Dynamics and Dynamical Systems

Jaime E. Villate. University of Porto, Portugal, 2019

List of symbols and notations

A, B… points in space, curves, surfaces and solids
A , B … matrices and operators
A, B… a, b… units
A , B … a , b … variables or modules of vectors
A , B … a , b …vectors
ˆ A , ˆ B … ˆ a , ˆ b …unit vectors
a · b scalar product of two vectors
a × b vector product of two vectors
d a d x derivative of variable a with respect to x
˙ a , ¨ a … derivatives of variable a with respect to time
¯ a average value of variable a
a acceleration
a acceleration vector
a n normal (centripetal) acceleration
a t tangential acceleration
a x , a y , a z cartesian components of the acceleration
C D drag coefficient
cm centimeter; also, center of mass
d moment arm of a force
e Euler's number (base of the natural logarithms)
E k kinetic energy
E m mechanical energy
ˆ e n , ˆ e t normal and tangential unit vectors
ˆ e s , ˆ e v unit vectors in phase space
F force
F e elastic force
F f frictional force
F g weight (gravitational force)
F k , F s kinetic and static frictional forces
F n , F t normal and tangential components of the force
F r resisting force in a fluid
g acceleration of gravity
H hamiltonian function
i imaginary number − 1
I impulse
I z , I cm moments of inertia with respect to the z axis and an axis through the center of mass
ˆ ı , ˆ , ˆ k cartesian unit vectors along the x , y and z axes
J jacobian matrix
J joule (SI unit of work and energy)
k elastic constant
kg kilogram (SI unit of mass)
L angular momentum
l effective length of a pendulum
m mass
m meter (SI unit of length)
M force moment of a couple
M P moment of a force about a point P
N newton (SI unit of force)
N normal force
N R Reynolds number
ˆ n unit vector perpendicular to a plane
p quantity of motion (momentum)
r position vector
r g radius of gyration
R curvature radius of a trajectory
R , θ , z cylindrical coordinates
ˆ R , ˆ e θ , ˆ k unit vectors in cylindrical coordinates
s position along the trajectory; also, deformation of a spring
s second (SI unit of time)
T period in a uniform circular motion or an oscillatory motion
u phase velocity
U potential energy
U e elastic potential energy
U g gravitational potential energy
V potential energy per unit mass
v velocity
v velocity vector
v x , v y , v z cartesian components of the velocity
W work
x , y , z cartesian coordinates
α angular acceleration
∆ a increase of variable a during an interval of time
∆ r displacement vector
∆ s displacement along the trajectory
η coefficient of viscosity
θ rotation angle of the normal and tangential unit vectors
λ eigenvalue of a matrix; also, Lagrange multiplier
µ s , µ k coefficients of static and kinetic friction
π ratio between the length of a circumference and its diameter
ρ density
ω angular velocity
Ω angular frequency
° degree (unit of angle)