#
2 definitions
by
**pinu7**

Top Definition

In classical mechanics, particularly Lagrangian Mechanics, a holonomic constraint is a special type of constraint of motion. It restricts the trajectory of a system of particles to a smooth manifold Q by the set smooth equations

a({x},t)=0

b({x},t)=0

.

.

.

Where

t=time

{x}= the set of 3N Cartesian coordinates for the system of N particles.

For N particles, the number of holonomic constraints must be less than 3N using the assumption that each equation has an explicit dependence to AT LEAST one coordinate.

a({x},t)=0

b({x},t)=0

.

.

.

Where

t=time

{x}= the set of 3N Cartesian coordinates for the system of N particles.

For N particles, the number of holonomic constraints must be less than 3N using the assumption that each equation has an explicit dependence to AT LEAST one coordinate.

A rigid body, defined by the constraint equations (using LaTeX) is

\left| {x_i - x_j } \right| - c_{ij} = 0

where i is not equal to j is a Holonomic Constraint.

\left| {x_i - x_j } \right| - c_{ij} = 0

where i is not equal to j is a Holonomic Constraint.

by pinu7
November 11, 2009

2

Lagrangian Mechanics is the reformulation of Newtonian Mechanics that utilizes the Lagrangian defined by

L=T-U

where

T= total kinetic energy of a system of particles

U= sum of the potential energy functions of a system of particles

In other branches of physics, the Lagrangian is defined as the function

L: TQ--> R

Where Q is the configuration space, a subset of R^3N

such that the action, defined as the functional,

A(q)= int(L) dt

when it reaches stationary value at {q(t)}, will male {q(t)} the equations of motion.

Note, int() means the integration notation with the limits of integration being positive an negative infinity, respectively.

It is shown, using the Calculus of Variations, that the equations of motion are in such a way that they satisfy the Lagrange's Equations of motion

D(d(L)/dv)-d(L)/dq=0

Where D is the differential operator with respect to time, d/dx stands for partial differentiation, and v is the generalized velocity.

L=T-U

where

T= total kinetic energy of a system of particles

U= sum of the potential energy functions of a system of particles

In other branches of physics, the Lagrangian is defined as the function

L: TQ--> R

Where Q is the configuration space, a subset of R^3N

such that the action, defined as the functional,

A(q)= int(L) dt

when it reaches stationary value at {q(t)}, will male {q(t)} the equations of motion.

Note, int() means the integration notation with the limits of integration being positive an negative infinity, respectively.

It is shown, using the Calculus of Variations, that the equations of motion are in such a way that they satisfy the Lagrange's Equations of motion

D(d(L)/dv)-d(L)/dq=0

Where D is the differential operator with respect to time, d/dx stands for partial differentiation, and v is the generalized velocity.

Bob: Why is general relativity so tough to learn?!

Doug: Cause' you don't know enough Lagrangian Mechanics!

Doug: Cause' you don't know enough Lagrangian Mechanics!

by pinu7
November 11, 2009