classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

, a physical system is termed a monogenic system if the force acting on the system can be modelled in a particular, especially convenient mathematical form. The systems that are typically studied in physics are monogenic. The term was introduced by Cornelius Lanczos in his book ''The Variational Principles of Mechanics'' (1970). In

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...

, the property of being monogenic is a necessary condition for certain different formulations to be mathematically equivalent. If a physical system is both a

holonomic system In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that d ...

and a monogenic system, then it is possible to derive Lagrange's equations from

d'Alembert's principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert ...

; it is also possible to derive Lagrange's equations from

Hamilton's principle In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, ...

Mathematical definition

In a physical system, if all forces, with the exception of the constraint forces, are derivable from the generalized scalar potential, and this generalized scalar potential is a function of

generalized coordinate In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...

s, generalized velocities, or time, then, this system is a monogenic system. Expressed using equations, the exact relationship between generalized force

\mathcal_i

and generalized potential

\mathcal(q_1,\ q_2,\ \dots,\ q_N,\ \dot_1,\ \dot_2,\ \dots,\ \dot_N,\ t)

is as follows: :

\mathcal_i= - \frac+\frac\left(\frac\right);

where

q_i

is generalized coordinate,

\dot

is generalized velocity, and

t

is time. If the generalized potential in a monogenic system depends only on generalized coordinates, and not on generalized velocities and time, then, this system is a

conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink o ...

. The relationship between generalized force and generalized potential is as follows: ::

\mathcal_i= - \frac

References

{{reflist Mechanics Classical mechanics Lagrangian mechanics Hamiltonian mechanics Dynamical systems

Mathematical definition

See also

References