Lagrangian Dynamics CSIR & GATE - S. N. Bose Physics Learning Center

Q.No:1 CSIR Dec-2014

The equation of motion of a system described by the time-dependent Lagrangian \(L=e^{\gamma t}\left[\frac{1}{2}m\dot{x}^2-V(x)\right]\) is

(1) \(m\ddot{x}+\gamma m\dot{x}+\frac{dV}{dx}=0\)

(2) \(m\ddot{x}+\gamma m\dot{x}-\frac{dV}{dx}=0\)

(3) \(m\ddot{x}-\gamma m\dot{x}+\frac{dV}{dx}=0\)

(4) \(m\ddot{x}+\frac{dV}{dx}=0\)

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Option 1

Q.No:2 CSIR June-2015

A particle of unit mass moves in the xy-plane in such a way that \(\dot{x}(t) = y(t)\) and \(\dot{y}(t) = -x(t)\). We can conclude that it is in a conservative force-field which can be derived from the potential

(1) \(\frac{1}{2}(x^2 +y^2)\)

(2) \(\frac{1}{2}(x^2 - y^2)\)

(3) \(x + y \)

(4) \(x - y\)

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Option 1

Q.No:3 CSIR Dec-2015

The Lagrangian of a particle moving in a plane is given in Cartesian coordinates as \[ L=\dot{x} \dot{y}-x^2-y^2 \] In polar coordinates the expression for the canonical momentum \(p_r\) (conjugate to the radial coordinate \(r\)) is

(1) \(\dot{r} \sin{\theta}+r\dot{\theta} \cos{\theta}\)

(2) \(\dot{r} \cos{\theta}+r\dot{\theta} \sin{\theta}\)

(3) \(2\dot{r} \cos{2\theta}-r\dot{\theta} \sin{2\theta}\)

(4) \(\dot{r} \sin{2\theta}+r\dot{\theta} \cos{2\theta}\)

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Option 4

Q.No:4 CSIR Dec-2016

The dynamics of a particle governed by the Lagrangian \(L=\frac{1}{2}m\dot{x}^2-\frac{1}{2}kx^2-kx\dot{x}t\) describes

(1) an undamped simple harmonic oscillator

(2) a damped harmonic oscillator with a time varying damping factor

(3) an undamped harmonic oscillator with a time dependent frequency

(4) a free particle

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Option 4

Q.No:5 CSIR Dec-2016

The parabolic coordinates \((\xi, \eta)\) are related to the Cartesian coordinates \((x, y)\) by \(x=\xi \eta\) and \(y=\frac{1}{2}(\xi^2-\eta^2)\). The Lagrangian of a two-dimensional simple harmonic oscillator of mass \(m\) and angular frequency \(\omega\) is

(1) \(\frac{1}{2}m[\dot{\xi}^2+\dot{\eta}^2-\omega^2(\xi^2+\eta^2)]\)

(2) \(\frac{1}{2}m(\xi^2+\eta^2)\left[(\dot{\xi}^2+\dot{\eta}^2)-\frac{1}{4}\omega^2(\xi^2+\eta^2)\right]\)

(3) \(\frac{1}{2}m(\xi^2+\eta^2)\left(\dot{\xi}^2+\dot{\eta}^2-\frac{1}{2}\omega^2 \xi\eta\right)\)

(4) \(\frac{1}{2}m(\xi^2+\eta^2)\left(\dot{\xi}^2+\dot{\eta}^2-\frac{1}{4}\omega^2\right)\)

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Option 2

Q.No:6 CSIR June-2017

The Lagrangian of a free relativistic particle (in one dimension) of mass \(m\) is given by \(L=-m\sqrt{1-\dot{x}^2}\) where \(\dot{x}=dx/dt\). If such a particle is acted upon by a constant force in the direction of its motion, the phase space trajectories obtained from the corresponding Hamiltonian are

(1) ellipses

(2) cycloids

(3) hyperbolas

(4) parabolas

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Option 3

Q.No:7 CSIR Dec-2018

The motion of a particle in one dimension is described by the Lagrangian \(L=\frac{1}{2}\left(\left(\frac{dx}{dt}\right)^2-x^2\right)\) in suitable units. The value of the action along the classical path from \(x=0\) at \(t=0\) to \(x=x_0\) at \(t=t_0\), is

(1) \(\frac{x_0^2}{2\sin^2{t_0}}\)

(2) \(\frac{1}{2}x_0^2 \tan{t_0}\)

(3) \(\frac{1}{2}x_0^2 \cot{t_0}\)

(4) \(\frac{x_0^2}{2\cos^2{t_0}}\)

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Option 3

Q.No:8 CSIR Dec-2019

Which of the following terms, when added to the Lagrangian \(L(x, y, \dot{x}, \dot{y})\) of a system with two degrees of freedom, will not change the equations of motion?

(1) \(x\ddot{x}-y\ddot{y}\)

(2) \(x\ddot{y}-y\ddot{x}\)

(3) \(x\dot{y}-y\dot{x}\)

(4) \(y\dot{x}^2-x\dot{y}^2\)

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Option 2

Q.No:9 Assam CSIR Dec-2019

The action \(S=\int dt \left[\frac{1}{2}m\dot{x}^2-gx^k\right]\) (where \(g\) is a constant) describes the motion of a particle of mass \(m\) in a one-dimensional potential. The action remains invariant under the scale transformation \(t\to \lambda^{\alpha} t\) and \(x\to \lambda^{\beta} x\), if

(1) \(k=-2, \alpha=1\) and \(\beta=2\)

(2) \(k=-2, \alpha=2\) and \(\beta=1\)

(3) \(k=1, \alpha=2\) and \(\beta=1\)

(4) \(k=1, \alpha=1\) and \(\beta=2\)

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Option 2

Q.No:10 CSIR June-2020

A point mass \(m\), is constrained to move on the inner surface of a paraboloid of revolution \(x^2+y^2=az\) (where \(a>0\) is a constant). When it spirals down the surface, under the influence of gravity (along \(-z\) direction), the angular speed about the \(z\)-axis is proportional to

(a) \(1\) (independent of \(z\))

(b) \(z\)

(d) \(z^{-2}\)

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Option c

Q.No:11 CSIR June-2023

A one-dimensional rigid rod is constrained to move inside a sphere such that its two ends are always in contact with the surface. The number of constraints on the Cartesian coordinates of the endpoints of the rod is

1) 3

2) 5

3) 2

4) 4

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Option 1

Q.No:12 CSIR June-2023

The trajectory of a particle moving in a plane is expressed in polar coordinates \((r,\theta)\) by the equations \(r=r_0e^{\beta t}\) and \(\frac{d\theta}{dt}=\omega\), where the parameters \(r_0\),\(\beta\) and \(\omega\) are positive. Let \(v_r\) and \(a_r\) denote the velocity and acceleration, respectively, in the radial direction. For this trajectory

1) \(a_r<0\) at all times irrespective of the values of the parameters

2) \(a_r>0\) at all times irrespective of the values of the parameters

3) \(\frac{dv_r}{dt}>0\) and \(a_r>0\) for all choices of parameters

4) \(\frac{dv_r}{dt}>0\), however, \(a_r=0\) for some choices of parameters

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Option 4

Q.No:13 CSIR Dec-2023

A Lagrangian is given by \[ L = \frac{1}{2}m(\dot{x}^2 + \dot{y}\dot{z} + \dot{z}^2) - \alpha(2x + 3y + z). \] The conserved momentum is

1) \( m[2\dot{x} + \dot{z}] \)

2) \( m[2\dot{x} + \dot{y} + \dot{z}] \)

3) \( m\left[\dot{x} + \frac{3}{2}\dot{y} + \frac{1}{2}\dot{z}\right] \)

4) \( m[2\dot{x} + 3\dot{z}] \)

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Option 4

Q.No:14 CSIR Dec-2024

A frictionless track is defined by \( z = z_{0} - \frac{x^{2}}{4z_{0}} \), as shown in the figure. A particle is constrained to slide down the track under the action of gravity. The tangential acceleration at position \((x, z)\) would be

1) \(\frac{2 g x}{\sqrt{x^{2} + 4 z_{0}^{2}}}\)

2) \(\frac{g x}{\sqrt{x^{2} + 4 z_{0}^{2}}}\)

3) \(\frac{g x}{2 z_{0}}\)

4) \(g \sqrt{\frac{x(x + z_{0})}{x^{2} + 4 z_{0}^{2}}}\)

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Option 2

Q.No:15 CSIR Dec-2024

The point of support of a simple pendulum, of mass \(m\) and length \(l\), is attached to the roof of a taxi as shown in the figure. The taxi is moving with uniform velocity \(v\). The Lagrangian for the pendulum is

1) \(L = \frac{1}{2} m l^{2} \dot{\theta}^{2} + \frac{1}{2} m v^{2} + m l v \cos\theta \, \dot{\theta} - m g l \cos\theta\)

2) \(L = \frac{1}{2} m l^{2} \dot{\theta}^{2} + \frac{1}{2} m v^{2} + m l v \cos\theta \, \dot{\theta} + m g l \cos\theta\)

3) \(L = \frac{1}{2} m l^{2} \dot{\theta}^{2} + \frac{1}{2} m v^{2} + m l v \sin\theta \, \dot{\theta} + m g l \cos\theta\)

4) \(L = \frac{1}{2} m l^{2} \dot{\theta}^{2} + \frac{1}{2} m v^{2} + m l v \sin\theta \, \dot{\theta} - m g l \cos\theta\)

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Option 2

Q.No:16 CSIR Dec-2024

A non-relativistic particle of mass \(m\) and charge \(q\) is moving in a magnetic field \(\vec{B}(x,y,z)\). If \(\vec{v}\) denotes its velocity and \(\{\cdot,\cdot\}_{\text{P.B.}}\) denotes the Poisson bracket, then \(\epsilon_{ijk}\{v_i, v_j\}_{\text{P.B.}}\) is equal to

1) \(-\frac{q}{m^{2}} B_k\)

2) \(0\)

3) \(\frac{2q}{m^{2}} B_k\)

4) \(\frac{q}{m^{2}} B_k\)

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Option 3

Q.No:17 CSIR Dec-2024

The Lagrangian of a system is \[ L = \frac{15}{2} m \dot{x}^{2} + 6m \dot{x}\dot{y} + 3m \dot{y}^{2} - mg(x + 2y). \] Which one of the following quantities is conserved?

1) \(12\dot{x} + 3\dot{y}\)

2) \(12\dot{x} - 3\dot{y}\)

3) \(3\dot{x} - 12\dot{y}\)

4) \(3\dot{x} + 3\dot{y}\)

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Option 1

Q.No:18 CSIR June-2025

A particle of mass \(m\) is subjected to a potential \[ V(x) = V_0 \,\theta(x) - kx, \] where \(V_0\) and \(k\) are positive constants and \(V_0\) is much larger than the energy of the particle. The function \(\theta(x) = 1\) for \(x \ge 0\) and equals 0 otherwise. The particle starts from rest at \(t = 0\) and \(x = -5\). In the limit \(V_0 \to \infty\), the graph for \(\dot{x}(t)\) is best represented by

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Option 2

Q.No:19 CSIR June-2025

A sphere of radius \(R\) is held fixed on the horizontal ground. A point particle of mass \(m\) slides without friction from the top under the action of earth’s gravity, as shown in the figure. The speed of the particle when it leaves the surface of the sphere is

1) \(\sqrt{\frac{2}{3} g R}\)

2) \(\sqrt{\frac{3}{4} g R}\)

3) \(\sqrt{2 g R}\)

4) \(\sqrt{g R}\)

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Option 1

Q.No:20 CSIR June-2025

A massless rod of length \(l\) is hinged at the extreme end of a vertical spring whose other end is fixed to the ground. A point mass \(m\) is fixed at the end of the rod, as shown in the figure. Assume harmonic motion of the spring given by \[ h(t) = h_0(2 + \cos \omega t), \] where \(h_0 >l\). The equation of motion of the mass (confined to the plane of the figure) is given by

1) \(l\ddot{\theta} + \omega^{2} h_0 \sin\theta \sin\omega t - g \sin\theta = 0\)

2) \(l\ddot{\theta} + \omega^{2} h_0 \sin\theta \cos\omega t - g \sin\theta = 0\)

3) \(l\ddot{\theta} + \omega^{2} h_0 \sin\theta \cos\omega t + g \sin\theta = 0\)

4) \(l\ddot{\theta} - \omega^{2} h_0 \sin\theta \sin\omega t + g \sin\theta = 0\)

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Option 2

Q.No:1 GATE-2012

A particle of mass \(m\) slides under the gravity without friction along the parabolic path \(y=ax^2\) as shown in the figure. Here \(a\) is a constant.

The Lagrangian for this particle is given by,

(A) \(L=\frac{1}{2}m\dot{x}^2-mgax^2\)

(B) \(L=\frac{1}{2}m(1+4a^2 x^2)\dot{x}^2-mgax^2\)

(D) \(L=\frac{1}{2}m(1+4a^2 x^2)\dot{x}^2+mgax^2\)

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Option B

Q.No:2 GATE-2012

A particle of mass \(m\) slides under the gravity without friction along the parabolic path \(y=ax^2\) as shown in the figure. Here \(a\) is a constant.

The Lagrange's equation of motion of the particle is

(A) \(\ddot{x}=2gax\)

(B) \(m(1+4a^2 x^2)\ddot{x}=-2mgax-4ma^2 x\dot{x}^2\)

(D) \(\ddot{x}=-2gax\)

Check Answer

Option B

Q.No:3 GATE-2012

A bead of mass \(m\) can slide without friction along a massless rod kept at \(45^{\circ}\) with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed \(\omega\). At any instant, \(r\) is the distance of the bead from the origin. The momentum conjugate to \(r\) is

(A) \(m\dot{r}\)

(B) \(\frac{1}{\sqrt{2}}m\dot{r}\)

(D) \(\sqrt{2}m\dot{r}\)

Check Answer

Option A

Q.No:4 GATE-2016

A particle of mass \(m=0.1 kg\) is initially at rest at origin. It starts moving with a uniform acceleration \(\vec{a}=10\hat{i} ms^{-2}\) at \(t=0\). The action \(S\) of the particle, in units of \(J\)-\(s\), at \(t=2 s\) is ____________. (Give your answer upto two decimal places)

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Ans 26.65-26.68

Q.No:5 GATE-2016

The Lagrangian of a system is given by \(L=\frac{1}{2}ml^2[\dot{\theta}^2+\sin^2{\theta}\dot{\varphi}^2]-mgl\cos{\theta}\), where \(m, l\) and \(g\) are constants. Which of the following is conserved?

(A) \(\dot{\varphi}\sin^2{\theta}\)

(B) \(\dot{\varphi}\sin{\theta}\)

(D) \(\frac{\dot{\varphi}}{\sin^2{\theta}}\)

Check Answer

Option A

Q.No:6 GATE-2017

If the Lagrangian \(L_0=\frac{1}{2}m\left(\frac{dq}{dt}\right)^2-\frac{1}{2}m\omega^2 q^2\) is modified to \(L=L_0+\alpha q\left(\frac{dq}{dt}\right)\), which one of the following is TRUE?

(A) Both the canonical momentum and equation of motion do not change

(B) Canonical momentum changes, equation of motion does not change

(D) Both the canonical momentum and equation of motion change

Check Answer

Option B

Q.No:7 GATE-2019

A projectile of mass \(1 kg\) is launched at an angle of \(30^{\circ}\) from the horizontal direction at \(t=0\) and takes time \(T\) before hitting the ground. If its initial speed is \(10 ms^{-1}\), the value of the action integral for the entire flight in the units of kg m^2 s^{-1} (rounded off to one decimal place) is __________ [Take \(g=10 ms^{-2}\)]

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Ans 33.2-33.4

Q.No:8 GATE-2020

If a particle is moving along a sinusoidal curve, the number of degrees of freedom of the particle is ______________________.

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Ans 1

Q.No:9 GATE-2021

A hoop of mass \(M\) and radius \(R\) rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass \(m\) slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is ___________.

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Ans 2

Q.No:10 GATE-2021

A uniform block of mass \(M\) slides on a smooth horizontal bar. Another mass \(m\) is connected to it by an inextensible string of length \(l\) of negligible mass, and is contrained to oscillate in the \(X\)-\(Y\) plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as \(x\) and \(\theta\), as shown in the figure.

If \(p_x\) and \(p_{\theta}\) are the generalized momenta corresponding to \(x\) and \(\theta\), respectively, then the correct option(s) is(are)

(A) \(p_x=(m+M)\dot{x}+ml\cos{\theta}\dot{\theta}\)

(B) \(p_{\theta}=ml^2\dot{\theta}-ml\cos{\theta}\dot{x}\)

(D) \(p_{\theta}\) is conserved

Check Answer

Option A_C

Q.No:11 GATE-2022

A particle of mass \(1 \hspace{1mm}\text{kg}\) is released from a height of \(1 \hspace{1mm}\text{m}\) above the ground. When it reaches the ground, what is the value of Hamilton's action for this motion in \(\text{J} s\)? (\(g\) is the acceleration due to gravity; take gravitation potential to be zero on the ground)

(a) \(-\frac{2}{3}\sqrt{2g}\)

(b) \(\frac{5}{3}\sqrt{2g}\)