Lagrange and Hamiltonian Mechanics, Physics tutorial

Introduction:

It may be hard or even impossible to get explicit expressions for forces of constraint by applying Newtonian procedure. To circumvent some practical difficulties which may arise in attempts to apply Newtonian mechanics to specific problems, alternative procedures are essential. This alternative methods are contained in Hamilton's principle and equations of motion that result from application of the principle; Lagrange's equations of motion. Lagrange's equations can be attained in the variety of ways. We are accustomed to thinking of mechanical systems in terms of vector quantities like force, velocity, angular momentum, torque, etc., but in Lagrangian formulation, equations of motion are attained completely in terms of scalar operations.

Frame of Reference and Constraints of Motion:

Constraints are limits imposed in free motion of the particle (or the system of particles). Imposing constraints on the system is just another way of defining that there are forces present in problem which can't be defined directly, but are known in term of their effect on motion of system. In order for Newton's laws of motion to have meaning, the reference frame (coordinate system) which is fixed in space with respect to distant fixed stars should be selected with respect to which motions of bodies can be estimated.

Frames of References:

The frame of reference may refer to the coordinate system or set of axes within which to compute position, orientation, and other properties of objects in it, or it may refer to the observational reference frame tied to state of motion of the observer. It may also refer to both observational reference frame and the attached coordinate system, as a unit.

Newton realized that for the laws of motion to have meaning, the reference frame (coordinate system) that is fixed in space with respect to distant fixed stars should be selected with respect to which motions of bodies can be estimated. The reference frame is known as inertial frame of reference if Newton's laws indeed hold in that frame.

If Newton's laws hold in one reference frame then they also hold in any other reference that is in uniform motion (that is it is not accelerating) with respect to first system.

Non-inertial frame of reference that is not fixed in space is the moving coordinate system like the one attached to the falling body or one that is rotating and therefore accelerating.

Constraints of Motion:

Constraints are limitations imposed in free motion of the particle (or system of particles).

E.g. The system of particles, inter particle distance is constant. Motion may be limited geometrically in the sense that it should stay on the certain definite surface or curve or to be along the specified path and motion is said to be constrained.

Total force acting on the particle moving under constraint is

mdv/dt = F + R

Where v is velocity, F is external force, R is force of constrained that is reaction of constraining agent.

There are two kinds of constraints:

1. Holonomic constraints are those which can be represented as the functions of position vector and time example Φ(r1, r2, r3.....rn,t)

2. Non - Holonomic constraints are those which can't be represented as functions of position vector and time example, x.

Generalized Coordinates:

The use of generalized coordinates may considerably simplify a system's analysis. They reduce the total number of degrees of freedom available to the system.The choice of generalized coordonates eliminates the need for the constraint force to enter into the resultant system of equations. When you describe a system in terms of generalized coordinates, you pick the coordinates with the goal of completely describing the motion of the system in the fewest number of coordinates.

Generalized Coordinates and Degrees of Freedom:

Consider the position of a particle p in Cartesian or polar coordinates or consider the coordinates of a system of particles as shown below;

All the three coordinate systems given above and several others are only special cases of what is known as generalized coordinate systems.

Transformation from one coordinate system to another coordinate is very possible; example from Cartesian (x, y, z) to polar (r, Θ)→ x = rcosΘ, y = rsinΘ and Θ = tan-1y/x

Definitions:

The minimum number n (designated q1, q2, q3...... qn) of coordinates is needed to state configuration of the given system. These coordinates are called as Generalized Coordinates.

Each independent way by which the system may obtain energy is known as degrees of freedom and number of coordinates n is called as number of degrees of freedom of the system. The particle explain by (x, y, z) has 3 degrees of freedom. The system of particles (having N number of particles) will have 3N degrees of freedom (for holonomic systems). Therefore generalized coordinates in given figure:

(a) Cartesian (q1, q2, q3) are normally represented as (x1, y1, z1)

(b) Polar (q1, q2) (r, Θ)

(c) System of particles (q1, q2) (R, Δ r) (Note Δ r = r1 - r2)

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