is explicitly constrained by time. The only independent variable is the distance of the bead along the wire. There is . Coordinate Transformation (Polar Coordinates):
: Features 250+ solved problems on planetary motion, oscillations, and Lagrangians. David Tong’s Lecture Notes (Cambridge)
(U = mgy) with (y = -L\cos\theta) gives (U = -mgL\cos\theta).
For multi-degree-of-freedom systems near stable equilibrium, the potential energy can be expanded as a Taylor series, leading to linearized equations of motion: lagrangian mechanics problems and solutions pdf
The constraint is the length of the rope. By defining the position of one mass as , the other is automatically , reducing the system to one degree of freedom. 3. Particle on a Rotating Hoop
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T=12m(R2θ̇2+R2ω2sin2θ)cap T equals one-half m open paren cap R squared theta dot squared plus cap R squared omega squared sine squared theta close paren Choosing the center of the hoop as zero height: V=−mgRcosθcap V equals negative m g cap R cosine theta Lagrangian ( ): is explicitly constrained by time
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Solved Problems in Lagrangian and Hamiltonian Mechanics (Springer) By defining the position of one mass as
T=12m(ẋ2+ẏ2)=12m(ṙ2+r2ω2)cap T equals one-half m open paren x dot squared plus y dot squared close paren equals one-half m open paren r dot squared plus r squared omega squared close paren Potential Energy (
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Such a coordinate is called a or ignorable coordinate . The corresponding generalized momentum, , is a constant of motion:
(T = \frac12 m_1(\dotx_1^2+\doty_1^2) + \frac12 m_2(\dotx_2^2+\doty_2^2)). For small angles, (\sin\theta\approx\theta,; \cos\theta\approx 1-\theta^2/2), and keep up to quadratic terms in (\theta,\dot\theta).