Question

The lagrangian for a particle constrained to move on a smooth horizontal table under the action of a spring​

Answer 1

Answer:

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Explanation:

Newtonian mechanics deals with force which is a vector quantity and therefore dif-

ficult to handle. On the other hand, Lagrangian mechanics deals with kinetic and

potential energies which are scalar quantities while Hamilton’s equations involve

generalized momenta, both are easy to handle. While Lagrangian mechanics con-

tains n differential equations corresponding to n generalized coordinates, Hamil-

tonian mechanics contains 2n equation, that is, double the number. However, the

equations for Hamiltonian mechanics are linear.

The symbol q is a generalized coordinate used to represent an arbitrary coordi-

nate x, θ, ϕ, etc.

If T is the kinetic energy, V the potential energy then the Lagrangian L is

given by

L = T − V (7.1)

Lagrangian Equation:

d

dt

dL

dq˙K

− ∂L

∂qK

= 0 (K = 1, 2 . . .) (7.2)

where it is assumed that V is not a function of the velocities, i.e.

∂v

∂q˙K

= 0. Eqn (2)

is applicable to all the conservative systems.

When n independent coordinates are required to specify the positions of the

masses of a system, the system is of n degrees of freedom.

Hamilton H = r

s=1 psq˙s − L (7.3)

where ps is the generalized momentum and q˙K is the generalized velocity.