Week 7
This week we will look at Lagrangian mechanics.
- Lagrangian mechanics uses geometry and the calculus of variations
- In most cases, the equations are those of Newtonian mechanics
- Energy, rather than Forces, play the principal role
An example
- Hooke's Law: $ F_{\mathrm{spring}} = - k x. $
- Newton's Equations: $ - k x = m \ddot{x} $
- Conservation: $ - k x \dot{x} = m \ddot{x} \dot{x} $ implies $$\ddt{t}{}(\frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2) = 0$$
- Hooke's Law: $ E_{\mathrm{spring}} = - \displaystyle\int_0^x F \D{x} = \frac{1}{2} k x^2. $
- Kinetic Energy: $ E_{\mathrm{kinetic}} = \frac{1}{2} m \dot{x}^2 $
- Newton's Equations: $ \ddt{t}{}\left(\didi{E_{\mathrm{kinetic}}}{\dot{x}}\right) + \didi{E_{\mathrm{spring}}}{x} = 0$
From Newton to Lagrange
Let a mechanical system have a configuration space $M$, an open subset of $\R^n$. Let the forces acting on the system, $F$, be conservative. Then Newton's equations for this mechanical system are equivalent to \[ \ddt{t}{}\didi{L}{\dot{x}} = \didi{L}{x}, \] where $L$ is the difference of kinetic and potential energy.
Let us suppose, without loss of generality, that $M$ is an open, convex set that contains $0$. Since $F$ is conservative, the work done is path-independent. Define a function \[ V(x) = -\int_0^x \ip{F(c(t))}{\D{c}} = \int_0^1 \ip{F(tx)}{x} \D{t} \] where $c(t) = tx$. The fundamental theorem of calculus shows that the gradient of $V$ at $x$ is $-F(x)$. Define \[ L(x,\dot{x}) = \frac{1}{2} m \norm{\dot{x}}^2 - V(x). \] Then \[ m \ddot{x} = \ddt{t}{} \didi{L}{\dot{x}} \qquad\text{and}\qquad F(x) = - \nabla V(x) = \didi{L}{x}. \]
Example - springs and masses
\begin{align*} F(\mathbf{q}) &= - A \mathbf{q} && V(q) = \int_0^1 \ip{A(t \mathbf{q})}{\mathbf{q}} \D{t} = \frac{1}{2} \ip{A \mathbf{q}}{\mathbf{q}}. \end{align*} \begin{align*} L(\mathbf{q}, \dot{\mathbf{q}}) &= \sum \frac{1}{2} m_i \norm{\dot{\mathbf{q}}_i}^2 - V(\mathbf{q}). \end{align*}
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| A spring-mass system. |
Constrained Systems
Let a mechanical system have a configuration manifold $M \subset \R^n$. We say the system is constrained when $M$ is a closed submanifold with $\dim M \lt n$.
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| A constrained mechanical system. |
Acceleration in a Constrained System
Let a mechanical system have a configuration manifold $M \subset \R^n$. A smooth curve $c : I \subset \R \to M$ has the derivative $$\D{{}_t} c = \dot{c}(t)$$ and acceleration $$\covder{t}{\dot{c}(t)} = \Pi_{c(t)}(\ddot{c}(t)),$$ where $\Pi_x : T_x \R^n \to T_x M$ is the orthogonal projection.
Kinetic Energy and Acceleration
Let $(M, g)$ be a Riemannian manifold, $v \in T_x M$. The kinetic energy of $v$ is $$T(x, v) = \frac{1}{2} g_x(v, v) = \frac{1}{2} \norm{v}_x^2.$$
Let $(M, g)$ be a Riemannian manifold, and $c : I \subset \R \to M$ be a $C^2$ curve. We define the acceleration of $c$ to be $$\nabla_{\dot{c}} \dot{c} \equiv \covder{t}{\dot{c}}.$$
