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The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2. If px denotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?
- a)H = 2x2 + 3x2 and px is conserved
- b)H = 2x2 + 3x2 and px is not conserved
- c)H = 2x2 - 3x2 and px is not conserved
- d)H = 2x2 - 3x2 and px is conserved
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The lagrangian of a system with one degree of freedom is given by L = ...
is not a cyclic coordiante and hence px is not conserved. Most Upvoted Answer
The lagrangian of a system with one degree of freedom is given by L = ...
Lagrangian and Canonical Momentum:
The given Lagrangian for the system is L = 2x^2 - 3x^2. To find the canonical momentum px conjugate to x, we can use the relation:
px = ∂L/∂x
Taking the derivative of the Lagrangian with respect to x, we have:
px = 4x - 6x
Simplifying, we get:
px = -2x
Hamiltonian:
The Hamiltonian H of a system is defined as the Legendre transformation of the Lagrangian with respect to the velocities. In this case, since we have a system with one degree of freedom, the Lagrangian does not depend on the velocity.
H = px * x - L
Substituting the expressions for px and L, we have:
H = (-2x) * x - (2x^2 - 3x^2)
Simplifying, we get:
H = -2x^2 - 2x^2 + 3x^2
H = -x^2
Conservation of px:
To determine whether px is conserved, we need to check if dpx/dt = 0. Taking the derivative of px with respect to time, we have:
dpx/dt = d/dt (-2x)
Since x is a function of time, we have:
dpx/dt = -2(dx/dt)
If dx/dt = v is the velocity, then dpx/dt = -2v. Since dx/dt = v, we can see that dpx/dt is not zero unless v = 0, i.e., the velocity is zero. Therefore, px is not conserved.
Correct Answer:
From the above calculations, we can conclude that the correct answer is option 'C': H = 2x^2 - 3x^2 and px is not conserved.
The given Lagrangian for the system is L = 2x^2 - 3x^2. To find the canonical momentum px conjugate to x, we can use the relation:
px = ∂L/∂x
Taking the derivative of the Lagrangian with respect to x, we have:
px = 4x - 6x
Simplifying, we get:
px = -2x
Hamiltonian:
The Hamiltonian H of a system is defined as the Legendre transformation of the Lagrangian with respect to the velocities. In this case, since we have a system with one degree of freedom, the Lagrangian does not depend on the velocity.
H = px * x - L
Substituting the expressions for px and L, we have:
H = (-2x) * x - (2x^2 - 3x^2)
Simplifying, we get:
H = -2x^2 - 2x^2 + 3x^2
H = -x^2
Conservation of px:
To determine whether px is conserved, we need to check if dpx/dt = 0. Taking the derivative of px with respect to time, we have:
dpx/dt = d/dt (-2x)
Since x is a function of time, we have:
dpx/dt = -2(dx/dt)
If dx/dt = v is the velocity, then dpx/dt = -2v. Since dx/dt = v, we can see that dpx/dt is not zero unless v = 0, i.e., the velocity is zero. Therefore, px is not conserved.
Correct Answer:
From the above calculations, we can conclude that the correct answer is option 'C': H = 2x^2 - 3x^2 and px is not conserved.
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The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer?
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The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer?.
The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The lagrangian of a system with one degree of freedom is given by L = 2x2 + 3x2.If pxdenotes the canonical momentum conjusate to x and if is the Hamiltonium then which anions the following statements is correct?a)H = 2x2 + 3x2 and px is conservedb)H = 2x2 + 3x2 and px is not conservedc)H = 2x2- 3x2 and pxis not conservedd)H = 2x2- 3x2 and pxis conservedCorrect answer is option 'C'. Can you explain this answer?.
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