What are the ignorable coordinates?
Often the Routhian approach may offer no advantage, but one notable case where this is useful is when a system has cyclic coordinates (also called “ignorable coordinates”), by definition those coordinates which do not appear in the original Lagrangian.
What is Rouths procedure?
[′rüths prə‚sē·jər] (mechanics) A procedure for modifying the Lagrangian of a system so that the modified function satisfies a modified form of Lagrange’s equations in which ignorable coordinates are eliminated.
How do you know if a coordinate is cyclic?
Momentum! -will be conserved!) If a particular coordinate does not appear in the Lagrangian, it is called ‘Cyclic’ or ‘Ignorable’ coordinate. A bead is free to slide along a frictionless hoop of radius R.
What are generalized coordinates in classical mechanics?
In analytical mechanics, 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.
What is cyclic coordinate?
A cyclic coordinate is one that does not explicitly appear in the Lagrangian. The term cyclic is a natural name when one has cylindrical or spherical symmetry. In Hamiltonian mechanics a cyclic coordinate often is called an ignorable coordinate .
What is cyclic or ignorable coordinate?
What is difference in cyclic and generalized coordinates?
A generalized coordinate that does not explicitly enter the Lagrangian is called a cyclic coordinate and the corresponding conserved quantity is called a constant of motion. Assume that a particle moves in a central potential, s.t.
What is the difference between generalized coordinates and Cartesian coordinates?
Usually, you start with Cartesian coordinates. These are the (x,y,z) coordinates that you learn about in high school. Generalized (or curvilinear) coordinates are other triplets of numbers which describe the same space, such as spherical or cylindrical coordinates.
Why do we use generalized coordinates instead of Cartesian coordinates?
Why do we use generalized coordinates instead of Cartesian coordinates? Usually employed in problems involving a finite number of degrees of freedom, the generalized coordinates are chosen so as to take advantage of the constraints of the system in reducing the total number of coordinates.
Why are cyclic coordinates called ” cyclic ” coordinates?
In Lagrangian formalism, when ∂ L ∂ q = 0, the coordinate q is called cyclic and a corresponding conserved quantity exists. But why is it called cyclic? In a finite-size system this situation usually corresponds to the rotational degree of freedom, that’s why it is called “cyclic”.
When do you use a cylindrical coordinate system?
They are sometimes called “cylindrical polar coordinates” and “polar cylindrical coordinates”, and are sometimes used to specify the position of stars in a galaxy (“galactocentric cylindrical polar coordinates”).
Which is the azimuth of a cylindrical coordinate system?
The azimuth φ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane. As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any integer.
Is the Lagrangian of a system cyclic or ignorable?
if the lagrangian of a system does not contain a given coordinate $q_j$ (although it may contain the corresponding velocity of $q_j$) then the coordinate is said to be cyclic or ignorable. The definition is not universal but is the customary one.