Holonomic and Non-holonomic Constraints

• Constraints in classical mechanics are restrictions on the possible motions of a system
• They reduce the number of degrees of freedom of the system

Holonomic Constraints

• Holonomic constraints can be expressed as an equation of the form f(r1, r2, r3, ..., t) = 0, where ri are the position coordinates of the particles in the system and t is time
• These constraints directly relate the coordinates of the particles
• Examples include a pendulum (fixed length), a particle moving on a surface, or a rigid body (fixed distances between particles)
• Holonomic constraints are integrable, meaning the constraint equation can be used to eliminate one or more variables from the system's equations of motion
• For a system with n particles and k holonomic constraints, the number of independent generalized coordinates is n - k

Non-holonomic Constraints

• Non-holonomic constraints cannot be expressed in the form f(r1, r2, r3, ..., t) = 0
• Typically, they are expressed as inequalities or non-integrable differential equations
• Examples include a particle moving on or outside a surface (inequality constraint) or a rolling wheel without slipping (differential constraint)
• These constraints cannot be used to directly eliminate variables from the equations of motion
• Rolling without slipping is a common example, where the velocity of the point of contact is zero, leading to a differential constraint
• The number of independent generalized coordinates is not simply determined by subtracting the number of constraints

Rheonomic Constraints

• Rheonomic constraints are time-dependent constraints
• The constraint equation explicitly contains time: f(r1, r2, r3, ..., t) = 0
• An example is a pendulum with a pivot point that is moving in time

Scleronomic Constraints

• Scleronomic constraints are time-independent constraints
• The constraint equation does not explicitly contain time: f(r1, r2, r3, ...) = 0
• An example is a simple pendulum with a fixed pivot point

Lagrangian Mechanics and Constraints

• In Lagrangian mechanics, constraints can be handled using various methods
• For holonomic constraints, the constraint equations can be used to reduce the number of generalized coordinates
• Alternatively, Lagrange multipliers can be introduced to incorporate the constraint forces into the equations of motion
• The Lagrangian is modified to L' = L + Σ(λi * fi), where λi are Lagrange multipliers and fi are the constraint equations
• The Euler-Lagrange equations are then applied to the modified Lagrangian, treating the Lagrange multipliers as additional variables
• This approach yields the equations of motion along with equations that determine the constraint forces

Generalized Coordinates

• Generalized coordinates are a set of independent coordinates that completely specify the configuration of a system subject to constraints
• The number of generalized coordinates is equal to the number of degrees of freedom of the system
• The choice of generalized coordinates is not unique and should be made to simplify the equations of motion
• For example, for a simple pendulum, the angle θ is a suitable generalized coordinate, rather than x and y

Degrees of Freedom

• The number of degrees of freedom of a system is the minimum number of independent coordinates required to completely specify the configuration of the system
• For a system of N particles in 3D space, the number of degrees of freedom is 3N
• Constraints reduce the number of degrees of freedom
• For a system with N particles and k holonomic constraints, the number of degrees of freedom is 3N - k

Constraint Forces

• Constraint forces are the forces that maintain the constraints
• They do no work on the system for ideal constraints (also called perfect constraints)
• Examples include the tension in a string of a pendulum or the normal force on a particle moving on a surface
• Using Lagrange multipliers, the constraint forces can be explicitly determined
• The constraint force associated with the i-th constraint is given by Fi = λi * ∇ri * fi, where ∇ri is the gradient with respect to the position coordinates

D'Alembert's Principle

• D'Alembert's principle states that the virtual work done by the constraint forces is zero for ideal constraints
• This principle is expressed as Σ(Fi ⋅ δri) = 0, where Fi are the constraint forces and δri are the virtual displacements
• Virtual displacements are infinitesimal, imaginary displacements that are consistent with the constraints at a given instant of time
• D'Alembert's principle provides a way to eliminate the constraint forces from the equations of motion

Examples

• Simple Pendulum: A mass attached to a fixed point by a massless, inextensible string is a holonomic and scleronomic constraint
• Particle on a Sphere: A particle constrained to move on the surface of a sphere is a holonomic and scleronomic constraint
• Rolling Wheel: A wheel rolling without slipping on a surface is a non-holonomic constraint
• Gas molecules in a container: The volume of the gas being constrained to equal to or less than the volume of the container is a non-holonomic contraint

Explanation of constraints in classical mechanics focusing on holonomic and non-holonomic constraints. Holonomic constraints can be expressed as an equation and are integrable. Non-holonomic constraints cannot be expressed in a simple equation.