor and tropomyosin receptor kinase B (TrkB) expression.

Research has been able to show the function of niacin in the pathway lipid metabolism. It is seen that this vitamin can decrease the synthesis of apoB-containing lipoproteins such as VLDL, LDL, IDL and lipoprotein (a) via several mechanisms: (1) directly inhibiting the action of DGAT2, a key enzyme for triglyceride synthesis; (2) influencing binding to the receptor HCAR2 thereby decreasing lipolysis and FFA flux to the liver for triglyceride synthesis; and (3) increasing apoB catabolism. HDL cholesterol levels are increased by niacin through direct and indirect pathways, such as by decreasing cholesterylester transfer protein activity and triglyceride levels, while increasing HDL cholesterol levels.

Pharmacokinetics

Niacin, serotonin (5-hydroxytryptamine), and melatonin biosynthesis from tryptophan

Biosynthesis

The liver can synthesize niacin from the essential amino acid tryptophan, requiring 60 mg of tryptophan to make 1 mg of niacin. Riboflavin, vitamin B₆ and iron are required for the process.

Physical and chemical properties

Laboratory synthesis

Several thousand tons of niacin are manufactured each year, starting from 3-methylpyridine.

Preparations

Niacin is available as a prescription product, and in the United States as a dietary supplement. Prescription products can be immediate release (Niacor, 500 mg tablets) or extended release (Niaspan, 500 and 1000 mg tablets). Dietary supplement products can be immediate or slow release, the latter including inositol hexanicotinate. The last has questionable clinical efficacy in reducing cholesterol levels.

Nicotinamide

Nicotinamide may be obtained from the diet where it is present primarily as NAD+ and NADP+. These are hydrolysed in the intestine and the resulting nicotinamide is absorbed either as such, or following its hydrolysis to nicotinic acid. Nicotinamide is present in nature in only small amounts, however it is the main form of vitamin B3 in plasma. In unprepared foods, niacin is present mainly in the form of the cellular pyridine nucleotides NAD and NADP. Enzymatic hydrolysis of the co-enzymes can occur during the course of food preparation. Boiling releases most of the total niacin present in sweet corn as nicotinamide (up to 55 mg/kg).

Nicotinamide may be toxic to the liver at doses exceeding 3 g/day for adults.

Extended release

A prescription extended release niacin, Niaspan, has a film coating that delays release of the niacin, resulting in an absorption over a period of 8–12 hours. The extended release formulations generally reduce vasodilation and flushing side effects, but increase the risk of hepatotoxicity compared to the immediate release forms.

A formulation of laropiprant (Merck & Co., Inc.) and niacin had previously been approved for use in Europe and marketed as Tredaptive. Laropiprant is a prostaglandin D2 binding drug shown to reduce vasodilatation and flushing up to 73%. The HPS2-THRIVE study, a study sponsored by Merck, showed no additional efficacy of Tredaptive in lowering cholesterol when used together with other statin drugs, but did show an increase in other side effects. The study resulted in the complete withdrawal of Tredaptive from the international market.

Inositol hexanicotinate

Inositol hexanicotinate

One form of dietary supplement is inositol hexanicotinate (IHN), which is inositol that has been esterified with niacin on all six of inositol's alcohol groups. IHN is usually sold as "flush-free" or "no-flush" niacin in units of 250, 500, or 1000 mg/tablets or capsules. It is sold as an over-the-counter formulation, and often is marketed and labeled as niacin, thus misleading consumers into thinking they are getting the active form of the medication. While this form of niacin does not cause the flushing associated with the immediate-release products, the evidence that it has lipid-modifying functions is disputed. As the clinical trials date from the early 1960s (Dorner, Welsh) or the late 1970s (Ziliotto, Kruse, Agusti), it is difficult to assess them by today's standards. One of the last of those studies affirmed the superiority of inositol and xantinol esters of nicotinic acid for reducing serum free fatty acid, but other studies conducted during the same period found no benefit. Studies explain that this is primarily because "flush-free" preparations do not contain any free nicotinic acid. A more recent placebo-controlled trial was small (n=11/group), but results after three months at 1500 mg/day showed no trend for improvements in total cholesterol, LDL-C, HDL-C or triglycerides. Thus, so far there is not enough evidence to recommend IHN to treat dyslipidemia.

Rename

In 1942, when flour enrichment with nicotinic acid began, a headline in the popular press said "Tobacco in Your Bread." So the Council on Foods and Nutrition of the American Medical Association approved of the Food and Nutrition Board's new names niacin and niacin amide for use primarily by non-scientists. It was thought appropriate to choose a name to dissociate it from nicotine, to avoid the perception that vitamins or niacin-rich food contains nicotine, or that cigarettes contain vitamins. The resulting name niacin was derived from nicotinic acid + vitamin.

History

Niacin was first described by chemist Hugo Weidel in 1873 in his studies of nicotine. The original preparation remains useful: the oxidation of nicotine using nitric acid. For the first time, niacin was extracted by Casimir Funk, but he thought that it was thiamine and due to the discovered amine group he coined the term "vitamine". Niacin was extracted from livers by biochemist Conrad Elvehjem in 1937, who later identified the active ingredient, then referred to as the "pellagra-preventing factor" and the "anti-blacktongue factor." Soon after, in studies conducted in Alabama and Cincinnati, Dr. Tom Spies found that nicotinic acid cured the sufferers of pellagra.

Niacin is referred to as vitamin B₃ because it was the third of the B vitamins to be discovered. It has historically been referred to as "vitamin PP", "vitamin P-P" and "PP-factor", that are derived from the term "pellagra-preventive factor". Carpenter found in 1951 that niacin in corn is biologically unavailable, and can be released only in very alkaline lime water of pH 11. In 1955, Altschul and colleagues described niacin as having a lipid-lowering property. As such, niacin is the oldest lipid-lowering drug.

Research

In animal models and in vitro, niacin produces marked anti-inflammatory effects in a variety of tissues – including the brain, gastrointestinal tract, skin, and vascular tissue – through the activation of NIACR1. Niacin has been shown to attenuate neuroinflammation and may have efficacy in treating neuroimmune disorders such as multiple sclerosis and Parkinson's disease. Unlike niacin, nicotinamide does not activate NIACR1; however, both niacin and nicotinamide activate the G protein-coupled estrogen receptor (GPER) in vitro.

In 2014, concurring with earlier work in 2001 by Arizona State University, researchers from Pennsylvania State University working with NASA found niacin, pyridine carboxylic acids and pyridine dicarboxylic acids inside meteorites.