Classical Mechanics Mcq PDF

Preview

Summary

Aslam o Alaikm hope so you will be fine I shared all of you Mcq hopefully your help in your examination. Thanks...

Description

M. CQS of Classical Mechanics (Lecture Number 31 & 33) Teacher Name: Muhammad Aamir University of Education Lahore, Multan Campus

1. Lagrange’s bracket is (a) Canonical invariant

(b) Canonical variant

(c) Non-invariant

(d) None of these

2. Which of the following is the answer of Lagrange’s bracket

(a)

j q, p

qi pi k pj

i

(b)

 pi qi q i p i  p  qk p j k i 

(c)

k

 q

  q i

 q ,p

  

pi qi p k  i

 q i

(d) All of these

3. Poisson’s bracket is (a) invariant under canonical transformation (b) variant under canonical transformation (c) Both A and B (d) None of these 4. Hamiltonian function is

(a)

(c)

H  L  p kq k

(b)

k

H   pk qk  L

qk

k

(d)

k

5. If any coordinate

H  pk qk  L

is cyclic, i.e not contained in H, then

H   pk qk  T k

H 0 (a)  qk

 (b) p k 0

(c) A and B both true

(d) A or B are true



6. As there are three generalized coordinates, then Hamilton’s canonical equations will be … in number (a) Three

(b) Four

(c) Five

(d) Six

7. The generalized momentum is also called (a) conjugate momentum

(b) canonical momentum

(c) Both A and B

(d) None of these

8. The Lagrange’s bracket

 qi pi

(a)

  F G 

(c)

  G , F q , p

i

 F , G q , p

is equal to

pi qi  F G 

9. The Lagrange’s bracket

 q p p q    i i  i i  (b) i  G F G F  (d) All are true

 q ,p k

j q, p

  kj

, then value of

kj

is

 0 ;k  j    (a) 1 ; k  j 

 1 ;k  j    (b)   1 ; k  j 

 1  (c) 1

1 ;k  j    (d)  0 ;k  j 

;k j  ; k  j

10. Which of the following answer is true for the Lagrange’s bracket



pk , pj 

(a)  1

(b) 1

(c) 0

(d) -2

11. If the Poisson bracket of a function with the Hamiltonian vanishes

(a) the function depends upon time (b) the function is a constant of motion (c) the function is not the constant of motion (d) None of these 12. If we make a canonical transformation from the set of variables variables

 pk , qk 

to new set of

 Pk , Qk  and the transformed Hamiltonian is identically zero, then

(a) the new variables are constant in time

(b) the new variables are not constant in time

(c) the new variables are not cyclic

(d) None of these

13. Hamiltonian H is defined as (a) the total energy of the system

(b) the difference in energy of the system

(c) the product of energy of the system

(d) All of these

14. Whenever the Lagrangian for a system does not contain a coordinate explicitly, (a) pk is cyclic coordinate (b)

pk

the generalized momentum, is a constant of motion

(c) q k is always zero (d) None of these 15. The dimensions of generalized momentum (a) are always those of linear momentum

(b) may be those of angular momentum

(c) may be those of linear momentum

(d) Both B and C are true

16. If the Lagrangian does not depend on time explicitly (a) the Hamiltonian is constant

(b) Hamiltonian not constant

(c) the kinetic energy is constant

(d) the potential energy is constant

17. Generelized coordinates

(a) depend on each other

(b) are independent of each other

(c) are spherical coordinates

(d) None of these

18. The Lagrangian equation of motion are……order differential equations. (a) first

(b) second

(c) third

(d) forth

19. In variational principle the line integral of some function between two end points is (a) zero

(b) infinite

(c) extremum

(d) one

20. If

f  y, y, x 

define a curve y  y ( x ) , then Euler-Lagrange’s equation is

d  L  f    0,  k 1,..., n   y k dx   yk 

(a)

(c)

d  L  f    0,  k 1,..., n   y k dx   yk 

f d  L    0,  k 1,..., n   yk dx   yk  (b)

(d) None of these

MCQs Classical Mechanics Lecture No. 29 and 32 i.

Poisson bracket is useful to find the (a) Integrals of motion (b) Equations of motion (c) Lagrange function (d) Hamilton’s function

ii. If (a)

F

does not depend explicitly on time, then for

∂F + { F , H }=0 ∂t

(b) { F , H }=0

F

to be integral of motion:

(c)

∂F =0 ∂t

(d) None of these

iii. The Poisson bracket operation is used in what type of space? (a) Phase space (b) Configuration space (c) Hilbert space (d) None of these

iv.

{F , q i }=¿ (a)

−∂ F ∂ pi

(b)

∂F ∂ pi

(c)

−∂ F ∂ qi

(d)

∂F ∂ qi

v. A transformation is said to be canonical, if (a)

pdq− PdQ =0

(b)

Pdq− pdQ =0

(c)

pdq− PdQ = dF

(d)

Pdq− pdQ = dF

vi. Poisson bracket are ------------- under canonical transformations (a) Invariant (b) variant (c) Equivalent to (d) None of these

vii. If the coordinates

(Q , P) satisfy the canonical Poisson bracket ------------ , then the

transformation

( q , p ) → ( Q , P ) is canonical (a)

{ Q ,Q} =0

(b)

{ P , P } =0

(c)

{ Q , P }=1

(d) None of these (e) All of these

viii.If (a)

F

depend explicitly on time, then for

F

to be integral of motion:

∂F + { F , H }=0 ∂t

(b) { F , H }=0 (c)

∂F =0 ∂t

(d) None of these

ix. The statement “If F(q, p, t) and G (q, p, t) are two integrals of motion. then [F, G] is also an integral of motion ” is called (a) Legendre’s transformations (b) Jacobi-Poisson’s theorem (c) Hamilton’s principle (d) None of these x. Dimension of space defined in ( pi ,q i ,t ¿

is

(a) 2n (b) (b) 2n+1 (c) (c) 4n (d) (d) 4n+1

xi. Which one is a form of Hamilton’s equations in terms of Poisson’s brackets (a)

− { q i , H } =´qi , − { pi , H }= p´ i

(b)

{ q i , H }=´qi

,

−{ pi , H }= p´ i

(c)

− { q i , H } =´qi ,

{ pi , H }= p´ i

(d) None of these

xii. Which one is called Jacobi identity for Poisson’s bracket

(a)

{ F ,{ G , H } }

(b)

dF ∂ F = +{F , H } dt ∂ t

(c)

d ∂ F ,G + ∂G { F ,G }= F, ∂t ∂t dt

+

{ G , {H , F } }

{

}{

+

{ H , { F ,G } }=0

}

(d) None of these xiii. The angular momentum is given by If Lx = ( y p z−z p y ) , (a)

{ Lx , L y }=0

(b)

{ Lx , L y }=−L z

(c)

{ Lx , L y }=L z

(d)

{ Lx , L y }=1

L´ =r´ × ´p .

L y =( z p x − y p z ) ,

Lz =( x p y − y p x ) , then

xiv. In general, which one must be true for a function F to be an integral of motion (a)

dF ∂ F +{F , H } = dt ∂ t

(b)

{ F , H }=0

(c)

dF =0 dt

(d) None of these xv. Which one is correct (a) The Poisson brackets do not give a complete solution of a system of motion but are very helpful in finding the Integrals of motion. (b) The Poisson brackets give a complete solution of a system of motion and are very helpful in finding the Integrals of motion. (c) The Poisson brackets do not give a complete solution of a system of motion and do not helpful in finding the Integrals of motion.

(d) The Poisson brackets give a complete solution of a system of motion but do not helpful in finding the Integrals of motion.

Classical Mechanics Dr. Muhammad Raza Ahmad

Lecture 27 1. The transformation of coordinates qi

Q i=Q i (q 1 ,q 2 , … , q n , t)

→ Qi

by equations of the type

is called ______.

(a)Point transformation

(b)Contact transformation

(c)Lagrange transformation

(d)Both a and b

2. The transformation of canonical coordinates (q, p) to (Q, P) preserving the form of Hamilton’s equations is called ______. (a)Hamilton transformation

(b)Canonical transformation

(c)Preserved transformation

(d)None of these

3. Identify the Hamilton’s canonical equations of motion.

−∂ H ∂H ∧ p´ = ∂ pi i ∂ qi

(b) q´ i =

∂H ∂H ∧ p´ = ∂ pi i ∂ qi

−∂ H ∂H ∧ p´ i= ∂ qi ∂ pi

(d) q´ i =

∂H ∂H ∧ p´ i= ∂ pi ∂ qi

(a) q´ i = (c) q´ i =

4. The modified Hamilton’s principle in original canonical coordinates (p, q) is given by

Pi Q´ i −K (Q , P , t ) ∑ ¿ dt i

(a)

¿ ¿ t2

Pi Q´ i + K (Q , P , t) ∑ ¿ dt i

(b)

¿ ¿ t2

δ∫¿

δ∫ ¿

t1

t1

pi q´ i + H (q , p , t) ∑ ¿ dt

pi q´ i−H (q , p , t) ∑ ¿ dt i

(c)

¿ ¿

i

¿ ¿

(d)

t2

t2

δ∫ ¿

δ∫ ¿

t1

t1

5. The modified Hamilton’s principle in transformed canonical coordinates (P, Q) is given by

Pi Q´ i −K (Q , P , t ) ∑ ¿ dt i

(a)

¿ ¿

Pi Q´ i + K (Q , P , t) ∑ ¿ dt i

(b)

t2

t2

δ∫¿

δ∫ ¿

t1

t1

pi q´ i−H (q , p , t) ∑ ¿ dt i

(c)

¿ ¿

¿ ¿

pi q´ i + H (q , p , t) ∑ ¿ dt i

(d)

t2

¿ ¿ t2

δ∫¿

δ∫ ¿

t1

t1

6. In the relation

dF

∑ pi q´i − H ( q , p , t )=∑ Pi Q´ i−K ( Q , P , t ) + dt i

, F is possibly a function of

i

____. (a)Old coordinates

(b)New coordinates

(c)Time and momenta

(d)All of these

7. In the relation

dF

∑ pi q´i − H ( q , p , t )=∑ Pi Q´ i−K ( Q , P , t ) + dt i

, the function F is called _____.

i

(a)Generating function

(b)Differential function

(c)Hamilton’s function

(d)None of these

8. The generating function F in canonical transformation can be written as _____. (a) F ( q , Q , t )

(b) F ( q , P , t ) ;

(c) F( p , P , t)

(d)All of these

F ( p ,Q , t)

9. Tick the correct option in case of type 1 generating function F1 (q ,Q , t ) .

(a) pi=

(c) pi=

−∂ F1 ∂ F1 (q ,Q , t ) ; Pi= (q , Q ,t ) ∂ qi ∂ Qi

(b) pi=

∂ F1 ∂F (q , P , t ) ; Pi= 1 ( q , P , t ) ∂ qi ∂Q i

−∂ F1 −∂ F 1 −∂ F1 ∂F ( q ,Q , t ) ; Pi= ( q , Q ,t ) (d) pi= ( q ,Q , t ) ; Pi= 1 ( q , Q ,t ) ∂ Qi ∂ qi ∂ Qi ∂ qi

10. Tick the correct option in case of type 2 generating function F2 ( q , P ,t ) .

∂ F2 ∂ F2 (q , P , t ) ( q , P , t ) ; Q i= ∂ qi ∂ Pi

(b) pi =

∂ F2 −∂ F2 (q , P , t ) ( q , P , t ) ; Q i= ∂ qi ∂ Pi

(d)

(a) pi =

(c) pi=

pi =

−∂ F 2 ∂ F2 (q , P , t ) (q , P , t ) ; Q i= ∂ qi ∂ Pi

−∂ F2 −∂ F 2 ( q , P , t ) ; Q i= (q , P , t ) ∂ qi ∂ Pi

11. Tick the correct option in case of type 3 generating function F3 ( p ,Q , t ) . (a) Pi =

−∂ F3 −∂ F 3 ( p , Q ,t ) ( p , Q ,t ) ; qi = ∂Q i ∂ pi

(b) Pi =

−∂ F 3 ∂ F3 ( p , Q ,t ) ( p , Q ,t ) ;qi= ∂Q i ∂ pi

(c) Pi =

−∂ F3 ∂F ( p , Q ,t) ; qi = 3 ( p , Q ,t ) ∂Q i ∂ pi

(d) Pi =

∂ F3 ∂F ( p , Q ,t ) ;qi= 3 ( p , Q ,t) ∂Q i ∂ pi

12. Tick the correct option in case of type 4 generating function F 4 ( p , P ,t ) . (a) qi =

−∂ F 4 ∂F ( p , P , t ) ;Q i= 2 (p , P , t) ∂ pi ∂ Pi

(c) qi =

∂ F4 ∂F ( p , P ,t ) ;Q i= 2 ( p , P , t ) ∂ pi ∂ Pi

qi =

(b) qi =

−∂ F 2 ∂ F4 (p , P , t) ( p , P ,t ) ; Qi= ∂ pi ∂ Pi

(d)

−∂ F 4 −∂ F2 (p,P,t) ( p , P , t ) ;Q i= ∂ pi ∂ Pi

13. If the generating function F does not contain t explicitly, then (a) K=H +

∂F ∂t

(b) K=H

∂F ∂t

(d)Both b and c

(c) K=H −

14. If K is the transformation of Hamilton H in a canonical transformation, then

(a) K=H +

∂F ∂t

(b) K=H

(c) K=H −

∂F ∂t

(d) K=

∂F −H ∂t

Lecture 28 1. Using Hamilton’s equations of motion, we can find equation of motion for any F(q, p) in terms of (a)Commutator brackets

(b)Poisson Brackets

(c)Lagrange brackets

(d)All of these

2. Poisson brackets provide a bridge between classical mechanics and ________. (a)Relativistic Mechanics

(b)Astrology (d)Quantum field theory

(c)Quantum Mechanics

3. Identify the expression representing the Poisson bracket (a)

∑

(∂∂ Fq ∂∂ Gp − ∂∂ Fp ∂G ∂q )

(b)

∑

(∂∂ Fq ∂∂ Gp + ∂∂ Fp ∂∂Gq )

(d)

i

(c)

i

i

i

i

i

i

i

i

[ F , G]q , p .

∑

∂ F ∂G ∂ F − (∂G ∂q ∂ p ∂ p ∂q )

∑

∂ F ∂ F ∂G − (∂G ∂ q ∂ p ∂ p ∂q )

i

i

i

i

i

i

i

i

i

4. Identify the fundamental Poisson brackets. (a)

[ q j , qk]q , p=0

(b)

[ p j , p k ] q , p =0

(c)

[ q j , p k]q , p=δ jk

(d)All of these

5. If F = G, then (a)

[ F , G ]=0

(b)

[ F , G ]> 0

(c)

[ F , G] ≠ 0

(d) [ F ,G]

does not exist

6. If F is a function of q and p; and C is a constant, then (a)

[ F , C] =0

(b)

[ F , C] >0

(c)

[ F , C] =C

(d)

[ F , C] =F

7. For 2 functions F and G such that F ≠ G, we’ve

i

i

(a)

[ F , G ]=− [G , F ]

(c)

[ F , G]=[ G , F ]

(b)

[ F , G] =0

(d)Both b and c

8. For 3 functions F, G and S of canonical variables q, p and time; (a)

[ F , G ]+ [F , S ]

(c)

[ F , G] S+G[ F , S]

(b)

[ F , G ]+ [G , S ] (d) [ F−G , S ]

9. For 3 functions F, G and S of canonical variables q, p and time; (a)

[ F , G]+ [ G , S ]

(c)

[ F , G] S+G[ F , S]

10. The expression

(b)

(d) [ F−G , S ]

[ F , [ G, S ] ] +[G , [ S , F ]] +[ S , [ F ,G ] =0

is called _______.

(b)Jacobi Identity

(c)Lagrange Identity

(d)Triple Identity

11. Time evolution of Poisson bracket

(c)

[ [

][ ][

dF dG ,G + F , dt dt dG dF ,G + F , dt dt

¿ ¿ [ F , GS] =¿ .

[ F , G ]−[ G, S ]

(a)Poisson Identity

(a)

¿ ¿ [ F , G+ S ] =¿ .

[ F , G ] yields _______.

] ]

(b)

(d)

[ [

][ ]

dF dG ,G − F, dt dt

]

dF dG , dt dt

13. Consider a function F of q’s, p’s and time t such that F = F (q , p , t ) , then

(

)

(

)

(a)

dF ∂F ∂F ∂F = q´i + p´ i + ∂ q ∂ p ∂t dt ∑ i i i

(c)

dF ∂F ∂F =∑ p´ q´ + dt i ∂ qi i ∂ pi i

(b)

(

)

dF ∂F ∂F ∂F = q´i + p´ i − ∂ q ∂ p ∂t dt ∑ i i i (d)

(

)

dF ∂F ∂F ∂F =∑ p´ i + q´i− ∂ p ∂ q ∂t dt i i i

14. The equation of motions can be represented using Hamilton’s equations in the form _____. (a)

dF ∂F = [ F , H ]+ ∂t dt

(b)

dF ∂F = [ F , H ]− ∂t dt

(c)

dF ∂H =[ F , H ] + dt ∂t

(d)None of these

MCQs Classical Mechanics Lec 25 and lec 26 1. In variational principle the line integral of some function between two end points is: (a) zero

(b) infinite

(c) extremum

(d) one

2. Hamilton’s principle is an example of a: (a) Force

(b) Lagrange multiplier

(c) stationary point

(d) variational principle.

3. The Hamiltonian can be constructed from the Lagrangian using the formula: (b) H = piqi – L

(a) H = ˙piq˙i – L 1/L

(c) H = ∂L/∂q˙i

(d) H =

4. The Hamiltonian function is given by: (a) H = T − V

(b) H = T/V

(c) H = T + V

(d) H = V − T

5. If ∂H ∂t = 0 , then (a) dH/dt = 0 and the Hamiltonian is not necessarily a constant of the motion, but it is necessarily the total energy of the system. (b) dH/dt = 0 and the Hamiltonian is not a constant of the motion, but it is necessarily the total energy of the system. (c) dH/dt = 0 and the Hamiltonian is a constant of the motion, but it is not necessarily the total energy of the system. 6. When i cyclic coordinates are present in a system, degrees of freedom in Hamilton’s formalism is: (a) n degrees of freedom (c) n-i degrees of freedom

(b) n-1 degrees of freedom (d) n+i degrees of freedom

7. Hamilton’s equations of motion can be derived from (a) Hamiltonian of system (c) Both a & b

(b) variation principle (d) None

8. The n-dimensional space is called ___________ space

(a) solar

(b) configuration

(c)real

(d) zero

9. Hamiltonian formulation is an alternative formulation of advanced classical mechanics that is -------- to the Lagrangian formulation. (a) Inferior in some respects aspect...