0
votes
0answers
20 views

Curvilinear Coordinates and basis vectors

In the following notes, http://www.maths.manchester.ac.uk/~wparnell/MT20401/MT20401_lecture3.pdf $\frac{\partial \vec{r}} {\partial q_i}$ forms a basis set for the vector space. How does this happen? ...
0
votes
1answer
30 views

Why is are these expressions in Leibniz notation not equivalent?

I have this problem, a projectile is fired into fluid with a rate of deceleration $a=-0.4v^3$ and in initial velocity $v_0=60$. We are to find how fast its going after $t=4$ seconds. If one starts ...
0
votes
1answer
45 views

Why is this integral not returning to the original equation when derived?

A projectile is fired into fluid at a rate of $60$ (nevermind the units on this one.) It decelerates such that $a=(-.4)v^3$. This is all fine and dandy. The book provides this solution. ...
1
vote
4answers
147 views

Mathematical modelling that involves projectile motion

I was asked to solve a mathematical differential equation to find the time taken by an object to reach the highest point and the time taken by the object to fall from its highest point to ground. I ...
-1
votes
1answer
37 views

The phase plane and potential energy

I think I have spotted a mistake in my notes, however I need help verifying my assumption: I am given the equation: $d^2s/dt^2=-s$ m=1 for simplicity I have recast it at a first order system: ...
1
vote
1answer
60 views

Finding the centre of mass and moment of inertia about a point of a laminate

A triangular laminate OAB of uniform area density,$\rho$, has vertices at (0,0), (0,2) and (1,0). The moment of inertia about an axis through O,perpendicular to the plane of the laminate is denoted by ...
0
votes
1answer
18 views

When finding the frequencies of normal modes, can you have a negative frequency?

Do you simply just consider the positive solutions? I tried a google search but didn't find anything quickly. The work I am studying is Lagrangian systems.
0
votes
1answer
41 views

Using conservation of energy on inclined plane [closed]

Hello, I'm trying to solve this using conservation of energy. So for (a) KE+PE at the top = KE+PE at the bottom and then relate to energy alone plane. However, I'm unsure how to state the KE and PE ...
0
votes
1answer
30 views

Conservation of Energy

Hello, I am trying to solve the following question using conservation of energy. Part (a) is fine, I have compared it with suvat questions and I have got: $z_{max} = v_0^2sin^2a/2g$ I am ...
2
votes
1answer
75 views

Mechanics - A particle moving in an ellipse

A and B were fine, for C I took componentwise vectors x(t) and y(t), differentiated twice and applied $1/2mv^2$. For (d) I have no idea, and for (e) I applied: $1/2m(v_2^2-v_1^2)$; From then on, ...
0
votes
4answers
118 views

Newtonian Mechanics - Differential equation

If we combine Newton's second law of motion i.e. $F=m\ddot{x}$ and Newton's law of gravity i.e., $$ F=G\frac{mM}{x^2}, $$ where $x$ is distance, we obtain the following equation: ...
0
votes
0answers
78 views

Line Integral Problem best or easier solved using geometry?

Does anyone have any recommendation on a line integral problem involving vector fields (aka work) such that evaluating the resulting line integral using parameterization would be significantly ...
0
votes
0answers
187 views

Short proof of Jacobi identity for the Poisson bracket — is this valid?

I've been trying to make a short proof for the Jacobi identity for the Poisson bracket on phase space. My idea goes like this, we know the following: $$ \frac{\mathrm{d}}{\mathrm{dt}} \{f,g\} = ...
4
votes
1answer
114 views

Am I correct? Distance when velocity depends on position

I hope I'm in the right place to be asking this: I'm looking for somebody that knows better than I who can verify whether or not I've done things correctly. In trying to implement a feature for a game ...
0
votes
1answer
51 views

How to find Equation of motions and Hamilton Function for this Lagrangian?

$$L = \frac{m \dot{x}^2}{2} - \exp(|x|) $$ I would appreciate if you could explain the steps needed to get the answer
0
votes
1answer
76 views

Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state ...
1
vote
1answer
153 views

Classical Mechanics and the Hamiltonian

This problem has to do with a canonical transformation and Hamiltonian formalism. Below is my attempt at solving it, but I am not too sure about it. Please help! Let $\theta$ be some parameter. And ...
1
vote
2answers
343 views

replace a differential equation by an equivalent system of first order equations:

If a particle of mass $m$ moves in the $x$-$y$ plane, then its equations of motion are $$m\frac{d^2x}{dt^2}=f(t,x,y)\space \space \text{and} \space \space m\frac{d^2y}{dt^2}=g(t,x,y).$$ Here $f$ ...
3
votes
0answers
244 views

system of implicit nonlinear differential equations

Here I have a system of nonlinear differential equations: $ (M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + ...
0
votes
1answer
1k views

Find the derivative of $F =$ $(GmM)\over r^2$

Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass M on a body of mass m is $F =$ $(GmM)\over r^2$ Where G is the gravitational constant and r is the ...
12
votes
1answer
316 views

Is it possible to formulate variational calculus geometrically?

In textbooks I've seen differential geometry is done with finite-dimensional manifolds. Is it possible to generalise to banach manifolds so as to formulate the calculus of variations within it, or ...
5
votes
0answers
1k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...